Description
An Mathematical Introduction to Population DynamicsUNDER CONSTRUCTION Howie Weiss Georgia Tech 08/10/10 Contents 1 2 Introduction 1.1 About these lectures . . . . . . . . . . . . . . . . . 1.2 What is a population and how does it change? . . 1.3 Why do biologists need mathematical models? . . 1.4 Limitations of mathematical models . . . . . . . . 1.5 Why do mathematicians need population models? 1.6 Confronting models with data . . . . . . . . . . . . 1.6.1 Model validation . . . . . . . . . . . . . . . 1.6.2 Model Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 6 7 7 8 8 9 9 Single Species Models 2.1 Discrete verses continuous population models . . . . . . . . 2.2 The determinants of population change . . . . . . . . . . . 2.2.1 How do microbes reproduce? . . . . . . . . . . . . . 2.3 Exponential growth paradigm . . . . . . . . . . . . . . . . . 2.4 Continuous growth models . . . . . . . . . . . . . . . . . . . 2.4.1 Logistic growth law . . . . . . . . . . . . . . . . . . The effects of r verses K selection . . . . . . . . . . The Allee effect . . . . . . . . . . . . . . . . . . . . . 2.4.2 Monod growth law for bacteria . . . . . . . . . . . . 2.4.3 Logistic growth with harvesting or predation . . . . Constant harvesting . . . . . . . . . . . . . . . . . . Holling Type I functional response . . . . . . . . . . Holling Type II functional response . . . . . . . . . . Holling Type III functional response . . . . . . . . . Arditi-Ginzburg functional response . . . . . . . . . 2.5 Case Study 1: Controlling the spruce budworm population 2.6 Discrete growth models . . . . . . . . . . . . . . . . . . . . 2.6.1 Discrete logistic model . . . . . . . . . . . . . . . . . 2.6.2 Beverton-Holt model . . . . . . . . . . . . . . . . . . 2.6.3 Ricker model . . . . . . . . . . . . . . . . . . . . . . 2.7 Stochasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Bacteria growth model (write this) . . . . . . . . . . 2.7.2 Natural catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 11 12 12 16 16 18 20 20 21 22 22 23 23 24 24 29 29 30 30 32 32 33 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 2.8 2.9 3 4 2.7.3 Genetic stochasticity . . . . . 2.7.4 Environmental stochasticity . 2.7.5 Demographic stochasticity . . The age-structured Leslie population Case Study 2: Saving the loggerhead 3 . . . . . . . . . . . . . . . . . . model . . sea turtle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 33 34 36 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 43 44 45 48 49 49 50 54 55 58 60 60 61 61 63 63 65 65 65 67 70 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 74 74 78 81 84 86 88 88 88 90 92 92 94 95 Models of Communities 3.1 Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The niche and competitive exclusion . . . . . . . . . . . . . . . . . . 3.1.2 The well-mixing hypothesis . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Lotka-Volterra competition model . . . . . . . . . . . . . . . . . 3.1.4 Competition between n species . . . . . . . . . . . . . . . . . . . . . 3.1.5 Discrete competition models . . . . . . . . . . . . . . . . . . . . . . 3.2 Predation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Lotka-Volterra predator-prey model . . . . . . . . . . . . . . . . . . 3.2.2 Inverted biomass pyramids . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Predator-prey model with logistic growth and Holling-type responses 3.2.4 Experiments with protist communities . . . . . . . . . . . . . . . . . 3.2.5 Experiments with bacteria and bacteriophages . . . . . . . . . . . . 3.2.6 A super-predator, predator, and prey community model . . . . . . . 3.2.7 Two predators and one prey community model . . . . . . . . . . . . 3.2.8 Canadian lynx and snowshoe hare . . . . . . . . . . . . . . . . . . . 3.3 Population dynamics in a chemostat . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Single species growth model . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Competiton: (write this) . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Predation: (write this) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mutualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Parasitoidism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Flour beetle model and chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Do real populations exhibit chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling the Spread of Infectious Diseases 4.1 SIR models . . . . . . . . . . . . . . . . . . . . . 4.1.1 Basic SIR model . . . . . . . . . . . . . . 4.1.2 The basic reproductive rate R0 . . . . . . 4.1.3 Examples . . . . . . . . . . . . . . . . . . 4.1.4 SIR model with vital rates . . . . . . . . . 4.1.5 Stochastic SIR model . . . . . . . . . . . 4.1.6 Time Series Stochastic SIR model . . . . 4.1.7 Estimating R0 from epidemiological data 4.2 SIS model . . . . . . . . . . . . . . . . . . . . . . 4.3 SIS criss-cross models . . . . . . . . . . . . . . . 4.4 SEIR models . . . . . . . . . . . . . . . . . . . . 4.4.1 Basic SEIR model with vital rates . . . . 4.4.2 Seasonally forced SIR and SEIR models . 4.4.3 Transmission models with time dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transmission coefficient . . . . . . . . . . . . . . 111 111 113 113 115 117 118 119 120 120 121 123 . . . .1 References . . . . . 5. . . . . . . .4. . . . . . . . . . . . . . . Evolution and transmission of infectious diseases . . .4 Routh-Hurwitz and Jury conditions . . . . . 6. . . . . . . . . . 4. . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Poincar´e-Bendixson theorem and closed orbits 6. . . . . . . . . . . . . Case Study 3: iSIR model with immunological threshold Mathematical Methods 6. . . . . .2 Invasion by a mutant . . . 4. . . . . . . . . . . . . .4 Agent based models . . . . . . . . . . . . . . . . . . . .2. . . . .6 Chemotaxis . . . . . .1 Local stability and bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7. . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . 5. . . . . . . . .6 4. . . . . . . . . . . . .2 Reaction-diffusion PDE equation models . . . . . . . . . . .7. . . . . . . . . . . . . .7 . . . . . . . . .8 4. . . . . . . . . . . . . . . . . . . .3 Network models . . . . . .2 Lyapunov functions . . . . . . 5. . . . . . . . . . . . . . . . . . . . .2. . . 7 Supplementary Material 134 7. . . . . . . . . . . . . . . . . . . . .2. . . . . .1 Infection with multiple strains . . . . . . 129 129 130 132 133 4.7. . . . .5 Why do bacteria move? (ADD MUCH MORE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Quasi-species models . .5 4. . . . . . . . . . . . . . . 134 . . . . . . . . . Modeling the spread of a computer virus . . . . . . . . . . . . .3 Modeling the spread of antibiotic resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-host model of viral infection . . 5. . . .1 Metapopulation models . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . . 4. . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 97 98 98 101 101 103 106 108 5 Spatial Population Models 5. . . . . . . . . .4 Modeling the spatial spread of rabies . . . . . . . . . .7 Further examples of spatial population models . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 The diffusion equation . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . 5. .3 The Fisher model and traveling waves . . . . . . . . . . . . . . . 4. . . . . . . . .7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . .2 Skellam’s model and the European invasion of muskrats 5. . . . . . . . . . . . . . . . . . . . . . . . . . . .9 6 Recovering the transmission coefficient from data . . . . . . . . . . . . . . . . most authors follow the geometric approach in [Rosenzweig and MacArthur. biology. it may appear that the topics in Chapters 2 and 3 (single species and community dynamics) are disjoint from the topics in Chapter 4 (disease transmission dynamics). Feedback from my students has been positive. ecology. For the latter. so that at the end of the semester the students can formulate and analyze their own population models. genetics. Some textbooks discuss the local analysis of equilibrium points. The majority of students are conducting thesis research on microbial population dynamics or transmission of infectious diseases. demography. engineering. and I try to choose models and applications with this in mind. Our treatment is considerably more analytical. and I introduce the main modeling tools used in these areas. and navigate the literature. immunology. epidemiology. fewer discuss bifurcations. and computing. This is definitely not the case. cell biology. developmental biology. The minimal course prerequisite is basic knowledge of differential equations. parasitology. and very few discuss global properties of the orbit structure. Within an 5 . Along the way I try to succinctly present the key biological ideas and to compare the predictions of models with actual lab or field data. Also.1 About these lectures These lecture notes follow the introductory course on mathematical biology that I teach at Georgia Tech. evolutionary biology.Chapter 1 Introduction c 2010 Howard Weiss 1. My students come from all corners of the campus including mathematics. although the ideal prerequisite is a second course in ordinary differential equations (ODEs) or dynamical systems. neuroscience. and several students suggested turning my lecture notes into a short monograph. infectious diseases. even though this is not their current research focus. This includes areas such as cancer biology. Our treatment of population dynamics is significantly more rigorous than is commonly found in textbooks. 1963] and use null-clines to study the geometry of solutions of systems of two ODEs. At first glance. The stress on population dynamics is not a large restriction since much of biology can be viewed as the study of populations. I have also written these notes for mathematicians who are contemplating starting to work on some applications to biology and wish to learn the basic population models and how to analyze them. My goals are to introduce and rigorously analyze the most commonly used population and infectious disease transmission models. physics. most students are also interested in applications of mathematical modeling to biodiversity and conservation biology. wildlife biology. conservation biology. etc. and a larger proportion of the population die. the vector population (deer ticks). Ross. Lotka. explain. If a refuge helps animals survive cold weather. He argued that unless the population is checked by moral restraint or disaster (e. The goals of population dynamics are to understand. Sometimes a factor that first appears to be density independent is actually density dependent. Interactions can be predatory. McKendrick. Other developers of the modern theory that we mention during these lectures include Gause. immigration rates. and Volterra. . etc. then when the population is small. Pierre Verhulst proposed his logistic model of population growth. Darwin read Malthus and concluded that exponential growth would lead to many mutations of offspring and would allow natural selection to operate and bring about evolutionary change. Although such complex models are beyond the scope of these notes. May. food supply. Others are embedded in the text and are preceded by a diamond symbol ♦. Some problems are explicitly stated as problems.2 What is a population and how does it change? A population is a group of individuals of the same species that occupy a particular area (see [Wells and Richmond. Many researchers date the modern era of population dynamics (sometime called population ecology) to 1798 and the publication by Malthus of his treatise “An Essay on the Principle of Population” [Malthus. disease. food supply. Kermack. 1. MacArthur. viruses such as HIV and hepatitis C compete for cells to infect. we present many of the basic components and tools to analyze them. and if there are limited refuges available. compete for new hosts to infect [Tillmann et al. In density independent regulation. each individual will have a higher probability of catching an infectious disease than if the individuals had been living farther apart. famine. This interplay is currently a major theme in population dynamics. Density independent factors include weather. natural catastrophes. and fierce competition would naturally ensue. commensural. burgdorferi). and predict the sizes and compositions of populations over time and space.g. death rates. Malthus believed that human populations grow exponentially while the food supply grows linearly.1) and disease ecologists view..1.. where population size is limited by a carrying capacity. Changes in population sizes and composition result from interactions between individuals of the same species. 2001]. the lecture notes contain a small number of exercises. mortality or fertility rates depend on the population density through factors such as predation.6 CHAPTER 1. with the distinction more typographical than pedagogical. but now most agree that both play a critical role. all find refuge and few die. competition. and emigration rates. interactions with the environment. but when the population size is large not all fit into the refuges. etc. and on the population level. There are two types of exercises.. INTRODUCTION infected host. say Lyme disease in New York State. interactions between individuals of different species. cooperative. and pollution. mortality or fertility rates are unaffected by population density. and the reservoir population (white-footed mouse). 1798]. widespread poverty and wars would inevitably result. There have many discussions in the literature about the relative importance of these two regulatory mechanisms. the pathogen population (B. Hutchinson views competition between species as resulting from niche overlap (3. 1995] for 13 definitions). disease. Populations are controlled by density independent and density dependent regulation. Finally. In 1838. as resulting from the niche overlap of the human population. or war). If a population is densely distributed. These changes are expressed in terms of birth rates. In density dependent regulation. Leslie. mutualistic. A severe flood can just as easily wipe out a large population as a small one. and infectious diseases. rather than measuring everything without a plan. Mathematics is all about identifying and classifying patterns. 2000]. expensive. diverse. 1976] 1. a model can realize at most two of the attributes. inaccessible. Levins argues that at best. 2009) A model is a simplification or abstraction of nature. A model that is realistic and general is sometimes called a conceptual model. 2004] “Doing research in population biology without mathematical and/or computer simulation models is like playing tennis without a net or boundary lines. only better..3.3 7 Why do biologists need mathematical models? “Mathematics is biology’s next microscope. but some are useful. numerous. or an epidemiologist wishing to control an outbreak of foot and mouth disease in a region. WHY DO BIOLOGISTS NEED MATHEMATICAL MODELS? 1. . Nobody will be upset if scientists start a smallpox outbreak in a large city and study the efficacy of various control strategies or kill half the female members of a grizzly bear population and measure the population recovery time. [Cohen. 4. A biologist interested in discovering a general principle such as whether macrophages can limit a bacteria population. 1966] defines the three major attributes of a population model: precision. Biologists will make much faster progress toward understanding nature by trying to verify or refute specific predictions. forests). dangerous. big. separating the important from the minor and irrelevant.4 Limitations of mathematical models [Levins. Management requires predictions and predictions require models. generality. “All models are wrong. small.g. For example. fisheries. unique.. A biologist could spend a lifetime measuring population sizes and distributions and have no idea about interactions and mechanisms. a wildlife biologist setting harvesting policy of a particular species in a particular ecosystem. requires realism and generality in her model. A model is precise if it simulates the system behavior in a quantitative realistic way. and is realistic if it simulates the system behavior in a qualitative realistic way. Mathematical models provide a way to design and evaluate protocols to manage and control animal populations. Thus mathematical models can elucidate biological mechanisms. and can sacrifice precision. 2.1. Models are used to approach questions too complex. and realism.” George Box [Box. Utility of mathematical population models 1. Mathematical models help generate testable predictions.g. requires realism and precision in her model. Mathematical models help biologists distinguish between different patterns they see in nature and different mechanisms that might cause these patterns.” Joel Cohen. natural resources (e. only better. It does not matter that her foot and mouth disease model can not be used to control outbreaks of bluetongue. deer population). 3. Models give direction and idea about the important things to measure and provide a means to interpret data. Mathematical models help biologists organize their thinking. if they do so using mathematical models instead of actual populations. Biology is mathematics’ next physics. mutable. Mathematical models provided guidance to the British government on controlling their 2001 outbreak of foot and mouth disease. Models also expose faulty assumptions. slow or fast to approach by other means [McKenzie. and infectious diseases. and can sacrifice generality. wildlife resources (e.” Bruce Levin (from the top of his web page. but are the least general. 000 generations of E.g. along with computer science are likely to be essential. innovation and application in other subdisciplines of mathematics. One must be cautious about going overboard with model realism.” Unfortunately. 1. error prone. but not both. Among these are the use of multiscale models incorporating diferential equations and stochastic elements. Agent based models are the most precise and realistic. which introduce stochasticity into the actual population numbers. counting vocalizations. and all too frequently. who proved the existence of a stable travelling wave of fixed velocity representing a wave of advance of the advantageous gene. While it may be easy to accurately count small numbers of sessile organisms. Estimation methods in the field include trapping (mark and recapture). This was simultaneously studied by Kolmogorov et al. this is impossible for animals that move or for large populations. fishing catch per effort.5 Why do mathematicians need population models? Population dynamics has already generated a considerable amount of new mathematics. geometry. “Science is a process for learning about nature in which competing ideas about how the world works [models!] are evaluated against observations. corrupted (e. although a remarkable exception is the Lenski lab’s long term evolution experiment that recently surpassed 50. etc. long term population and epidemiological data are rare. pellet counting. but we recognize some current trends. involving hundreds of parameters. The system may also exhibit chaotic behavior (like for weather forecasting models) that would severely limit the reliability of long term predictions. .. We do not yet know the shape that mathematical biology will take two decades hence. 1. INTRODUCTION Thus models can be used for general understanding of biological principles or for precise population predictions. Clearly a model that is “too realistic” has no practical use. and many believe it will be one of the main driving forces of new mathematics during this century. In the short and medium term. including algebra. population data tend to be noisy.8 CHAPTER 1. and each simulation may require years to run on a super computer. Fisher’s study of the spread of advantageous genes in a population led him to what we now call the reaction diffusion equation with a logistic reaction term. Any such model would likely require hundreds of nonlinear partial differential equations with time lags. I believe that population biology will generate many new examples of dynamical systems for mathematicians to study and will provide new questions about both old and new examples. It may also take many years to develop and implement a reliable computer algorithm to run simulations. According to Richard Feynmann. coli. Few population time series have 30 generations.. For one example. Even when available. Imagine devising a population model which is a faithful reflection of an ecosystem in all of its complexity. Many of these parameters would take years to measure. percentage ground cover. Turing also used reaction diffusion equations to understand pattern formation and morphogenesis problems in developmental biology. In these cases population sizes are estimated using statistical methods. even when seeming essential. For the latter problem. the mathematics of nonlinear diffusion equations received much of its impetus from biology. and the ability to identify and classify patterns on many scales within enormous data sets.6 Confronting models with data The use of data to evaluate models is fundamental to science. large amounts of missing data). and topology. Many authors present a model sensitivity analysis. one is actually learning little from such models. In a theoretical model. A good fit is a necessary. Given a probability model for the data. There are also Baysian methods of parameter fitting.6. This technique requires maximizing the likelihood function over a high-dimensional space. See [Cooper. 1979] distinguish between types of validity for a predictive model: models that mimic data already acquired and used to parametrize the model.1 9 Model validation [Caswell.” However. then the model is likely to be missing at least one important component. 1. the focus is on inference about “truth”. for statistically independent and normally distributed data). and models that not only reproduce observed new data. or fitting parameters from data. which measures the intensity of response of the model solutions to small perturbations of the parameters. and a popular algorithm to estimate the maximizer is based on the Markov Chain Monte Carlo (MCMC) simulation method [Diaconis. over all values of the parameters. In a predictive model. Although the first type of validity is ubiquitous in the scientific literature. The quality of results depends on the minimization method. which is usually whether the model mimics some aspect of the real world sufficiently well. and the data.6. the sum of squares error (SSE) between the data and the model predictions: SSE(~λ) = N X 2 yi − predi (~λ) . These days maximum likelihood methods seem much more popular than least squares methods. The least squares method involves minimizing the difference between the model outputs and the data. “truth” is not the main issue. 1978].1. models that accurately reproduce new observed data. is usually done using the least squares or maximum likelihood methods [Myung.2 Model Parameterization Parametrizing a model from data. 2009]. but truly reflect the way in which the real system operates to produce this behavior (the truth).. rather validation involves determining whether the model is acceptable for its intended use. a deficient model is more helpful than a model that fits the data.g. The maximum likelihood method selects the value(s) of the model parameters that make the observed data more likely than any other parameter values. and validation focuses on attempts to invalidate the theory [Holling. in general the two methods produce different parameterizations from the same data set.6. Beware that graphical comparisons can be extremely unreliable. “with four parameters I can fit an elephant and with five I can make him wiggle his trunk. 1976] argues that model validation is different for a predictive model than for a theoretical (hypothesis generating) model. one tries to minimize. in the sense that the former has a smaller SSE(~λ) or likelihood value? This does not mean that the former model does a better job of capturing the underlying biological process. . which allow you to incorporate prior information into the fitting procedure. but not a sufficient. Although in very special cases the two methods produce the same output (e. the likelihood function is the probability that the model outputs the observed data given a set of parameters. In this sense. 2007] for a short general introduction. [Zeigler and Oren. i=1 where λ denotes a vector of parameters and predi (~λ) denotes the models prediction for the i − th data point using the parameter set λ. More precisely. condition for such a conclusion. What does it mean when one model fits the data better than another model. CONFRONTING MODELS WITH DATA 1. if the output of a model does not mimic the data used to parametrize it. According to John von Neumann. A complex model with many parameters will require a large data set to parametrize. 2003]. the minimization criterion. . Many population researchers use the open source (free!) R program language for modeling. According to the publisher: “Ecological Models and Data in R is the first truly practical introduction to modern statistical methods for ecology. .. the book teaches ecology graduate students and researchers everything they need to know in order to use maximum likelihood. simulation. and Bayesian techniques to analyze their own data using the programming language R. Wilson. 2004. quantify how well a model fits the data and adds penalties for extra parameters. Wilson] for short expositions. In step-by-step detail. information-theoretic.” . It is also a good idea to compare a model’s data fitting ability with model variants having a greater number and fewer number of parameters. 2008].5 contains an elementary example that clearly illustrates some dangers of overfitting a model. See [Johnson and Omland. shows how to choose among and construct statistical models for data..10 CHAPTER 1. and interpret the results. Model selection tools. and data analysis. INTRODUCTION This is because a model can achieve a superior fit to its competitors for reasons that have nothing to do with the model’s fidelity to the underlying process.. Section 4.. and with other models in the literature. such as the Akaike information content (AIC). estimate their parameters and confidence limits.. I recommend the new book “Ecological Models and Data” [Bolker. in natural time units k. time. The Poincar´eBendixson theorem precludes an autonomous system of one or two ODEs from exhibiting complicated dynamics (see Section 6. food availability and quality. the population change is given by dN = B + I − D − E. E are functions of population size. The adults die shortly afterwards. a one dimensional discrete model can exhibit complicated dynamics (e. The eggs of brown trout in central Pennsylvania streams hatch once per year in the Spring. 2. Many insects and annual plants are obvious examples.Chapter 2 Single Species Models 2.2 The determinants of population change There are only four determinants for population change: births (B). B. I. and the eggs hatch into caterpillars in mid summer and re-emerge the next spring as butterflies. dt (2. etc. environment. In general. Population models for such animals give rise to difference equations or discrete dynamical systems.g. 11 . For a discrete model. chaos). unless otherwise stated. A system of one or two ODEs is generally easier to analyze than a one-dimensional discrete system. (2. immigration (I). deaths (D). D. Because of the time lag. Continuous population models are useful for large populations where births can occur at any time.3). The Baltimore checkerspot butterfly (state insect of Maryland) breed once per year and lay eggs in early summer. where births and deaths are ignored. A major exception will be our discussion of metapopulations. the population at time k + 1 is related to the population at time k by the following balance equation N (k + 1) = N (k) + B + I − D − E.1 Discrete verses continuous population models Discrete population models are useful when the generations do not overlap or all births occur at fixed intervals.1) For an ODE model. which contain a natural time lag.. and immigration and emigration are the major players.2) For these lectures. Moose produce offspring once per year in the Spring. we model closed systems where I = E = 0. as with humans. and emigration (E). vbi.12 2. Three different modes of exchange have thus far been identified in bacteria: transformation. although a few do so by binary fission.vt.wikis] 2. The plasma membrane then invaginates and splits the cell into two daughter cells.e. coli strain 0157:H7 [fromci.. Each circular strand of DNA then attaches to the plasma membrane. A virus particle or virion is a cellular parasite and can not reproduce without the help of a living cell. B = bN .3) . [b] E.1(b) shows E. Most viruses exit the cell by making the cells burst.1: (a) Bacterial fission [from http://www. To reproduce. such as HIV. however. This begins when the DNA of the cell is replicated. are released more gently by a process called budding. the end result is that a bacterium contains a combination of traits from two different parental cells.jpg]. where b and d are constants. i. transduction. Most yeasts reproduce asexually by budding. the DNA of bacteria has a relatively high mutation rate. a process called lysis. coli. The discrete exponential growth model is N (k + 1) = N (k) + bN (k) − dN (k) = (1 + b − d)N (k) = (1 + r)N (k) and the ODE analog is (2. The cell elongates.uic. bacteria also have mechanisms for exchanging genetic material (plasmids). Although different from sexual reproduction.edu].edu/classes/bios/bios100/lecturesf04am/binfission.1(a)). Binary fission theoretically results in two identical daughter cells.3 Exponential growth paradigm The exponential growth model assumes that the birth and death rates are constant.2. Figure 2. which yields two identical cells in each replicating cycle (see Figure 2. Similar to more complex organisms. Other viruses.1 CHAPTER 2. D = dN . Unlike bacteria which replicate by producing two cells in each replicating cycle. the replication of viral DNA or RNA is explosive and frequently yields many hundreds of new virions. strain 0157:H7 with different cells in different stages of reproduction. SINGLE SPECIES MODELS How do microbes reproduce? Most prokaryotes (bacteria and archaea) reproduce asexually by binary fission. The newly created viral proteins and nucleic acid combine to form hundreds of new virions. Once the genetic material has entered the cell. it will hijack the cellular machinery (ribosomes and enzymes) to replicate itself. and conjugation. Protozoan also reproduce by fission. a virion binds to the host cell membrane and injects its genetic material (DNA or RNA) into the cell. causing the two chromosomes to separate. Figure 2. (c) Virus reproduction [from mrcovingtonsciencepage. The exponential growth rate r is a common measure of the fitness of the population.e. Reproduction begins immediately after birth (no time lags). Figure 2. Do some populations grow exponentially? Yes.3. US population from 1650 to 1800 (see Figure 2. invasive species when they first arrive. at least initially. Solving these equations is trivial. and species that are rebounding from a population crash. or size.2: Population of E. EXPONENTIAL GROWTH PARADIGM dN = bN − dN = (b − d)N = rN. 5. sex. The invasive Monk parakeet in US 1976-1994 (see Figure 2. N (k) = (1 + r)k N (0). Inoculation of bacteria into fresh medium after initial lag phase (see Figure 2. 2.. and if b = d the population does not change in time. 4. The population is closed. I = E = 0. Thus if b > d the population grows exponentially.2. N (t) = N (0) exp(rt). and thus the per capita growth rate of the population is constant and equal to r. The rates b and d never change.3(a)). We note that the expression dN/dt = rN is equivalent to N 0 /N = r. In the discrete case.2).uc. Thus there are no differences in the birth and death rates b and d due to age. i.3(b)).clc.4) where r = b − d. 2. Some examples include (notice log scale on y-axis): 1. The following is a list of major assumptions behind the exponential growth model: 1. while for the ODE. coli grown in different media http://biology.edu/fankhauser/labs/microbiology/growth curve/growth curve. dt 13 (2. Species exist as single panmictic population (all individuals are potential partners). 3.htm] [from . if b < d the population decays to zero exponentially. This occurs for species colonizing a new habitat. 3. 3: (a) Population of Monk Parakeet in the US during 1976-1992 [Van Bael and Pruett-Jones. 1999] .4: Exponential growth rate r as a function of body size [Manuel and Molles.14 CHAPTER 2. SINGLE SPECIES MODELS Figure 2. 1996] (b) Population of the United States during 1650-1800 Figure 2. Then comes the exponential growth phase (sometimes called the log phase) which is eventually limited by the exhaustion of available nutrients and accumulation of inhibitory metabolites or end products. Since bacteria can evolve in real time. Zambrano and Kolter. N 0 = rN 2 . See Figure 2. we are learning that death allows new life. Through genetic analysis of bacteria during the final stages.5..g. 1996]. Problem 1. prior to their resumption of division. Example 1. would. N (0) . there is an initial lag phase where the individual bacteria are synthesizing RNA. and other molecules they need for growth. [Cohen. The Doomsday Population Model.e. e. Compare Turchin’s statement with Newton’s first law (law of inertia): an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. are thriving [Siegele and Kolter. and the solution is N (t) = The population becomes infinite in finite time. enzymes. waves of new mutants. Suppose the population growth is even faster and is proportional to the square of the population size. This separable ODE can be easily integrated. Penicillin and other beta-lactam antibiotics are most effective against rapidly growing bacteria. in what is known as the death phase. [Turchin. Show that in a single day. EXPONENTIAL GROWTH PARADIGM 15 In general. divide every twenty minutes. coli could produce a super-colony equal in size and weight to the entire planet earth. while the vast majority of bacteria are dying.5) . under ideal circumstances.2.4 For bacteria transfered from one medium to another. i. Eventually the bacterial population shrinks.. 2001] has formulated a fundamental law of ecology: a population grows exponentially as long as the environment experienced by all individuals remains constant.5: The four growth phases of bacteria The efficacy of antibiotics is sometimes growth phase dependent. 1995]: The exponential model assumes that N 0 = rN . 1 − N (0)rt (2. This phase with slower growth is called the stationary phase.3. Sometimes this can go on for months. one cell of E. Figure 2. which are better and better able to thrive in the noxious and nutrient depleted soup. smaller organisms have larger exponential growth rate r as illustrated in Figure 2. A single cell of the bacterium Escherichia coli. 1992. The latter statement means that N (t) = K is a constant solution. Figure 2.6: Examples of density dependence of fertility rates [Krebs. called the carrying capacity. mates. light. Boonstra. the solution N (t) grows exponentially at a rate approximately r. and the buildup of noxious wast products in habitat.1 CHAPTER 2. where f (N ) = r(1 − N/K)N . The simplest ODE satisfying these conditions is N0 = r(1 − N/K) N (2.. territory. 2001] We now construct a phenomenological ODE model (meaning there is no theoretical underpinning) where.16 2. for small population sizes.6) or equivalently. etc. and K is an attracting equilibrium point. but due to competition for food.4 2. The competition for limited resources (including food. water. territory.4. and limt→∞ N (t) = K for any solution with N (0) > 0. and Sinclair. The leveling off of the population is a consequence of intraspecific (same species) competition. Boutin. In the logistic model the per capita population growth . oxygen) decreases the fertility or survival of individuals (see Figure 2. The carrying capacity is the maximum number of individuals that the environment can stably support (see [Cohen.6). N 0 = f (N ). SINGLE SPECIES MODELS Continuous growth models Logistic growth law No population can grow exponentially for all time. 1995] for 26 definitions). Many populations initially grow exponentially. their population size level off after some time to a stable size K. 10) Figure 2. and many models include logistic growth terms. natural populations rarely if ever reach an equilibrium population K. it is considered to be a reasonable qualitative model for many populations. (2. Natural populations are exposed to many more factors than are assumed in the model. but rather constantly fluctuate.8) and integrating both sides. . Although there appears to be no mechanistic derivation of the logistic growth law.2.4. Unlike the exponential growth model. Since f 0 (0) > 0 and f 0 (K) < 0 we obtain from elementary linear stability analysis that N = 0 corresponds to an unstable solution and N = K corresponds to an attracting solution. Some frequently cited examples exhibiting logistic growth for some period of time include bacteria colonies and yeast colonies (see Figure 2. Another objection is that the logistic model does not take into account time lags in the density dependence of the birth rate. it is separable since it can be rewritten as dN = rdt. For insects. or it approaches the carrying capacity K as t → ∞. and assume that increased population results in increased crowding which linearly depresses the birth date and linearly increases the death rate.9(a) for logistic growth of Tasmanian sheep and 2. The equilibrium points N of an ODE N 0 = f (N ) are those points where f (N ) = 0. A major objection of the logistic model is that although it does a good job describing laboratory populations grown under strict conditions. However. it may take weeks or months for larvae to develop into mature individuals. This population size dependence is called density dependence. Problem 2. we obtain log(N ) − log(N − K) = rt + C. CONTINUOUS GROWTH MODELS 17 rate is a decreasing function of the population size.7 illustrates the behavior of representitive solutions. and for the logistic ODE.9) Assuming that N (0) 6= 0. Either a solution is identically zero. it is evident that N = 0 and N = K are equilibrium points.7) dN. the closed form solution is K N (t) = 1+ K N (0) . this ODE is nonlinear. The resulting ODE N 0 = (b − pN )N − (d + qN )N is logistic with r = b − d and K = (b − d)/(p + q). Another “derivation” of the logistic ODE is obtained by considering density dependent birth and death rates.8). One does not need to explicitly solve the ODE to determine the asymptotic behavior of solutions. 2001](see also Problem (66)). The solutions with initial conditions greater than the carrying capacity are rarely seen in nature (♦ why is this true?).9(b) for AIDS cases in the US. Using this type of analysis show that solutions of any ODE of the form N 0 = f (N ) can not oscillate. − 1 exp(−rt) (2. These correspond to constant solutions. See also Figure 2. We note that the logistic ODE is an an example of a Bernoulli ODE and can be transformed into a linear ODE [Boyce and DiPrima. (2. N (1 − K )N We can rewrite the left hand side as 1 1 − N N −K (2. Start with the exponential growth model N 0 = bN − dN . short generation time. as the ability to compete successfully for limited resources is crucial. See [Gadgil and Bossert. and fecundity are believed to have evolved subject to many trade-offs in the allocation of individual’s resources. breeding that starts early in life. effort. and energy used for one purpose diminishes the time effort. insects.5 2. Most pests are r-strategists. Populations of K-selected organisms are thought to be typically close to their carrying capacity. long life spans. high fecundity. Stearns. as the ability to reproduce quickly with as many offspring as possible is crucial. many life history parameters such as adult body size. Traits that are thought to be characteristic of K-selection include: large body size. Similarly. early maturity onset. combines the notions of quantity of offspring with quality of offspring. called fitness in evolutionary biology. resources spent growing to a larger body size cannot be spent increasing the number of offspring. age at weaning. In diverse and stable communities. Traits that are thought to be characteristic of r-selection include: short life spans. litter size. 1970. The idea behind MacArthur and WIlson’s [MacArthur and Wilson.7: Solutions of logistic ODE. and there is little advantage in competing with other organisms because the environment is likely to change again.18 CHAPTER 2. and poor maternal quality. 1977. For example. while also possessing r-selected traits such as high fecundity.5 1.0 Figure 2. For example. and good maternal quality. 2001] r verses K selection theory is that evolutionary pressure works in two directions. Time.0 0. elephants. In unstable or unpredictable environments r-selection predominates. There is actually a r verses K continuum. Organisms with K-selected traits include include large organisms such as humans. breeding that starts later in life. trees possess K-selected traits such as large body size and long life span.0 1. low fecundity. The resources in a particular environment are finite. Organisms with r-selected traits include bacteria. 1980]. SINGLE SPECIES MODELS Solutions of logistic ODE with K=5 10 8 6 4 2 0 0. The effects of r verses K selection How many offspring should an individual have to ensure that as many of its genes as possible enter the next generation? This measure of reproductive success. . and whales. small body size. K-selection predominates. and energy available for another. and rodents. 1936].com/aids. 1939].9: (a) The figure on the left shows the logistic growth of a population of Tasmanian sheep [from www. CONTINUOUS GROWTH MODELS 19 Figure 2. . (b) The figure on the right shows the logistic growth of AIDS cases in the US [from www.au/Journals/TRSSA/TRSSA V062/trssa v062 p141p148.pdf].nlreg.8: The figure on the left shows the growth of a laboratory population of Paramecium caudatum fitted to a logistic equation [Gause.htm].gov. Figure 2.2.samuseum.sa.4. The figure on the right shows the growth of a laboratory population of yeast cells [Pearl. for every t ≥ 0.4. and N = a is repelling. Since eN˙ + R˙ = 0. Thus the solution N (t) for an initial condition N (0) < a decays to zero as t → ∞ while the solution N (t) for an initial condition N (0) > a converges to K as t → ∞. Monod [Monod.e.11) and (2. R) = eN + R is conserved.14) It is easy to see that the equilibrium points are of the form (0..11) The function ψ(R) is called the Monod function. the maximum per capita population growth rate occurs when the population N is very small (zero). Based on experiments with bacterial populations. K +R = N (2.e. Combining (2.13) (2. the maximum per capita population growth rate occurs for an intermediate population size.12) where 1/e is the amount of resource needed to produce one bacterium. R∗ > 0 and thus the equilibrium points are non-isolated and are the union of two lines. dN dR = −e dt dt. N dt K +R (2.11) to obtain the one-dimensional ODE ν(eN (0) + R(0) − N ) N˙ = f (N ) = N . K + eN (0) + R(0) − N (2. The simplest model is N 0 = rN (N − a)(1 − N/K). some populations employ cooperative hunting strategies or cooperative protection strategies from predators which are ineffective when the population size is small. For such populations. R∗ ) and (N∗ . which we can substitute into (2. Unlike the . eN (t) + R(t) = eN (0) + R(0). Problem 3. R(0)). members of some populations find it difficult to find mates when the population size is small. An easy calculation shows that f 0 (0) > 0 and thus N = 0 is a repelling equilibrium point. Also. or in other words.20 CHAPTER 2.15) The equilibrium points of this ODE are obtained by setting N˙ = f (N ) = 0 and are N = 0 and N = eN (0)+R(0). the function F (N (t). This is a degenerate system of ODEs. Thus ν is the maximum growth rate and K is the concentration of the limiting resource when the growth rate is half the maximum. However. Another calculation shows that f 0 (eN (0) + R(0)) = −ν(R(0) + eN (0))/K < 0 and thus limt→∞ N (t) = eN (0) + R(0). the quantity F (N. Show that N = 0 and N = K are attracting equilibrium points (bi-stability). 0) where N∗ . 1949] proposed an empirical growth law that relates the per capita growth rate of microbial populations to the limiting resource (nutrient) concentration via the formula νR 1 dN = ψ(R) = . R(t)) = F (N (0). This implies that R(t) = eN (0) + R(0) − N (t).2 Monod growth law for bacteria Microbial populations frequently increase until nutrients are exhausted. SINGLE SPECIES MODELS The Allee effect For the logistic model. (2.. i. 2. i. It is monotonically increasing and approaches ν as R → ∞. Monod also found that that bacterial population growth is proportional to the resource depletion.12) yields the coupled system of ODEs N˙ R˙ νR K +R νR = −eN < 0. All equilibrium solutions correspond to population extinction. Although the limiting value does not depend of K. coli fed on glucose. using kinetic.3 Logistic growth with harvesting or predation It is useful to simple construct models that combine logistic population growth with various forms of harvesting or predation.4. 1949] There have been some recent attempts to provide theoretic justifications for the Monod model. (2.10) is from Monod’s original paper and shows how the model fits his population data of E. K +R = N (2. Argue in the phase plane that R˙ < 0 precludes closed orbits. Problem 4. but it is easy to show that the equilibrium points are (0.2. the Monod system becomes N˙ R˙ νR − dN K +R νR = −eN < 0. and thus the population always goes extinct.17) It is not possible to reduce this system to a one-dimensional ODE as we did when d = 0. There is no bacteria death in the Monod system (although it is not totally clear what bacterial death means). The is a manifestation of the degeneracy of the system of ODEs. this parameter effects how quickly solutions approach their limiting value (how?). so this is again a degenerate system of ODEs.18) dt . R∗ ) where R∗ > 0. Figure (2. Figure 2.16) (2. thermodynamic. With a linear death rate term. the limiting value of the Monod system depends on the initial values N (0) and R(0). 2.4.10: Monod growth law from [Monod. such as dN = rN (1 − N/K) − g(N )P. and transport approaches [Liu. CONTINUOUS GROWTH MODELS 21 long-term behavior of the logistic ODE where all solutions with N (0) > 0 approach the same limiting value (the carrying capacity). 2007]. the largest yield/catch that can be taken from a species’ stock over an indefinite period. which one should equate with extinction. Holling introduced the following family of functional responses [Holling. the population (not starting at the unstable equilibrium point) either approaches the attracting equilibrium point or tends to −∞. Problem 5. 1949]. The harvesting yield is Y (t) = HN (t). where E is the fishing effort and q is the catchability of the fish. Why is this harvesting strategy safer than quota harvesting? What .22 CHAPTER 2. the rate of prey consumption increases linearly with the prey population size. If this is modeling a fishery. the population growth rate is fastest. Thus in this model the population can become negative. if there are only five tons of fish left in a certain area of the ocean.20) dt For predators with a Type I functional response. SINGLE SPECIES MODELS where N (t) denotes the population of prey. Show that the MSY is rK/4 when N = K/2. constant harvesting does not make sense when the population is very small. (2. this functional response is called the Schaefer short-term catch equation and is written as P g1 (N ) = qEN . Constant harvesting The model is dN = rN (1 − N/K) − H. This harvesting strategy requires monitoring N . but it is no longer considered a safe management strategy and has fallen into disuse. 1970] based on the assumption that the instantaneous consumption of prey depends only on prey availability and not on the consumer or predator abundance as well. In simpler terms. Biologically. What are the real life dangers with this type of harvesting strategy? Holling Type I functional response The functional response g1 (N ) = HN . For example. Many fishery managers employed this harvesting strategy in the past. In population ecology and economics. independent of their population. the population approaches −∞. Show that for r < HP the equilibrium point N = 0 is globally attracting and the population goes extinct. In the fisheries literature. (2. and for H < rK/4. which can be expensive. Problem 6. there is a single semi-stable equilibrium point at N = K/2. Predators with such unlimited appetites are rarely found in nature. Thus the MSY is H = rK/4. P denotes the predator populations size. A > 0 and the logistic ODE with proportional or constant rate harvesting is dN = rN (1 − N/K) − HN P. The equilibrium point N = 0 undergoes a transcritical bifurcation at r = HP . the maximum sustainable yield or MSY is. and the functional response g(N ) denotes the number of prey eaten per predator per unit of time [Solomon. prey are harvested at the same rate H. and at the equilibrium population. then harvesting ten tons per day makes no sense. Rewrite the ODE as a logistic ODE. the maximum of N 0 clearly occurs at N = K/2 and the maximum value is rK/4. 1959b. This is called Graham’s Theory of Sustainable Fishing (1935): if the fish population is maintained at half its carrying capacity. Y (t) = HK(1 − P H/r). and the sustainable yield is greatest. for the logistic ODE N 0 = rN (1 − N/K). the predators will eat four times as much per day. Show that there is a saddle-node bifurcation at H = rK/4 such that for H > rK/4. Thus for r > HP . At H = rK/4.19) dt In constant or quota harvesting. the fisherman catch the same number of fish every day. If the number of prey quadruple. theoretically. a non-zero population approaches the attracting equilibrium point N = K(1 − P H/r). Predation involves two tasks: searching for prey and consuming the prey (chasing. A predator spending time Ts searching for prey. Thus the required handling time to consume H prey is HTh . There exists a saddle node bifurcation at P2 > P1 . and when N = B the value of the Type III predation function is A/2.4. there can be either one. It is unstable for P < P1 and attracting for P > P1 . B > 0 and the logistic ODE with Holling Type II response is dN AN = rN (1 − N/K) − P.23) The parameter B is known as the switching value. and captures Ha = aeHTs . The origin N = 0 is always an equilibrium point. their feeding rate increases as the prey population increases. where H is the prey density and e is the hunting efficiency. but when the prey density is above the threshold. This response is characteristic of organisms that require non-trivial amounts of time to capture and ingest their prey. depending on initial conditions. 1970. For P > P2 the equilibrium point N = 0 is globally attracting and the population becomes extinct. Holling Type II functional response The Holling Type II functional response is g2 (N ) = AN/(B + N ). do not eat much of the prey. (2. and digesting). Show that there is a transcritical bifurcation at P = P1 . Other names for this functional response are the Monod response or Michaelis-Merten response. the population will reach a positive equilibrium or become extinct. and at high prey densities. Transforming from prey density to prey population gives the desired functional form. Thus Ts = Ha /(aeH). two. This response is characteristic of predators that below a certain prey density threshold. Show that depending on the parameters. Holling Type III functional response The response g3 (N ) = AN 2 /(B 2 + N 2 ). the larger equilibrium point is attracting and the smaller one is unstable.22) 1 + aeHTh At low prey densities. . eating. This is similar to the solutions with Allee effect. where A. A.21) dt B+N For predators with a Type II functional response. For P1 < P < P2 . but saturates at some maximum level A. Murdoch and Oaten. killing. but eventually levels off to an asymptote. predators spend most of their time on handling prey. searches an area of size aTs . (2. predators spend most of their time searching for prey. dt B + N2 (2.2. the rate of prey consumption increases with the prey population size. or three equilibrium points. It follows that the time T required for a predator to search and consume Ha prey is Ha /(aH) + Ha Th . 1975]. Holling gave a simple mechanistic explanation of this functional response. CONTINUOUS GROWTH MODELS 23 are the dangers with this type of harvesting strategy? This harvesting strategy is commonly used by fisheries and wildlife managers. This functional response seems to be the most common and is well documented in empirical studies [Holling. There is a fixed handling time Th associated with each prey eaten that is independent of the number of prey. Problem 7. Solving for Ha yields aeHT Ha = . B > 0 and the logistic ODE with Holling Type III response is dN AN 2 = rN (1 − N/K) − 2 P. exactly one-half its maximum. Thus for P1 < P < P2 . Sketch the bifurcation diagram. There may be more than one way to nondimensionalize an ODE. Define new (and dimensionless) coordinates x = N/B.g. and begins with a dimensional analysis of the problem. The method is a little ad hoc. The parameters A and P can be trivially combined. and sometimes one can exploit the presence of a small parameter (e. since g30 (N ) = 0. This could occur for several reasons.24 CHAPTER 2. The spruce budworm consumes the leaves of coniferous trees. τ = AP t/B. Holling’s three functional responses reflect different types of hunting and feeding behavior of predators. and excessive consumption damages and kills trees (see Figure (2. at high prey densities.g. Predators may lose their “search image” of the prey. which reduces the number of parameters in an ODE and makes it more tractable. which is an increasing function of the ratio of prey density to predator density. At low prey density. Multiple applications of the chain rule yield the equivalent ODE x2 dx = ρx(1 − x/κ) − . Arditi-Ginzburg functional response The above-mentioned functional responses assume that the prey eaten per predator per unit of time is a function of prey abundance alone. Now is a good time to introduce the concept of nondimensionalizing an ODE. it is known that predator density can also influence individual consumption rate. the paradox of enrichment. As for the Type II response. and κ = K/B. at low prey densities. 1989] proposed modeling the functional response using g(N ) = AN/(N + BP ). Observations have shown approximate . 1992]. In these cases the authors [Arditil and Ginzburg. dτ 1 + x2 (2. or four equilibrium points. there can be either two. The logisitic term captures the population dynamics on a much longer time scale. We now show that a simple linear change of coordinates will “eliminate” two more variables. Thus the time scale is short: roughly hours to at most days.5 Case Study 1: Controlling the spruce budworm population The eastern spruce budworm (Choristoneura fumiferana. There could be a small number of refuges which hide the small number of prey from predators. The condition g30 (N ) = 0 ensures that the equilibrium point N = 0 is always unstable. SINGLE SPECIES MODELS This response captures two feeding effects. Finally. Thus the prey population never goes extinct. However.. which do not appear with the Arditi and Ginzburg functional response [Berryman. One then carries out a linear change of variables for each variable and then rewrites the ODE in terms of the new variables. e. Depending on the parameters. one makes intelligent choices of the scaling constants to simplify the problem.24) We will study solutions of the logistic ODE with Holling’s Type III response in detail in Case Study 1. 2. predators have a difficult time finding prey. an effect termed predator dependence. ρ = rB/(AP ).11a)) is a serious defoliating pest of spruce-fir forests in the eastern US and Canada. It also allows for a direct comparison of the magnitude of parameters.11b)). apply singular perturbation theory). predators spend most of their time on prey handling. Also. three. Such predator dependence (usually a functional response that is decreasing with increasing predator density) has been observed in many vertebrate and invertebrate species. see Figure (2. This incongruity of time scales leads to some problems. predators may consume alternate prey. The logistic ODE with Holling’s Type III response has five parameters.. 11: a) Spruce budworm [from www. The authors model the foliage growth S 0 with another ODE. CASE STUDY 1: CONTROLLING THE SPRUCE BUDWORM POPULATION 25 40 year cycles of outbreaks. but we first study the budworm population dynamics on the shorter time scale. (2. Since the birds will eat other prey when few budworms are available. The authors model the budworm population in the absence of predation using a logistic growth term. Jones. where the carrying capacity KS depends on average leaf area per tree S. which tend to be synchronous over large areas of susceptible forests.23).gif] The budworm population can increase several hundred fold in a few years.2. 1978] devised an ODE model to gain an understanding of the mechanism causing the outbreaks. It is assumed that the main limiting factors of the budworm population are food and the effects of predators and parasites. It takes the spruce trees about 7 − 10 years to completely replace their foliage. This is precisely the ODE in (2. have been tried without success.carleton. devastating the forests. who eat many other insects as well. so a characteristic time scale for the budworm is several months. [Ludwig. r is the intrinsic growth rate of the budworms. In an attempt to control the outbreaks. and the lifespan of the trees in the absence of predators is 100 − 150 years.26) .5. Since this is an autonomous first order ODE.25) dt KS N0 + B 2 where P is the population density of predatory birds. So the characteristic time scale for the foliage is decades. including pesticides. solutions can not oscillate. and then returns to low levels. and N0 is the switching value. 1977]. and Holling. The budworms are eaten primarily by birds. Both models incorporate two widely-separated time scales: the budworm density is the fast variable and the foliage quantity is the slow variable. this ODE can be expressed as x2 dx = ρx(1 − x/κ) − . May later gave a simplified version of the model [May. assuming that the foliage and bird populations are constant. Thus the ODE for the budworm larvae population density B is dB B βB 2 = rB 1 − − 2 P.org/DOCREP/ARTICLE/WFC/XII/0562-B3-1. The budworm population explodes.ca] b) Budworm damage to a Canadian forests [from www. We have seen that after nondimensionalizing. which lacks a key predictive feature. the authors use a Holling Type-III predation term to represent the per capita predation. and the birds’ feeding saturates at high worm population levels.fao. The loss of timber represents a significant cost to the wood products industry and various pest management techniques. Figure 2. dτ 1 + x2 (2. 12(b). or three intersection points. There is a globally attracting equilibrium point for 0 < ρ < ρ1 . which are the slopes of the two dashed blue lines in Figure 2. To find the bifurcation values analytically. one sketches the graphs of f (x) = ρ(1 − x/κ) and g(x) = x/(1 + x2 ) on the same axes for various values of ρ (see Figure 2.28) (2. the equilibrium point occurs for small x (low worm population). three equilibrium points for ρ1 < ρ < ρ2 .12(a) ).27) ♦ Clearly x = 0 is always an equilibrium point. (2. two equilibrium points for ρ = ρ2 . which exhibits a cusp catastrophe. For 0 < ρ < ρ1 . two equilibrium points for ρ = ρ1 . there are two attracting steady states. one sets Figure 2. The equilibrium points are solutions of the equation x ρ(1 − x/κ) − x 1 + x2 SINGLE SPECIES MODELS = 0. There are two saddle node bifurcations at ρ = ρ1 and ρ2 . and a globally attracting equilibrium point for ρ > ρ2 . In the case of three equilibrium points. Geometrically. 1 − x2 (2. dx 1 + x2 (2. One equilibrium point occurs for small x and the other for large x. There is a major difference between the globally attracting equilibrium point for 0 < ρ < ρ1 and for ρ > ρ2 . Show that this equilibrium point undergoes a transcritical bifurcation. two are attracting and one is repelling. For ρ1 < ρ < ρ2 . two. is show in Figure 2.12(b). One can search for further bifurcations geometrically or analytically.29) and obtains the bifurcation points x implicitly as ρ= 2x3 (1 + x2 )2 and κ = 2x3 .30) This bifurcation diagram.12: a) Equilibrium points for budworm model b) Bifurcation diagram of cusp catastrophe in the (ρ. and observes that there can be either one. κ) plane ρ(1 − x/κ) = d (ρ(1 − x/κ)) dx = x 1 + x2 d x .26 CHAPTER 2. while for ρ > ρ2 the the equilibrium point occurs for large x (high worm population or outbreak). whether the steady state population is small or large depends on the initial condition. In this case. . Then equation (2.. the spruce trees die and the forest is taken over by birch trees. 0) is an unstable equilibrium point. But the birch trees are outcompeted by the new or remaining spruce trees. Care must be taken since Equation (2. 2. S increases. the leaf density S decreases. the algebra required to effect a local analysis for (B ∗ . S is small. The authors next allow the leaf density S to evolve. i. and thus the initial condition will be close to the equilibrium point for ρ / ρ1 .32) is undefined for S = 0.32) is replaced by . Then ρ= rm S βP and κ = l S. the worm population is always close to equilibrium. Then an outbreak occurs and the budworm population explodes. Computer simulations [May. 1. Now. S ∗ ) is an unstable spiral and also indicate the existence of an attracting limit cycle. Then (0. and eventually the spruce forest returns. Sketching the two nullclines (see Figure 2. However. 4. and ρ passes through ρ1 . the budworm population remains small. lS m S + B2 S qS 1 − − cB.13) shows a counter-clockerwise circulation in the first quadrant.33) where Smax is the maximal leaf density. even after nondimensionalizing. In the long term. the equilibrium budworm population increases. but stays at non-threatening levels. As the forest slowly grows. Smax (2. CASE STUDY 1: CONTROLLING THE SPRUCE BUDWORM POPULATION 27 Let S > 0 denote the average leaf area per tree and assume that K and N0 (from (2. and some algebra yields an interior equilibrium point (B ∗ . 3. Clearly this simplified model does not capture the approximate 40 year boom and bust forest-worm cycle. the budworm population remains at outbreak level because the new initial condition will be in the basin of the outbreak equilibrium point. S ∗ ). indicating the presence of a spiral equlibrium point or limit cycle. As the forest further matures. Suppose one tries to reduce the budworm population by applying an insecticide that kills a proportion λ of the budworm population per unit time. KS = lS and N0 = mS. However. until ρ passes through ρ2 . As the trees start dying. Since the worm population grows much faster than the foliage.25)) are both proportional to S. and eventually ρ drops back below ρ2 .31) This simple model makes the following predictions about the budworm population and the health of the forest. When the forest is very young. The cycle repeats. depending on initial conditions.5.2. This is called a hysteresis effect. there are two possible steady states: either the budworm population remains low or there is an outbreak. They assume that S has logistic growth and the budworm predation by birds satisfies a Holling Type I functional response. βP (2. and thus ρ < ρ1 .e. 1977] indicate that (B ∗ . The forest starts to defoliate. The corresponding system of ODEs is dB dt dS dt = = B βB 2 rB 1 − − 2 2 P. S ∗ ) is formidable. In real life.32) (2. The budworm population is controlled by the birds and remains small. Fernandez-Cancio. However. [Bonilla.34) (2. SINGLE SPECIES MODELS Figure 2. 1995] for results on global attracting). Show that if one applies sufficiently large amounts of insecticide in perpetuity.36) Using realistic parameters.25) to obtain the time delay ODE dB B(t − T ) βB(t)2 = rB(t) 1 − − 2 P. they show the existence of a periodic solution with period about 40 years. Simulations show that a Hopf bifurcation occurs at some 0 < λ < r that eliminates the attracting limit cycle (in some parameter region) and makes the interior equilibrium point attracting (see [Hsu and Huang. and Velarde. Thus the effect of spraying is to decrease both the initial exponential growth rate r of the budworms and their carrying capacity. the environmental and monetary costs would be enormous. the budworm population will be eradicated. 1982] added a time delay to the simplified model (2. but their analysis is significantly more sophisticated.35) where r0 = r − λ and l0 = l(r − λ)/r. 1977] dB dt B βB 2 = rB 1 − P − λB − 2 2 lS m S + B2 B βB 2 = r0 B 1 − 0 − 2 2 P. . such that λ > r. dt KS N0 + B(t)2 (2.13: Nullclines of budworm system [May. lS m S + B2 (2.28 CHAPTER 2. The resulting steady state would still result in a permanent budworm presence and the indefinite application of large amounts of insecticide. To account for the time interval of seven to ten years for the trees to completely replace their foliage. Problem 8. Time-delayed ODEs can exhibit much richer dynamics than ODEs. If 1 ≤ µ < 3. The map fµ3 has a period doubling bifurcation at µ3 . there is an attracting period 23 orbit. This infinite period doubling cascade is illustrated on the bifurcation diagrams in Figures 2. . 1. . then N (k + 1) is negative. √ 5.544090 . the fixed point x = 0 is unstable. but will concentrate on the dynamics of the first model.564407 .14. there is an attracting period 22 orbit. 1] is defined by fµ (x) = µx(1 − x). 7. 1] → [0. If µ2 = 1 + 6 = 3.37) K Notice that if N (k) is sufficiently large. If 3 ≤ µ < 1+ 6. The map fµ has a transcritical bifurcation at µ = 1. .1 Discrete logistic model The most naive way to discretize the logistic ODE is to discretize the derivative. The limit limk→∞ xk = {x21 . . x22 } for 0 < x0 < 1. √ 4. There is an infinite increasing sequence of parameters µk → µ∞ = 3. so that the difference equation becomes xk+1 = µxk (1 − xk ). (2. . If µ3 = 3. Although extremely simple in form. The fixed points x = 0 and x = (r − 1)/r are both unstable. The two fixed points are x = 0 and x = (r − 1)/r (for r > 1). (r − 1)/r. 2005]. . See Figure 2. . Mathematicians usually apply the coordinate change xk = (r/(1 + r))N (k)/K and µ = 1 + r. i. the applications to real populations are much less so.14.38) We can view this difference equation as the dynamical system xk+1 = f (xk ). .2. . the fixed point x = 0 is globally attracting. this one parameter family of dynamical systems exhibits a wide range of complicated dynamical behaviors (including chaotic behavior) and a rich bifurcation structure. The map fµ has a period doubling bifurcation at µ = 2. All the previous mentioned periodic points of period {1.544090 · · · ≤ µ < µ4 = 3. The map fµ2 has a period doubling bifurcation at µ2 . 2003].e. . See Figure 2. 2004]. 2. Thus limk→∞ xk = (r − 1)/r for 0 < x0 < 1.6 29 Discrete growth models We now discuss three natural discretizations of the logistic ODE [Turchin. limk→∞ xk = 0 for all x0 . . While the mathematics of this family of models is rich. 2k−1 } are unstable.5699456 . See Figure 2. 2. (2. 2. If 0 ≤ µ < 1. 6. including [Weisstein. where one obtains the nonlinear difference equation N (k) N (k + 1) = N (k) + rN (k) 1 − . DISCRETE GROWTH MODELS 2.6. . . The following facts can be found in many places. .6. and it is difficult to find examples of populations which are well-modeled by logistic maps. See Figure 2. 22 . All the previous mentioned periodic points are unstable.449490 · · · ≤ µ < µ3 = 3. This model was introduced into the population dynamics literature by May in [May. and the fixed point x = (r − 1)/r is attracting. x22 } such that f (x21 ) = x22 and f (x22 ) = x21 . such that fµk has a period doubling bifurcation at µk which creates an attracting period 2k orbit. For this reason I do not spend much time discussing this model. there is an attracting period 2 orbit {x21 . x0 6= 0. This is biologically silly. 3. All the previous mentioned periodic points are unstable.14.14.15.. where fµ : [0. 10. 2. which together are dense in the unit interval.42) The Ricker model possesses the same rich dynamical structures and chaotic behavior as the logistic model. (2. n = 1. 1] via the conjugacy h(x) = sin2 (πx/2) and is semi-conjugate to the one-sided shift map on two symbols. SINGLE SPECIES MODELS 8.40) x0 + (K − n0 )R−n which is precisely the time-n map of the logistic ODE (see Equation (2.16. It has unstable period orbits of orders 2n . It has the closed form solution Kx0 xn = . Letting t → n + 1. which contains the closure of the unstable periodic points. .41) for 0 ≤ n ≤ t < n + 1.2 Beverton-Holt model Another way to discretize the logistic ODE is to find a difference equation that is the time-one map of the logistic ODE. 2. . . These maps have positive topological entropy. An interesting question is whether there exist real populations which exhibit period doubling bifurcations or chaos. The restriction of fµ∞ to Λ∞ is minimal and topologically equivalent to an adding machine.30 CHAPTER 2.3 Ricker model A third way to discretize the logistic ODE is to start with the modified ODE x = rx(t)(1 − [x(t)]/k). 1975] that the associated mappings fµ are chaotic.6. This fractal set attracts the orbit of almost every point in [0. We will return to this question when we discuss modeling the population of flour beetles in Section 3. Thus the discrete Beverton-Holt modelf exhibits the same long term dynamics as the logistic ODE. 3. one obtains the Ricker difference equation xn+1 = xn exp (r(1 − xn /K)). The map f4 is smoothly conjugate to the tent map f (x) = 2x mod (1) on [0. 9. These persist until µ = 4. (2. . (2.39) xn+1 = 1 + xn /M where R is the proliferation rate per generation and K = (R − 1)M is the carrying capacity of the environment. .10)). The map fµ∞ is usually called the Feigenbaum map. in the sense that they exhibit periodic points of all periods and uncountably many points with sensitive dependence on initial conditions. Integrating from n to n + 1 yields x(n + 1) = x(n) exp (r(1 − x(n)/K)). It follows from a result of Li and Yorke [Li and Yorke. 1957] and can be written as Rxn . 2. This model was introduced by Beverton and Holt in their study of fisheries [Beverton and Holt. See Figure 2. 1]. (2.6. Thus f4 is chaotic in the strongest sense.829 which creates a stable/unstable pair of period three orbits.6. The map fµ∞ also has a non-chaotic strange attractor Λ∞ . where [x] denotes the greatest integer less than or equal to x. Thus it has zero topological entropy. There is a saddle node bifurcation at µ = 3. 60001 0.7 0.4 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Figure 2.9 0.5 0.png] .1 period 2 orbit 0.easypedia.6 0. DISCRETE GROWTH MODELS Μ=2.15: Bifurcation diagram for quadratic family [from www.14: Periodic orbits of members of the quadratic family Figure 2.6.65 0.8 0.56 Period 23 orbit 0.60002 0.2.4 0.8 0.59998 0.60 0 2 4 6 8 10 12 14 Μ=3.75 0.7 0.6 0.59999 0.gr/el/images/shared/7/7d/Logistic Map Bifurcation Diagram.48 period 22 orbit 0 2 4 6 8 10 12 14 Μ=3.5 0.60000 0.5 31 Μ=3.70 0. 32 CHAPTER 2. Μ=4 SINGLE SPECIES MODELS Chaos x_0=.22000 and y_0=.22001 1.0 0.8 0.6 0.4 0.2 0.0 0 5 10 15 20 Figure 2.16: Chaotic behavior of f4 . Notice the sensitive dependence on initial conditions. 2.7 Natural catastrophes, genetic stochasticity, environmental stochasticity, and demographic stochasticity All populations exhibit random fluctuations. However, for small populations, these fluctuations are sometimes major determinants of population change and causes of extinction. The field of conservation ecology addresses population dynamics issues associated with the small population sizes of rare species, where the phenomena discussed in this section are crucial considerations. Some investigators build stochasticity into their population models. They argue that adding stochasticity provides added flexibility to better fit real data, that invariant probability distributions of stochastic processes can provide additional insights, and that the simulation of stochastic models may be easier than for deterministic models. 2.7.1 Bacteria growth model (write this) From Norris’ book on Markov chains Food shortages, disease, and extreme weather can all devastate populations. The following sad tale from [Shaffer, 1981] is illustrative. The heath hen’s original range was the northeast coast of the U.S., from Maine to the Carolinas. It was generally found in the vicinity of oak trees, where its diet consisted of acorns and berries. They were extremely easy to hunt. During colonial times, the heath hen had been found in such abundance that ”servants stipulated with their employers not to have Heath Hen brought to the table oftener than a few times a week.” By 1876 the bird had vanished from even the most remote woods on the mainland, and less than 100 heath hens could be found on the island of Martha’s Vineyard off the coast of Massachusetts. In 1907 a portion of the island was set aside as a refuge for the birds, and a program of predation control was instituted. The population responded to these measures, and by 1916 the population reached 800 hens. Later that year a fire destroyed most of the remaining nests and habitat, and during the following winter the hens suffered unusually high predation from goshawks. The combined effects of these events reduced the population to 100-150 individuals. In 1920, after the population increased to 200, disease took its toll and the population went again below 100. The 1927 count was below 30, and it was discovered that most were sterile males, and soon the females became sterile. One male survived until 1932, and he was last seen on March 11, 1932. 2.7. STOCHASTICITY 2.7.2 33 Natural catastrophes Natural catastrophes, such as floods, fires, droughts, or meteor strikes occur infrequently, and can cause the death of a large proportion of individuals. For the heath hens, the 1916 fire that destroyed most nests and habitat was an example of a natural catastrophe. 2.7.3 Genetic stochasticity For small populations the amount of genetic material available for natural selection is small. Thus if conditions change, there may be less genetic variability for natural selection to act, which could result in extinction. For the heath hens, the sterility of the last survivors was an example of genetic stochasticity. 2.7.4 Environmental stochasticity Environmental stochasticity is the variation in vital rates from one season to the next in response to weather, disease, predation, competition, or other factors external to the population. Environmental stochasticity can affect large populations, as well as small. For the heath hen, the excessive predation by goshawks in 1917 and disease in 1920 are examples of environmental stochasticity of survival rates. The following is for students who have taken an advanced course in probability. A common way that authors incorporate environmental stochasticity is to add additive noise on the log scale. The simplest such extension of the exponential growth model is the stochastic ODE (SODE) dNt = rNt dt + Nt dwt , (2.43) where wt is a Wiener process, and the meaning of this expression is in the integral sense. I thank my Georgia Tech colleague Ionel Popescu for his patience in explaining to me the basic ideas behind analyzing SODEs. Integrating both sides one obtains Z Nt = N0 + t t Z rNs ds + Ns dws , 0 (2.44) 0 Taking expected values of both sides, and using the fact that the second integral is a martingale and the expected value of a martingale is zero, one obtains Z E[Nt ] = E[N0 ] + r t E[Ns ]ds. (2.45) 0 This integral equation can be easily solved E[Nt ] = E[N0 ] exp (rt). (2.46) Thus the expected value of the population size at time t coincides with the solution of the exponential growth model. To compute the population variance, we need to apply Ito’s formulam which is a funny looking version of the chain rule. It states that the process Nt2 can be represented as dNt2 = 2(r + 1)Nt2 dt + 2Nt2 dwt . Integrating both sides one obtains dNt2 = Z 0 t 2(r + 1)Ns2 ds + Z t 2Ns dws , 0 (2.47) 34 CHAPTER 2. SINGLE SPECIES MODELS Taking expected values of both sides, one obtains E[Nt2 ] = E[N02 ] + 2 Z t (r + 1)Ns2 ds. (2.48) 0 This integral equation can also be easily solved E[Nt2 ] = E[N02 ] exp (2(r + 1)t). (2.49) It follows that the variance V [Nt ] = E[Nt2 ] − E[Nt ]2 (2.50) E[N02 ] exp (2(r + 1)t) − E[N0 ]2 exp (2rt). p As t → ∞, the coefficient of variation V [N (t)]/E[N (t)] grows like q E[N02 ] exp (t). = (2.51) (2.52) Thus the population fluctuations become relatively greater as time gets larger and larger. The SODE (2.43) actually has a closed form solution. It follows from Ito’s formula that Nt = exp ((r − 1/2)t) exp (w(t)). (2.53) . This explicit formula allows easy computer simulations of the population process. Problem 9. Consider the SODE modeling exponential population growth with random migration dNt = rNt dt + dwt . (2.54) Imitate the above calculations line by line to show that E[Nt ] V [Nt ] 2.7.5 = E[N0 ] exp (rt) = V [N02 ] exp (2rt) and (2.55) + (exp (2rt − 1))/r. (2.56) Demographic stochasticity Demographic stochasticity is the variability in vital rates arising from random differences among individuals in survival and reproduction within a season. Large populations are highly unlikely to experience significant variation of these averages. For example, if one assumes that a vital rate of individuals varies independently, then the variance is inversely proportional to the population size, and thus the fluctuations are negligible. However, for very small populations of 100 or fewer members, demographic stochasticity could cause extinction. For example, a long run of male births (think ten successive heads of a coin flip) could skew the sex ratio and substantially increase the risk of extinction by almost eliminating the number of breeding females. An important study of demographic stochasticity [Jones and Diamond, 1976] involved breading pairs of birds in the California Channel Islands over an 80 year period. The investigators found that on islands with over 1000 breeding pairs of birds, none went extinct, on islands with 10-100 breeding pairs of birds, 10% went extinct, and on islands with 10 breeding pairs of birds, 39% went extinct. 2. or the population at time t to be n + 1 and for there to be one death. 0) = N (0) βt β = µ. x) = . For ∆t small. the population is certain to go extinct. 2001] and [??. We briefly present the simpliest linear version of this process [Kot. yielding: N (0) µ(1−x) exp(rt)−(µ−βx) β 6= µ β(1−x) exp(rt)−(µ−βx) F (t. The probability that the population will go extinct in time t is N (0) µ(exp(rt)−1) β 6= µ β exp(rt)−µ p0 (t) = F (t.59) The initial conditions are pn (0) = 1 for n = N (0) and P∞pn (0) = 0 for n 6= N (0). Also. During this interval. Thus pn (t + ∆t) = (n − 1)βpn−1 (t)∆t + (n + 1)µpn+1 (t)∆t (2. t + ∆t]. t + ∆t] and has probability µ∆t + o(∆t) of dying during time [t. we assume that each individual has probability β∆t + o(∆t) of giving birth to a single offspring during time [t. β (2. x) = n=0 pn (t)xn and then derives a PDE for F (t. STOCHASTICITY 35 McArthur and Wilson first used a continuous birth-death Markov process to model demographic stochasticity.57) + (1 − nβ − nµ)pn (t)∆t + o(∆t) (2. dt (2. To solve this system of equations. x) that can be easily solved. (1+βt)−βtx where r = β − µ.60) N (0) βt+(1−βt)x β = µ.7.58) Dividing by ∆t and letting ∆t → 0 yields the system of ODEs dpn (t) = (n − 1)βpn−1 (t) + (n + 1)µpn+1 (t) − (nβ − nµ)pn (t). the expected value of the population size at time t grows exponentially . 1+βt (2. For the population to have size n at time t + ∆t requires the population at time t to be n − 1 and for there to be one birth. few multiple birth and death events occur. (2. all].61) and thus the asymptotic probability of extinction lim p0 (t) = t→∞ µ N (0) β>µ 1 β ≤ µ. which is a monotonically decreasing function of the initial population size. and do so with probability o(∆t). Let pn (t) denote the probability that the population size at time t is n. or the population at time t to be n and no births or deaths occurring.62) If β ≤ µ. Even if β > µ there is some positive probability of extinction. one introduces the generating function F (t. ∂F . . 63) ∂x . E[N (t)] = = N (0) exp rt. (2. x=1 It is easy to show that . ∂F 2 . . ∂x2 . = E[N 2 (t)] − E[N (t)]. 64) .x=1 (2. the coefficient of variation ∂F ∂2F + − ∂x2 ∂x p ∂F ∂x 2 !. SINGLE SPECIES MODELS Thus. the variance of the population size is V [N (t)] = = As t → ∞.36 CHAPTER 2. . . x=1 ( β+µ N (0) β−µ exp(rt)(exp(rt) − 1) 2N (0)βt (2.65) β= 6 µ β = µ. V [N (t)]/E[N (t)] grows as s β + µ p N (0). β−µ (2.66) (2.67) For large initial populations, the population fluctuations are relatively very small for large time. 2.8 The age-structured Leslie population model Our next example of a single species population model is the age-structured Leslie population model [Leslie, 1945, 1948]. This is demographers’ main tool for forecasting human populations and most world-wide demographic forecasts are based on this simple linear model. The standard reference is [Caswell, 2001]. The demographers’ Leslie model assumes that females comprise half of the total population and only considers the female population. The population is decomposed into m five year age groups. Associated to age group i are two vital rates. The survival probability pi is the probability that an individual in the ith age group will survive to enter the (i + 1)th age group, and the per capita fertility rate fi is the average number of offspring an individual has while a member of the ith age group. We assume that pm = 0. These vital rates are compiled by demographers and worldwide statistics can be found in [Keyfitz and Flieger, 1990]. More generally, let Ni (k) be the population size of the ith age group at “time” k (measured in units of five years). The basic Leslie model makes two main assumptions: 1. Individuals in age group i at time k enter age group i + 1 at time k + 1, provided they survive the ith age group. Thus N2 (k + 1) = p1 N1 (k), N3 (k + 1) = p2 N3 (k), . . . , Nm (k + 1) = pm−1 Nm−1 (k). 2. During time k, each individual in age group i has, on average, fi offspring. Thus the individuals in age group i at time k produce fi Ni (k) female off-spring, which enter the first age-group at time k + 1. Thus N1 (k + 1) = f1 N1 (k) + f2 N2 (k) + · · · + fm Nm (k). We also assume there is no immigration or emmigration, although if these are known they are not difficult to add into the model. Combining these assumptions yields the following discrete system of difference equations: f1 N1 (k) + f2 N2 (k) + · · · + fm Nm (k) N1 (k + 1) = N2 (k + 1) = p1 N1 (k) N3 (k + 1) = .. . = p2 N2 (k) .. . Nm (k + 1) = pm−1 Nm−1 (k). . i... . which is called the dominant eigenvalue or spectral radius. The matrix A has non-negative entries. y ∈ C + for a. y) = d(y... fm 0 0 0 37 N1 N2 N3 .70) yi (k + 1) Ail yl (k) yj (k) yj (k) l Ail yl (k) j j where pij > 0 and P j pij = 1. y ∈ C + max(xi /yi ) xi yj d(x. y) for any x. and zero entries except for the first row and main sub-diagonal. 1957]. Proof. is strictly greater in magnitude than any other eigenvalue. There exists a real positive eigenvalue λ1 that is a simple root of the characteristic equation of A.. THE AGE-STRUCTURED LESLIE POPULATION MODEL It is convenient to rewrite this linear system in matrix form: f1 f2 f3 f4 N1 p1 0 0 0 N2 0 p2 0 0 N3 (k + 1) = 0 0 p3 0 . .. j yj (k) yi (k + 1) yj (k) (2. x).71) unless all the xj (k)/yj (k) are equal. d(x.. This immediately implies that d(Ax. y) where 0 < τ < 1.. y) ≥ 0.. and the triangle inequality. (2.. Theorem 1. d(Ax.69) i. we write x(k + 1) = Ax(k) and y(k + 1) = Ay(k)...e. One then applies a version of the contraction mapping principle to obtain a unique fixed point w ∈ C + which by definition is an eigenvector with positive entries.8. Ay) ≤ τ (A)d(x.. We briefly sketch Birkhoff’s proof of this theorem based on the Hilbert projective pseudo metric [Birkhoff. and thus the following theorem of Perron and Frobenious applies. . y) = log = max .2. Let us assume that all the vital rates are positive.. Nm pm } (k) (2. then the population vector at time n is given by N(n) = N(0)An . . ... by) = d(x. (2. Ay) ≤ τ d(x. The associated right eigenvector w and left eigenvector v are both real and are the only strictly positive right and left eigenvalues of A. . . and d(ax. . y) where 0 ≤ τ (A) < 1. To verify the contraction property. It immediately follows (♦ why?) that min j xj (k) xi (k + 1) xj (k) < < max . This eigenvalue.68) The matrix A is called the Leslie matrix or projection matrix. If N(0) denotes the initial population vector. Also show that d satisfies the following additional properties: d(x.. Nm 0 0 0 0 | {z A . . Then one can show that all entires of the matrix Am are positive. . The main claim is that A : C + → C + is a contraction mapping with respect to d.j xj yi min(xi /yi ) Problem 10.. . . ♦ Fill in the details. We define a pseudo-metric d on the cone of positive vectors C + in Rn as follows: for x. . . Then P xi (k + 1) Aij xj (k) X A y (k) xj (k) X xj (k) P ij j = P = = pij . y) = 0 ⇔ y = cx where c is a positive constant. Show that d satisfies most usual properties of a metric: d(x.. . . .. b positive constants. 72) 2.00000 0 0 0 0 0 0 0 0 0 Figure 2.99748 0 0 0 0 0 0 . Show that the right eigenvector is proportional to the stable age distribution. the population does not change.98167 0 .38 CHAPTER 2. The eleven age groups for females are 0 − 4.” Theorem 2.75) The limit limk→∞ (1/λk1 )Ak N (0) = c1 w1 = hN (0). λk1 (2. the population grows exponentially. 1990].17 contains the Leslie matrix for the US in 1985 [Keyfitz and Flieger. . .34008 + .19291 0 0 0 . wm }. if the 1985 vital rates persisted. This vector may be scaled so that its first element is one.74) (2.00148 0 . and can be rescaled to give either the proportion or the percentage of individuals in each age class. 5 − 9.71128i). If we write an initial population vector N (0) = ci wi . . If λ1 > 1. (2. that is the number of offspring that an individual may expect to have in the future at their current age. Problem 11. the population of the US would slowly decrease to zero (at a rate given by λ2 = . If λ1 = 1. This says that without immigration. the proof of Theorem 2 is easy. w1 iw1 .03357 0 0 0 0 0 0 0 . SINGLE SPECIES MODELS The Perron-Frobenious theorem allow us to determine the long-term behavior of the population.06330 0 0 .99721 0 0 0 0 0 . 15 − 19.000023 0 0 0 0 0 0 0 0 0 .00000 . and lim k→∞ 1 k A N (0) = hN (0). the population decays exponentially. .00000 . The spectral radius of the 1985 Leslie matrix for the US is λ1 = . . and lim Ak N (0) = 0.21560 0 0 0 0 0 . Consider the ordered spectrum of eigenvalues λ1 > |λ2 | ≥ |λ2 | ≥ · · · ≥ |λm P| along with the basis of normalized eigenvectors {w1 .99537 0 0 0 0 . w1 iw1 .98872 0 0 . . 1. 45 − 59. 50 − 54.26426 0 0 0 0 .99842 0 0 0 0 0 0 0 0 0 . then 1 k A N (0) λk1 c1 λk1 vw1 c2 λk2 w2 cn λkn wn + + ··· + k k λ1 λ1 λk1 k k λ2 λn = c1 w1 + c2 w2 + · · · + cn wn . .73) k→∞ 3.976.00500 0 0 0 0 0 0 0 0 . λ1 λ1 = (2. .99834 0 0 0 0 0 0 0 . The rate of convergence is exponential with rate (λ2 /λ1 )k determined by the second largest eigenvalue in absolute value.99305 0 0 0 . If λ1 < 1. .99901 0 0 0 0 0 0 0 0 . Show that the left eigenvector is the reproductive value of the population. Demographers refer to this result as the “fundamental theorem of demography.11127 0 0 0 0 0 0 . Given Theorem 1. Problem 12. 10 − 14.17: Leslie matrix for the US in 1985 Figure 2. . 9. Federal Endangered Species Act. 2005] is a labor intensive process where the nests are collected and the eggs transferred to hatcheries. Statistics collected since 1998 indicate that in 2007 Florida had the lowest nesting levels in 17 years. 2. There are three strategies currently used to reverse the decline of the loggerhead population: corralling. Forrester. with some estimates at over $125 per turtle. Corralling involves moving nests with eggs to better protected beach areas. They require soft sandy beaches that are dark in the evening.” Although sea turtles live most of their lives in the ocean. After 8-24 months the young turtles are released into the ocean. A sensitivity or elasticity analysis studies the dependence of the spectral radius λ1 on the individual vital rates. and thirty years of studies seem inconclusive. 2. Sometimes nests are covered with mosquito netting to protect them against parasitic fly larvae infestation. with the loggerhead (see Figure 2. 1987. Several authors consider λ1 as a measure of the population’s fitness.S. and adults. and show that the elasticities eij = ∂λ aij ∂ log λ = . Head starting is extremely expensive. and the largest concentration of loggerhead nests are in south Florida. Crowder. three such stages might be larvae. and drowning in shrimp trawling nets (see Figure 2. Although sea turtles spend almost all . Coastal lighting and housing developments disorient hatchlings and make it difficult for them to find the ocean. and turtle exclusion devices (or TEDs). poaching and depredation of eggs results in only a very small percentage of eggs that hatch. the more exposed it is to predators. All other sea turtle species are classified as either “Threatened” or “Endangered. 2005. For insects. On some beaches. adult females must return to beaches to lay their eggs.76) ∂aij hw.000 loggerhead and Kemp’s ridley turtles each year. 1997]. The Council estimated that during the 1980s shrimp trawling drowned 44.084 in 2007. where nesting rates declined from 85. predation of nests. The loggerhead is listed as “Threatened” under the U. pupae. Loggerheads nest along the atlantic coast. where they are incubated and the hatchlings are well nourished. Some authors claim that head starting is detrimental to turtles. head starting. CASE STUDY 2: SAVING THE LOGGERHEAD SEA TURTLE 39 Problem 13.18 (b)). Head starting [Fontaine and Shaver.77) The age classes in the Leslie model can be replaced by developmental stages. The are six species of sea turtles that nest in the United States.2. incidental capture.18(b)). vi where aij is the (i. The longer a hatchling spends on the beach searching for the ocean. Threats to the loggerhead include loss of nesting habitat due to coastal development. 1. A 1990 National Research Council report concluded that shrimp trawling killed more sea turtles in US waters than all other human means combined (see Figure 2. Head starting is also controversial. Adult sea turtles have few natural predators. Grand and Beissinger. and Caswell. Show that the sensitivities ∂λ vi wj sij = = .18(a)) being the most common. j) entry of the Leslie matrix A.988 nests in 1998 to approximately 45.9 Case Study 2: Saving the loggerhead sea turtle Good references for this case study are [Crouse. ∂ log aij ∂aij λ (2. mostly large sharks. (2. 40 CHAPTER 2. SINGLE SPECIES MODELS Figure 2.18: (a) A Loggerhead sea turtle [a.abcnews.com/Technologypopupid=3865610&contentIndex=1&page =2&start=false]. (b) A dead loggerhead sea turtle caught in fishing net [http://www.seaturtlefoundation.org/wpcontent/uploads/2008/05/olive-ridley-caught-in-ghost-net-ian-bell-2004-web.jpg]. their lives submerged, they must breathe air for the oxygen needed to meet the demands of vigorous activity (especially while working to escape from a fishing net). The nets usually kill large juvenile, subadult, and adult turtlea. 3. In the 1970s and 1980s, the National Marine Fisheries Service developed turtle excluder devices, or TEDs. A TED is a grid of bars with an opening either at the top or the bottom of the trawl net (see Figure 2.19). The grid is fitted into the neck of a shrimp trawl. Small animals such as shrimp pass through the bars and are caught in the bag end of the trawl. When larger animals, such as turtles and sharks are captured in the trawl they strike the grid bars and are ejected through the opening. Studies show that trawl nets equipped with properly functioning TEDs could lead to a 97% reduction in sea turtle net entrapment. Figure 2.19: Sea turtle escaping through a TED (on display at The Georgia Sea Turtle Center on Jeckyll Island and photographed by the author). 2.9. CASE STUDY 2: SAVING THE LOGGERHEAD SEA TURTLE 41 What is the best way to use limited resources to prevent the extinction of loggerhead turtles? [Crouse, Crowder, and Caswell, 1987] and [Grand and Beissinger, 1997] constructed Leslie population models for the loggerhead population with no interventions, with interventions to protect the eggs and hatchlings, with TEDs, and with combinations. They assumed a closed population (which other authors question). [Crouse, Crowder, and Caswell, 1987] used a Leslie model with seven developmental stages, which was simplified to five stages by [Grand and Beissinger, 1997] and other authors. The approximations of vital rates came from [Frazer, 1983], and were based on his thesis study of sea turtles on Little Cumberland Island, Georgia. Frazer acknowledged that his parameter estimates are somewhat uncertain, especially his survival rates in the earlier stages. Figure 2.20: Estimated vital rates of loggerhead sea turtles on [Frazer, 1983] The five life stages are eggs/hatchlings, small juveniles, large juveniles, subadults, and adults. The Leslie matrix in figure 2.21 assumes that the probability that the eggs survive to become small juveniles is p and TEDs are not used. L1 = 0 p 0 0 0 0 0 4.665 0.703 0 0 0.047 0.657 0 0 0.019 0.682 0 0 0.061 61.896 0 0 0 .809 Figure 2.21: Leslie matrix for the loggerhead turtle population with first stage survival equal to p. [Crouse, Crowder, and Caswell, 1987] chose p = 0.6747, in which case (assuming a 1:1 sex ratio) the spectral radius λ1 of L1 is 0.95. Thus the population slowly goes extinct. Even with 100% effective head starting (p = 1), λ1 = 0.97, and the population still goes extinct. [Crouse, Crowder, and Caswell, 1987] argue that without any human assistance, p = 0.185, and with corralling, p = 0.580. These two numbers will be important in the following discussion on TEDs. The Leslie matrix in Figure 2.22 assumes that the probability that the eggs survive to become small juveniles is p and TEDs are used to prevent accidental drownings of older turtles. With p = 0.185, the spectral radius of L2 is λ1 = 0.968, and with p = 0.580, the spectral radius λ1 = 1.02423. Thus the use of TEDs alone will not forestall extinction, but the use of TEDs together with strategic use of corralling may prevent extinction. 42 CHAPTER 2. L2 = 0 p 0 0 0 0 0 5.992 0.717 0 0 0.033 0.749 0 0 0.029 0.761 0 0 0.077 67.075 0 0 0 0.877 SINGLE SPECIES MODELS Figure 2.22: Leslie matrix for the loggerhead turtle population with first stage survival equal to p and use of TEDs. In practice, TEDs, especially in their early generation, reduced the fishermen’s catch. There was strong opposition by the shrimp trawl industry against them. However, in 1992, the federal government required all U.S. shrimp trawlers in the Atlantic Ocean and Gulf of Mexico to use TEDs in all waters, during all seasons. Later, summer flounder trawls were also required to use TEDs. However, currently, not all trawl fisheries are required to use them. The use of TEDs, in combination with other conservation measures, appear to be partially successful in helping to recover sea turtle populations. In 2000, the Turtle Expert Working Group found that the population of (the Endangered) Kemp’s ridley turtles is increasing exponentially. However, this same report found that of the four genetically distinct subpopulations of loggerhead turtles, only one is stable or increasing, the status of two are unknown, and the northern subpopulation is still declining. Another application of Leslie matrices to conservation biology can be found in [Fujiwara and Caswell, 2001], which models the population of the Highly Endangered northern right whale. Finally, [Starfield, 1997] contains a useful discussion the use of mathematical models in conservation biology. while an example of intraspecies competition is taller trees in a forest receiving more sunlight than the shorter trees in their shadow. some individuals are denied access to resources by the often aggressive actions of others. and it is not uncommon for these complex relationships to vary over time. 2003] defined the niche as the organism’s place in the biotic environment in relation to food and predators. Darwin and many latter evolutionary biologists view interspecific and intraspecific competition as the driving force of adaptation. The general ecology texts [Begon. commensalism.. and the term symbiosis describes the interactions between different species.1 3. a range of prey size it can eat. Harper.1. In direct competition. individuals have free access to resources but use of those resources by some individuals diminishes their availability to other individuals. Later. habitat formation. 1957] gave the seemingly more “rigorous” definition as the n-dimensional hypervolume. 1996. limited. and community ecology studies the distribution. and Townsend. etc.1 Competition The niche and competitive exclusion Competition is the interaction between organisms. and reproduction. as well as a discussion of the basic models. or the evolution of adaptations.Chapter 3 Models of Communities Communities are collections of coexisting populations. and limiting resource or territory. These include including competition. 43 . It is believed that populations have evolve to perfectly fit their niche. survival. A plant will have a range of soil pH it can tolerate. and ultimately of evolution. In indirect competition. demography. 1971] contain excellent discussions of the relevant biology. Species can interact in several ways. Competition can be between members of different species (interspecific competition) or between members of the same species (intraspecific competition). The niche is a useful heuristic guide that underlies much thinking about communities. where every point in which corresponds to a state of the environment which would permit the organism to exist indefinitely. an organism will have a range of temperatures it can tolerate. 3. etc. mutualism. etc. An example of interspecies competition is lions and hyenas competing for antelope carcasses. Odum and Odum. a range of precipitation it can tolerate. Competition can be direct and indirect. The idea is that an organism’s niche consists of all the environmental factors that influence its growth. Competition usually results in a reduction or elimination of one or both competitors. abundance. or groups of organisms. that use a shared. and interactions between populations. a range of sunlight it can tolerate. predation. [Elton. [Hutchinson. For example. 1969] of competition support this conclusion (see Problem (3. and this implies that members of both populations are homogeneously distributed in space and do not . Complete competitive exclusion is rarely observed in natural..44 CHAPTER 3. or the mean field assumption. [Gause.1: Laboratory experiment of Gause showing competitive exclusion of two species of bacteria [Gause. often with cascading trophic effects. and analyze. This assumption is called well-mixing. leading to a loss of native species diversity. and zebra mussels (Eastern Europe and Russia). homogeneous mixing.” Nelson Hairson 3. Elaborations of niche theory [MacArthur.1. Exotic species are often seen to competitively exclude native species with similar niches. 1936] states that if two species have almost completely overlapping niches. MODELS OF COMMUNITIES In Hutchinson’s niche theory. However. Figure 3. a cornerstone of theoretical ecology. 1936].2. The principle of competitive exclusion.2 The well-mixing hypothesis Most multiple species and infectious disease transmission models assume that the rate of encounter between members of two different populations is proportional to the product of the population sizes. undisturbed ecosystems. Most mathematical models and microcosm experiments [Gause. they cannot coexist at constant levels indefinitely. simulate. 1970] provide mechanistic models of competition for explicit resources. including Kudzu (Asia).4). Whereas some ecologists continue to see value in the concept. “Ecology’s love-hate relationship with the niche concept has been long and not especially pretty. starlings (Europe). The well-mixing hypothesis implies that doubling the size of either population results in twice as many encounters. 1936.. it can often be observed when invasive “exotic” species are introduced into an ecosystem where they have no native predators or pathogens to control their populations. Such spatially explicit models are usually much more difficult to parametrize. This main utility of this assumption is that it allows the use of ODEs instead of PDEs (Section 5. tiger mosquitos (Asia). .7)) . interspecies competition is due to niche overlap.2) or agent based models (Section 5. There are many examples in the US. mass-action mixing. others have despaired of ever finding an expression that is both general and non-circular. Vandermeer. while the Lotka-Volterra models are phenomenological models (with no theoretical underpinning). 0). If αij = r. and exponentiate.1. α ¯ ij = αij (Kj /Ki ) above. In this case the law of mass action states that the reaction rate (the rate of encounter between members of the different populations) is proportional to the product of their masses (the product of the population sizes).1) by N1 .1. N2 (0) ≥ 0. (3. (3. and that the growth rates are reduced due to interspecific competition determined by well-mixing of the populations. intraspecies competition is more important than interspecies competition. u2 ) (3. An easy calculation shows there are four equilibrium points: (0. then each individual of species j depresses growth of Ni by the same amount as adding r individuals of species i. α21 > 1.4) . The validity of the well-mixing hypothesis is very rarely addressed (see [Regoes et al.2) where the competition coefficients αij > 0. and children typically have many more contacts than seniors. Wilson and Worcester [Wilson and Worcester. When used to model infectious diseases. Even in supposedly well-mixed chemostats (see Section 3. biofilms form on the vessel wall that can act as a refuge and invalidate the assumption. u2 = N2 /K2 . while if α12 < 1. When I feel ill.3) = ρu2 (1 − u2 − α ¯ 21 u1 ) = f2 (u1 . N2 (t) ≥ 0 for all t ≥ 0.3 The Lotka-Volterra competition model The Lotka-Volterra competition model assumes logistic growth for the population of each species to account for the intraspecies competitions. (Invariance of positivity) Prove that if N1 (0). (0. this assumption implies that infected individuals interact with every susceptible individual in the entire population. and an interior equilibrium point (N1∗ . 1945] argue that the the real utility of the wellmixing assumption must be found from the a posteriori observation that with the proper choice of constants the model describes the data.3. integrate. I try to stay home and not interact with anybody. Hint: Divide both sides of (3. ρ = r2 /r1 . such conditions are believed to be rare for both free living and parasitic bacteria. The mathematical model for two competing species is dN1 dt dN2 dt = r1 N1 (1 − N1 /K1 ) − α12 N1 N2 (3. interspecies competition is more important than intraspecies competition. Notice also that if α12 > 1. 2003] for an example). neighbors. We nondimensionalize to reduce this number to three. But we all know that humans have contacts with only a small fraction of individuals in their community. The American physical chemist Lotka and Italian mathematician Volterra introduced the well-mixing hypothesis to ecology. Most models of bacteria growth and evolution treat the bacteria as planktonic cells in a well-mixed. K2 ). α21 < 1. COMPETITION 45 mix mostly in any smaller subgroups. and classmates.1) = r2 N2 (1 − N2 /K2 ) − α21 N1 N2 . N2∗ ). (K1 . we obtain du1 dτ du2 dτ = u1 (1 − u1 − α ¯ 12 u2 ) = f1 (u1 . then N1 (t). Problem 14. There are six parameters in this model. However. τ = r1 t. mass culture. u2 ).3). Substituting u1 = N1 /K1 .. 0). i 6= j measure the per capita effect of species j on the population growth of species i. believing that populations could be treated as particles interacting in a homogeneously mixed gas. The following problem treats an important property of all population models. are more likely to have contacts with family members. 3. (u∗1 . u∗2 ) = 1−α ¯ 21 1¯ α21 . (0. It follows from Dulac’s criterion that this system has no closed orbits. trace J(1. 0) = J(1. Show that there are saddle node bifurcations at α ¯ 12 = 1 and α ¯ 21 = 1. 2. 1994] implies there are no closed orbits contained in the first quadrant. Since det J(1. 0) = ρ > 0. hf2 ) = 1 1 ∂(hf1 ) ∂(hf2 ) + =− − < 0. and ∆ = trace2 J(0. 1) = −¯ α12 . Since det J(0. and the last equilibrium point corresponds to steady state coexistence of both species. A simple calculation shows that O · (hf1 . Dulac’s criterion [S. The community matrix or Jacobian matrix for the system is ! ∂f1 ∂f1 1 − 2u1 − α ¯ 12 u2 −¯ α12 u1 ∂u ∂u 1 2 J(u1 . trace J(0. it follows that (0.5) Biologically. u2 ) = = . MODELS OF COMMUNITIES All orbits are bounded since both components of the vector field are negative for u1 and u2 sufficiently large. u∗2 ) = . 1) is a saddle for α12 < 1 and an attracting node for α12 > 1. 0) is an unstable node. ρ−α ¯ 21 ρ 1−α ¯ 12 −ρ¯ α21 0 −ρ . 1−α ¯ 12 α ¯ 21 1 − α ¯ 12 α ¯ 21 . ∂f2 ∂f2 −ρ¯ α21 u2 ρ − 2ρu2 − α ¯ 21 u1 ∂u ∂u 1 2 Thus the Jacobian matrices at the boundary equilibria points are J(0. 0) − 4 det J(1. and ∆ = trace2 J(1. u∗2 ) is 1 1−α ¯ 12 (1 − α ¯ 12 )¯ α12 J(u∗1 . The Jacobian matrix at the fourth equilibrium point (u∗1 . The requirement that the last equilibrium point has positive coordinates is either α ¯ 12 < 1. Simple algebra yields four equilibrium points: (0. and ∆ = trace2 J(1. trace J(1. (¯ α21 − 1)α ¯ 21 ρ ρ−α ¯ 21 ρ α ¯ 12 α ¯ 21 − 1 . it follows that (1. 0). Strogatz. 0) is a saddle for α21 < 1 and an attracting node for α21 > 1. the first equilibrium point corresponds to extinction of both species. 0). 0) = ρ(1 − α ¯ 21 ) − 1. u2 > 0. ∂u1 ∂u2 K1 u1 K2 u2 for u1 . 1. 0) − 4 det J(1. (3. Since this divergence has constant sign throughout the first quadrant. 1). 0) = ρ(α ¯ 21 − 1). We now determine the nature of the equilibria points using the method of linearization or local stability analysis. define the Dulac function h(u1 . u2 ) = 1/(u1 u2 ). 0) − 4 det J(0. 0) = 1 + ρ > 0. the second and third correspond to extinction of one species. α ¯ 21 < 1 or α ¯ 12 > 1. J(0. Since det J(0. (1. 3. 1) = ρ(¯ α12 − 1).46 CHAPTER 3. To see this. Problem 15. it follows that (0. 0) = 2 (1 + (α ¯ 12 + ρ) ≥ 0. 0) = (−1 + (α ¯ 21 − 1)ρ)2 ≥ 0. 0) = 1 − α ¯ 12 − ρ. α ¯ 21 > 1. 0) = 1 0 −1 0 0 ρ . 0) = (ρ − 1)2 ≥ 0. every orbit must approach an attracting equilibrium point. and since there are no interior saddle points.1. For the Lotka-Volterra competition system (3. 1. We obtain the following theorem. If α ¯ 12 > 1 and α ¯ 21 < 1. Theorem 3. and thus it is globally attracting. Since all orbits are bounded. u∗2 ) is a saddle for α ¯ 12 > 1.1) and (3. or heteroclinic connection. COMPETITION 47 Since det J(u∗1 . u∗2 ) = ρ(¯ α12 − 1)(1 − α ¯ 21 ) α ¯ 12 α ¯ 21 − 1 (1 − α ¯ 12 )(1 + ρ) . then species 1 goes extinct and the size of species 2 approaches its carrying capacity K2 . α ¯ 21 > 1 and an attracting node for α ¯ 12 < 1. We verified that there are no limit cycles. attracting limit cycle. the orbits of all interior initial conditions must approach the attracting equilibrium point. 2. If α ¯ 12 < 1 and α ¯ 21 > 1.2: Four types of phase portraits We apply the Poincar´e-Bendixson theorem to prove that in each case the unique attracting equilibrium point is globally attracting. u∗2 ) = trace J(u∗1 . α ¯ 21 < 1. Figure 3.2). α ¯ 12 α ¯ 21 − 1 and it follows that the equilibrium point (u∗1 . consider initial conditions in the interior of the first quadrant.3. then species 2 goes extinct and species 1 approaches its carrying capacity K1 . the theorem states that asymptotically. . Show the function V (x. b. Problem 20. This implies that if a periodic orbit exists.6) = dy + exy + f y 2 . compete for grass on the same field. 2006] constructed an example with four species having a strange attractor. then x(t).1. species 1 goes extinct and the size of population 2 approaches K1 . f [Hofbauer and Sigmund. Thus to survive. y) = xp y q (A + Bx + Cy) is a first integral of motion for suitable choices of parameters. Suppose two populations.. i. MODELS OF COMMUNITIES 3. In case 4. If α ¯ 12 < 1 and α ¯ 21 < 1. the analogous model for n competing species can exhibit complicated dynamics. u∗2 ) is globally attracting by verifying that the function V (u1 . 1998].4 Competition between n species Unlike the Lotka-Volterra competition model for two species. and Sprott. it is not isolated. The equilibrium point (N1∗ . 1975]. c. provide an alternate proof that (u∗1 . 2003]. if x(0). May and Leonard. This is an example of K-selection (see Section (2. population i would want to minimize its competition coefficient aij with the other population or increase its carrying capacity. 1982. Problem 16.e. Wildenberg. one of a large animal species and the other of a small animal species. Suppose a few individuals of a new species arrive and compete with an established species with population at its carrying capacity. y(t) > 0 for all t > 0. Prove that for any general Lotka-Volterra system of the above form.4. c3 . Hirsch and Smith.1)). . no “general Lotka-Volterra system” of the form dx dt dy dt = ax + bxy + cx2 (3. According to the Lotka-Volterra model. Anderson. This model shows that sufficiently high interspecies competition results in the extinction of one of the species. More generally. y(0) > 0. what conditions would ensure that the new population starts growing? What conditions would ensure the new species becomes established? Problem 19. e. or species 2 goes extinct and the size of population 1 approaches K1 . 1988.7) can possess a limit cycle for any choice of parameters a. (3. Models with three species can have limit cycles and heteroclinic connections [Lu and Luo. [Vano. for almost every initial condition (off the stable manifold of the saddle). solutions with positive initial conditions stay positive for all time. which by the Poincar´e-Bendixson theorem must contain an equilibrium point in its interior. u2 ) = c1 (u1 − u∗1 )2 + c2 (u1 − u∗1 )(u2 − u∗2 ) + c3 (u2 − u∗2 )2 is a global Lyapunov function (see 6.48 CHAPTER 3. Problem 17. N2∗ ) is globally attracting. as well as a zoo of global bifurcations. who survives? Problem 18. then depending on initial conditions. 2003. but no strange attractors [Hirsch. c2 . then there is stable coexistence. The size of population 1 approaches N1∗ and the size of population 2 approaches uN2∗ . Suppose they are equally competitive and their intraspecies competition is as strong as their interspecies competition.2) for suitable choices of c1 . Assume that a limit cycle exists. 3. If α ¯ 12 > 1 and α ¯ 21 > 1. Using Theorem 3. d. Noel. 4. 2). Problem 21. They used the following system of difference equations with Beverton-Holt type intraspecies competition terms.2 Predation Predation is the consumption of one organism (prey) by another (predator). PREDATION 3. 1999. 1977. Compare with the equilibrium points of (3. Lenski 1988a). 0) is globally attracting for b1 < 1 and b2 < 1. 1985] Chao et al. The local stability analysis combined with this result gives the global dynamics. Lu and Wang. The same is true for discretizations of the Lotka-Volterra competition model. See [Cushing. and chaos xk+1 = xk exp (r1 (K1 − α11 xk − α12 yk /K1 )) (3.1. This discrete model exhibits the same four dynamical scenarios as the Lotka-Volterra ODE model. Levarge. and Henson.2. We begin with the classical predator-prey model of Lotka and Volterra. 2001] show that all orbits approach a fixed point as t → ∞. 1987. Prove that (0. 1 = b1 x k 1 + c11 xk + c12 yk 1 = b2 yk . the following system with Ricker type intraspecies competition terms exhibits a zoo of complicated orbit structures. Lenski and Levin 1985. Hutson. More recently. Predators lower the fitness of their prey. while fishing was reduced (see Problem 2).8) (3. although they also played an important role in the historical development of the competitive exclusion principle.9) After nondimensionalizing. On the other hand. 3. 2004] for complete details. Chitnis.1) and (3. and Jansen. bacteria and bacteriophages (virus that attack bacteria) have been proposed as ideal experimental systems for studying predator-prey dynamics [Lenski and Levin. Park and Leslie combined laboratory experiments and mathematical modeling to study competition of flour beetle species. resulting in various anti-predator adaptations.5 49 Discrete competition models Discrete competition models are not as common in the literature as ODE models. global bifurcations. Recall that there are several natural discretizations of the logistic ODE which exhibit very different dynamics. Wendi and Zhengyi. (3. Using the theory of discrete monotone flows.11) See [Hofbauer. Levin and Lenski 1983. the authors [Liu and Elaydi.10) yk+1 = yk exp (r2 (K2 − α22 yk − α21 yk /K2 )) . Selective pressures has led to an evolutionary arms race between prey and predator. and thus reduce the prey’s chances of survival and reproduction. In the 1960s. Find the fixed points of the nondimensionalized system and classify their local stability. See ****** .3. 1999] for rigorous results concerning permanence of solutions and the existence of a globally attracting fixed point in certain special parameter ranges. one can assume that c11 = c22 = 1. 1 + c22 xk + c22 yk xk+1 yk+1 (3. Volterra devised this model to explain the large increase in the percentage of selachians sold in the fish markets at Italian ports during World War I. P ) = . or center for the linearized system.13) to obtain the separable ODE V (r − αP ) dV = dP P (−q + βV ) or −q + βV r − αP dV = dP. and the populations are well-mixed.1 CHAPTER 3. The Jacobian matrix for the system is r − αP −αV J(V. If β is small. (3.g.50 3. Such a system is called a conservative system. The method of linear stability analysis and the Hartman-Grobman theorem are not applicable in this case. e. and without food.3). 0) < 0. instead of modeling population size.g. The term βV is called the numerical response of the predator population.14) Integrating. and thus the extinct equilibrium point (0. Without predators. the predator population decreases exponentially and become extinct. Assume that the predator’s sole food source is the prey. α > 0 denotes the capture efficiency. e. Again. The trace of J(q/β. MODELS OF COMMUNITIES Lotka-Volterra predator-prey model Let V (t) denote the prey (victim) population and P (t) the predator population at time t. thus the eigenvalues are imaginary. a seed consumed by a bird. Clearly having more prey around increases the probability of a predator catching a prey. V P (3. a single prey item contributes substantially to the predator population. r/α) = 0 is positive. The determinant of J(0. r/α). the prey population increases exponentially. The parameter β measures the ability of predators to convert a new prey into per capita growth of the predator population. 0) and (q/β. the prey’s food source is unlimited (so there is no competition among the prey). βP −q + βV The Jacobian matrices at the equilibrium points are r 0 J(0.2. a single prey item does not contribute much to the predator population. It is easy to find a conserved quantity for this system. The term αV is called the functional response of the predator population and measures the rate of prey capture by a single predator as a function of prey population. 0) is a saddle. one obtains that the function F (V.. The Lotka-Volterra predator-prey model is dV dt dP dt = rV − αV P (3. For a more realistic model this should indeed include a saturation term (see Section 3.15) .13) where r is the prey growth rate. Thus (q/β.. 0) = . r/α) is a linear center. a moose captured by wolves. Divide (3. these equations may describe population density or biomass. There are two equilibrium points: (0. If β is small. P ) = −q log V + βV − r log P + αP is constant along orbits.2.12) by (3. (3. β α = 0 q −v 0 . q is the predator death rate. The capture efficiency α measures the effect of a predator on the per capita growth rate of the prey population.12) = −qP + βV P. r/α) = 0 and the determinant of J(q/β. and measures the per capita growth rate of the predator population as a function of prey population size. 0 −q J q r . and β > 0 is the product of the capture efficiency and the biomass conversion efficiency. we introduce logorithmic coordinates x = log V and y = log P . In recent work [Gidean. This orbit . and rewrite (3. Figure 3. PREDATION 51 This predator-prey system is also a Hamiltonian system.20) Problem 22. Strogatz.3: Integral curves of F which are orbits of the ODE The fact that all orbits are closed curves or a single equilibrium point means that the predator and prey populations oscillate. Conservative systems in population modeling are not common. ∂H ∂x ∂H ∂y = = dy dt dx − .2.3 and 3. The most elementary way is to sketch the level curves of F and see that all orbits other than the two equilibrium points are closed curves. which states that for a Hamiltonian system of two ODEs. with amplitude depending on the initial conditions (see Figures 3..19) (3. One can quote the Lyapunov center theorem.18) is a Hamiltonian for this system. There are several ways to prove the interior equilibrium point is a nonlinear center. Ugarcovici. 1994]. a linear center is actually a nonlinear center [S. To see this.13) as dx dt dy dt = r − α exp(y) (3.16) = β exp(x) − q.4). y) = α exp (y) − qx + β exp (x) − ry (3. Can you find a biological interpretation of H? Please email me if you can. and Weiss. Meiss. 1994]. dt (3. but there are some well known examples that can be transformed into conservative systems. i.17) It is easy to check that the function H(x.12) and (3. 2009] we apply methods from KAM and twist map theory to study the dynamics of some of these models.e. One can also quote the analogous theorem for conservative systems [S.3. (3. Strogatz. Thus the interior equilibrium point is surrounded by closed orbits and must be a center. 2. and this is one of several major criticisms of this model. Figure 3. y(t))dt = . Predators go hungry and declines in number. Show that the orbit averages Z Z 1 T q 1 T r and V (x(t). D’Ancona asked his future father-in-law to explain this. T 0 β T 0 α (3. Such an orbit structure is called structurally unstable. Humberto D’Ancona. With fewer predators. which are not desirable for humans to eat) caught (see Figure 3. and Volterra devised and used the predator-prey model with proportional rate fishing: dV dt dP dt = rV − αV P − f V (3. 3. which is followed by growth in V . The victim population peaks before the predator population. Figure 3. which is followed by a crash in P . 1. and Venice during and shortly after WWI. For any initial condition besides the equilibrium point.4: Oscillation of solutions of predator-prey system Problem 23.5).23) . Predators eat prey and reduces their numbers. prey survive better and their population increases. In other words. and the cycle continues. 4. but there was an large increase in the percentage of selachians (predators such as sharks and rays.22) = −qP + βV P − f P. Trieste. Consider a closed orbit (x(t). Increasing prey populations allow predator population to increase 5.21) Example 2. (3. y(t)) of period T . Structurally unstable systems can not be observed in nature. Go to 1. During the war there was significantly reduced fishing. MODELS OF COMMUNITIES structure is non-generic and can be significantly altered with an arbitrarily small smooth perturbation.52 CHAPTER 3. which can change the center to a sink or source.3 (noting the direction of the arrows) shows that growth in V is followed by growth in P (and decay in V ). was a marine biologist who studied the numbers of fish sold at the fish markets of three Adriatic sea ports: Fiume. y(t))dt = P (x(t). (Volterra’s Principle) Volterra’s son-in-law. The authors [Evans and Findley. Although every second order autonomous ODE can be written as a first order system of ODEs.25) Show that addition of the refuge stabilizes the system by making the interior equilibrium point attracting. This is an example of Volterra’s Principle: an intervention in a prey-predator system that removes prey and predators in proportion to their population increases prey populations. .ac. for f < r. the average fish populations are q+f r−f .13) and that the previous results apply. the converse is false. In fact. and the indiscriminate use of the pesticide killed both scales and ladybugs. the pest was already partially controlled by a predator. In particular. PREDATION 53 Notice this system of ODEs has the same form as system (3.12) and (3. 1999] show that the predator-prey system (3. a citrus pest.24) = −qP + βV P.manchester. β α . that an increasing in fishing will lead to an increase in the selachian food population. Figure 3.2. yet the scale population increased. The model predicts that the decreased fishing should result in an smaller average prey population and a larger average selachian population. Such a system can be modeled by the system dV dt dP dt = r(V + S) − αV P (3. (3. (Predator-prey with prey refuge) Consider the predator prey system with a simple prey refuge that hides (a constant) S prey from predators and is always full.uk/ mrm/Teaching/MathBio/ Problems/probSet2. and show that solutions of the predator-prey system are related to elliptic integrals.pdf] Problem 24. as was observed. Note that this is somewhat counterintuitive.13) is equivalent to a second order ODE. This is considered the first application of “sophisticated mathematics” to environmental modeling.3. Another example is the Australian cottony cushion scale insect. the ladybug. DDT was used in an attempt to eradicate this pest.5: The fraction of selachians brought into the port at Fiume from 1914-1923 [from http://www.12) and (3. that was introduced to the United States in 1868.maths. and that inverted biomass pyramids can occur for suitable immigration rates.54 3. 2009].27) where c denotes the biomass conversion efficiency. Adding a constant immigration i to (3. must consume 90. Let V denote the biomass of the aggregate prey class and P the biomass of the aggregate predator class. 2008).28) = −qP + βV P. then at equilibrium. It seems obvious that biomass pyramids must be bottom-heavy. Obura. The first problem explores a very simplifed model of how an inverted biomass pyramid can occur in such communities. Tyler Miller’s example: Three hundred trout are needed to support one man for a year. The next problem explores two simple models of how an inverted biomass pyramid can occur in such ecosystems. Problem 25. pers. i. 000 frogs.. Malay.30) = −q(P + i) + βV P. at equilibrium.2 CHAPTER 3. 000 tons of grass. Donner. comm. that must consume 27 million grasshoppers that live off of 1. and Weiss. (3. the biomass ratios are increasing functions of the immigration rate. Singh. Friedlander. Show that if c · r > q. Consider the predator-prey system dV dt dP dt = rV − αV P (3. (3. We have recently used mathematical modeling to identify several mechanisms that can cause inverted biomass pyramids [Wang. Smith. I like G.13) consistent with the two scenarios yields the resulting systems of ODEs dV dt dP dt = rV − αV P + i (3. Consider two types of immigration: (i) immigrating prey fish stay in the reef and adapt to survive in the new habitat. Reef ecologists have observed significant immigration of prey fish in a North Carolina reef that appears to have an inverted biomass pyramid (M. Konotchick.26) = −qP + αcV P. Hay. or inverted biomass pyramids.. et al. 2008].31) Show that in both immigration cases. The authors of the .29) and dV dt dP dt = rV − αV P (3. In some aquatic ecosystems. The trout. Recently.12) and (3.. MODELS OF COMMUNITIES Inverted biomass pyramids An biomass pyramid is a graphical representation designed to show the biomass of organisms at each trophic level in a given ecosystem. an inverted biomass pyramid has been observed at a couple of pristine coral reefs [Sandin.e. there is an inverted biomass pyramid. but there are a few examples of top-heavy. they provide additional food to predators but do not add to the local prey population. DeMartini.2. Maragos. Dinsdale. in turn. (3. Problem 26. (ii) immigrating prey fish leave the reef if they are not eaten by hungry predators. the biomass of phytoplankton may be lower than the biomass of zooplanktin. which consume the phytoplankton. Morrison. Consider the predator-prey system that includes a Holling Type II predation response. and thus there are no closed orbits contained in the first quadrant. This implies that the population will decrease super-exponentially. Show that d dt α V + P β < 0. With realistic parameter values. the predator population decreases exponentially. In a recent manuscript [Singh. (3. The corresponding system of ODEs is dV dt dP dt = = αP V V +γ βP V −f P + . Ginzburg.2. 2000]. Problem 28. and thus (d/dt)(N 0 /N ) < 0. Wang. PREDATION 55 study found 80% of the biomass in apex predators (reef sharks. The following problem shows the simple chemostat system can be expressed as a type of predator-prey system. P (t)) lies in a closed and bounded set for t ≥ 0. Problem 27. Remark 1. However.35) The following problem analyzes the dynamics of this model. This equivalence is extensively discussed in [Jost.3 in this context.32) (3. solutions exhibit an inverted biomass pyramid. in his book [Ginzburg and Colyvan. We are using the model to examine the role of fishing in degrading a pristine reef.33) Observe that this system is identical to the basic chemostat equations (3. Morrison. 2.). their internal energy level will decline over time.36) . [Ak¸cakaya.58). The prey were small fish hiding in coral refuges. 1988] conducted experiments with Hydra in the 1980s and observed a super-exponentially population decrease. 1. 3. 2009] we devise a refuge-based model of the fish biomass structure at a pristine coral reef. 2004] Ginzburg argues that if an organism continues to metabolize energy in the absence of food. and constant rate prey and predator emigration f . for V > K and thus every orbit (V (t).34) = −qP + βV P. if there is zero prey.2. and Slobodkin.13) to include logistic growth of the prey population. Thus N 0 /N = r < 0. V +γ f0 − f V − (3.12) and (3. Show that h(V. P ) = 1/(V P ) is a Dulac function. snappers. and our model shows that any type of fishing is harmful in the sense that it works to “undue” the inverted biomass pyramid.3. (3. constant prey immigration f0 . According to the predator-prey model. The corresponding system of ODEs is dV dt dP dt = rV (1 − V /K) − αV P (3.57) and (3.3 Predator-prey model with logistic growth and Holling-type responses We begin by modifying the predator-prey system (3. and Weiss. etc. Several reef ecologists speculate that an inverted biomass pyramid is a sign of a natural and healthy coral reef. Slice. Interpret our analysis in Section 3. It should follow that their net rate of reproduction N 0 /N will decline over time. The model which does not assume mass-action mixing of predators and prey and has a refuge of explicit size. 5. over the long term the two population coexist. Deduce using the Poincar´e-Bendixson theorem that for K > q/β the equilibrium point (V ∗ . P ∗ ) is globally attracting. Thus if the carrying capacity of the prey is sufficiently large. Deduce using the Poincar´e-Bendixson theorem that for K < q/β the equilibrium point (K. 4. P ∗ ). Verify that the function .56 CHAPTER 3. Show there is a saddle node bifurcation at K = q/β that for K > q/β creates an attracting interior equilibrium point (V ∗ . 0) is attracting and K > q/β is a saddle. Show that for K < q/β the equilibrium point (K. MODELS OF COMMUNITIES 3. 0) is a saddle equilibrium point. Show that (0. 6. 0) is globally attracting. . . . . V . α . P . q ∗ ∗ . . P ) = (V − V ) − log . W (V. ∗ . + ((P − P ) − P log . . ∗ . . P ∗ ) is globally attracting. The corresponding system of ODEs is dV dt dP dt = rV (1 − V /K) − = −qP + βP V . 7. The general theory of ODEs tells us that solutions of ODEs depend continuously on the parameters. This was first studied in [Rosenzweig and MacArthur.13) is a limiting case of this system. If there is a unique interior equilibrium point. V +γ αP V V +γ (3.42) Problem 29. y) = dξ + c dη. (3. Compare the orbit structures of the modified predator-prey system and the original predator-prey system.41) (3.37) is a global Lyapunov function (see 6.39) where p(x) is a smooth function satisfying p(0) = 0 and p(x) > 0 for 0 ≤ x ≤ K. The reader might wonder about the origin of the Lyapunov function W . Let K.12) and (3. All Hollings type responses satisfy these properties for p(x). This provides another proof that (V ∗ . β V β P ∗ (3. 1963]. so the solutions of the MacArthur and Rosenzweig model converge to the solutions of the Lotka-Volterra model.38) = (p(x) − d)y. We now further modify the predator-prey system to include a Holling Type II predation response. α. ♦ Prove that the function Z x Z y p(ξ) − d η − y∗ W (x.40) p(ξ) η x∗ y∗ is a global Laypunov function. .2). then it is globally. (3. γ → ∞ such that α/γ → c and show that the Lotka-Volterra model (3. Consider the following Gauss-type predator-prey system: dx dt dy dt = rx(1 − x/K) − cp(x)y (3. Conclude that the equilibrium point (x∗ . Show that (0. (1 + x∗ )(1 − x∗ /κ)) = .40) to construct a Lyapunov function to prove that for c/(1 − c) > κ the locally attracting spiral (x∗ . c = q/β. κ = K/γ to obtain the simplified system dx dt dy dt Problem 30. 4. τ = t/r. 0) is attracting and is a saddle for c/(1 − c) < κ.44) 1.3. y ) = trace J(x∗ . P = rγ/α. 1+x (3. Show that for c/(1 − c) > κ the equilibrium point (κ. = x(1 − x/κ) − = −cdy + xy 1+x (3. but one needs to check the non-degeneracy conditions in the Hopf bifurcation theorem to ensure that a Hopf bifurcation actually occurs. y ∗ ) is a locally attracting spiral for 1 + c − κ + cκ < 0 and an unstable spiral for 1 + c − κ + cκ > 0. Show that d dt 1 x+ y d < 0. y ∗ ) ∗ c(1+c−κ+cκ) (c−1)κ d 1 − c(1+κ) κ ∗ J(x .43) dxy . y(t)) lies in a closed and bounded set for t ≥ 0.2. 3. 5. for x > κ (3. the eigenvalues of J(x∗ . y ∗ ) = c(1 + κ) cd 1 − κ c(1 + c − κ + cκ) . y ∗ ) are ±i. y ) = −c 0 ! . 1 − c κ(1 − c)2 (3. Deduce using the Poincar´e-Bendixson theorem that for c/(1 − c) > κ the equilibrium point (κ. y ∗ ) = (x∗ . 0) is a saddle equilibrium point.46) with Jacobian matrix at (x∗ . 0) is globally attracting. d = β/r.45) and thus every orbit (x(t). y ∗ ) − 4 det J(x∗ . The symbolic manipulation capability of Mathematica is ideally suited for this task. This is indeed the case. y ∗ ) is globally attracting. Use (3. 2. y ∗ ) < 0. Show that for c/(1 − c) < κ there exists an interior equilibrium point c (κ − κc − c) (x∗ . (c − 1)κ and do the tedious calculation to show that ∆ = trace 2 J(x∗ . . PREDATION 57 We can nondimensionalize this system by defining new coordinates V = γx. which indicates a Hopf bifurcation and the creation of an attracting limit cycle. Show that ∗ ∗ det J(x . When 1 + c − κ + cκ = 0. Modify the predator-prey system (3.58 CHAPTER 3. 1989] contains a thorough discussion of the dynamics of this model. The existence of a closed orbit then follows from the Poincar´eBendixon theorem. including the system (3. However. Problem 32.demonstrations. Use the geometry of the nullclines to construct a trapping region (see Section 6.4 Experiments with protist communities In the 1970s. See Figure (3.42).8 shows that this system does a reasonable job at predicting the qualitative outcome of the experiment and has the correct number of oscillations.3). Harrison used Luckinbill’s data to test the predictive ability of several predator-prey models. MODELS OF COMMUNITIES Here. In one medium the populations oscillated for 33 days (see Figure 3. More food availability does not lead to higher prey biomass.41) and (3. x∗ does not depend on κ and thus the equilibrium prey population does not depend on its carrying capacity. Biologically. Rosenzweig called this the “paradox of enrichment [Rosenzweig. which means that one of the populations becomes extinct.7).2.6: Limit cycle for MacArthur and Rosenzweig model [from http.6).” Computer simulations show that as the carrying capacity increases. at some point the equilibrium point becomes unstable. Luckinbill conducted predator-prey experiments with protists. . He grew Paramecium aurelia together with is prey Didinium nasutum in various mediums [Luckinbill. 1995].13) to include a Hollings Type III predation response and analyze the possible phase portraits. the limit cycle increases in size and gets closer to the coordinate axes. one might assume that increasing the carrying capacity of the prey (called enrichment) “increases the stability” of the system. In [Harrison. 1971]. 1973].12) and (3. Problem 31. Figure 3. com/PredatorPreyDynamicsWithTypeTwoFunctionalResponse/] 3. Figure 3. Consider parameters for which a limit cycle exists.wolfram. [Yodzis. 3.8: Comparison of the population density of the predator Paramecium in experiment and model .2.7: The population cycles of Paramecium and Didinium Figure 3. PREDATION 59 Figure 3. x+k (3. The super-predator eats the predator which eats the prey.60 CHAPTER 3.53) 2b1 a2 − b2 d2 a2 − b2 d2 This fourth equilibrium point corresponds to long-term coexistence of all three species. If a1 > b1 d1 .2. 1996]. 0) and (1. 0. y ∗ .49) = (3. . Antia and Lipsitch.6 A super-predator. The system always has the equilibrium points (0. 1 + b1 x = x(1 − x) − (3. and prey community model MacArthur and Rosenzweig proposed a model of a community consisting of three species: super-predator. (3. The Tc curves are saddle node bifurcations and the H curves are Hopf bifurcations. 1997] the authors model the response of the immune system to a growing microparasite population. numerics indicate that period doubling cascades of limit cycles generate strange attractors with chaotic behavior. z ∗ ) = q 1 b1 d2 a1 x∗ y ∗ b1 − 1 + (b1 + 1)2 − a4a − d ∗ 1 d2 2 −b2 d2 1+b1 y .5 Experiments with bacteria and bacteriophages (WRITE THIS) 3. MODELS OF COMMUNITIES Problem 33. there is a third equilibrium point d1 a1 − d1 (b1 + 1) . 0 . There is an zoo of local and global bifurcations that are carefully described in [Kuznetsov and Rinaldi. Analyze this model. and both predators have Holling (or Monod) predation responses.2. for some parameter values the three species coexist and their populations cycle periodically.47) (3. Here the immune system is the predator and the microparasites are the prey. . predator. In particular. After nondimensionalizing.. Figure 3. a1 − b1 d1 (a1 − b1 d1 )2 and in certain parameter ranges contained in a2 > b2 d2 there is a fourth equilibrium point (x∗ .52) . the system becomes dx dt dy dt dz dt a1 xy 1 + b1 x a1 xy a2 yz − d1 y − 1 + b1 x 1 + b2 y a1 xy − d2 z.50) = (3.51) where d1 is the death rate of the predators and d2 is the death rate of the super-predators. . predator. 3. Their basic model is of the form dx dt dy dt = rx − cxy = s x y − dy. 0). Globally. 1994. and prey. (3.9 shows part of the bifurcation diagram as a function of the two parameters d1 and d2 . In [Antia et al. 0.48) where x is the parasite density and y measures the immune response. 8 Canadian lynx and snowshoe hare In boreal forests.7 Two predators and one prey community model Several authors proposed and studied a model of a community consisting of two predators and one prey [Hsu et al. Both predators are competing for the same food.11 contains a plot of the pelt data from 1845 to 1935. and it kept meticulous records on the numbers of lynx and hare they purchased from trappers in Canada over a 220 year period.3. the Canadian lynx (a wild cat similar to the bobcat) is a specialist predator of the snowshoe hare (see Figure 3. 1978a.1. 1982] rigorously established the existence of Hopf bifurcations that result in attracting limit cycles. and there are issues about the validity of this assumption.10). and it would seem that the Principle of Competitive Exclusion (3. and which guarantees the long-term coexistence of both predators. 1996] 3. [Smith. Figure 3. the system becomes dx dt dy dt dz dt = = = x(1 − x) − a1 xy a2 xz − 1 + b1 y 1 + b2 z a1 xy − d1 y 1 + b1 y a2 xz − d2 z. PREDATION 61 Figure 3. Find all the equilibrium points for this system. that the HBC’s pelt records provide a reasonable proxy for the actual lynx and hare populations during this time.2. The data show that both populations oscillate with 1 The HBC still exists with department stores around Canada.2. . You will notice that no equilibrium points correspond to long-term coexistence of both predators. However. I recently purchased a pair of Levis at the The Bay in Banff.9: Local bifurcation diagram for three tropic level predation model [Kuznetsov and Rinaldi. After nondimensionalizing. The Hudson’s Bay company (HBC) 1 was the de facto government in northern parts of North America before European-based colonies and nation states existed.56) Problem 34. The HBC was the main purchaser of lynx and hare pelts.2.54) (3.. for some parameter regions.55) (3. Let’s assume. 1 + b1 z (3.1) would prevent their long-term coexistence. ♦ Does this violate the Principle of Competitive Exclusion? 3.b]. They now call themselves “The Bay” and are owned by a US-based private equity firm. Then. This suggests that fluctuations in the hare population may be due to factors other than predation. and Sinclair. while during low years of the hare cycle. The data during 1875-1904 and the model seem to suggest that the hares prey upon the lynx [Gilpin. But [Weinstein. and although the model shows it is possible. [Krebs. In those poor years. 2001] conducted manipulative studies in the southwesten . the numbers would drop to less than one hare per hectare. More recent studies have shown that one of the hare’s favorite foods. Boonstra. 1973].com/Pictures/Nature/Naturep/Animals/BigCats/ Canada%20Lynx] Can the oscillation of the lynx and hare populations be explained by the Lotka-Volterra predator-prey model? Figure 3. In Gilpin’s article. More recently. there is no evidence of such an epidemic. although a fur bearer-fur trapper cycle is possible. The arrows go in different directions.12). 1977] suggests that trappers could ill afford to bypass the poor hare years. Figure 3. the populations of hares crash. In winter. Boutin. the hare population did not peak before the lynx population. is always available in significant quantities during the winter. During a peak year in Alberta. Perhaps the hares were carriers of an infectious disease which they transmit to the lynx? Gilpin extended this model to test this hypothesis. but merely artifacts of the fur records. hares are absent. usually with a one or two year delay in the rise and fall of the predator population behind the prey population. Another hypothesis is that hare cycles may be caused primarily by winter food shortages. Also the direction of the arrows in the plane plot of the data and compare with the direction of the arrows along the closed orbits of the predator-prey model (see Figure 3. During two ten year cycles. Thus. [Keith. the need for food might drive the trappers into areas of low lynx densities. Consequently. To thicken the plot. 1963] observed that on Anticoti Island in Canada. once in the field. they may spend much of their time trying to catch the more profitable and rare lynx. Trappers.12 contains a time series plot and a “phase plane plot” of the data from 1875 to 1904.lindewurth. until the cycle starts over again.10: Canadian lynx chasing a snowshoe hare [from www. This could possibly explain the hare oscillations on lynx-free Anticosti Island. may sit out poor years with few hares. hares feed extensively on woody vegetation. the top predator. it is also possible that the cycles are not even real. MODELS OF COMMUNITIES period about ten years. but the population of introduced lynx cycles in sync with mainland populations. and predator abundance rather than food supply becomes the most important factor in regulating the hare population.62 CHAPTER 3. population estimates were 23 hares per hectare. he makes a passing comment that maybe the cycles are caused by an interaction between the Canadian fur trappers and the lynx and hare. As the population of hares increases. the grey willow. the supply of this food becomes depleted. and return to the woods only when the hare population has become abundant. the bacteria soon die.3 Population dynamics in a chemostat In a Petri dish or liquid culture.html] 3. POPULATION DYNAMICS IN A CHEMOSTAT 63 Yukon to determine the cause of the ten year hare cycle.3.duke.math. Hares do not usually die directly from starvation or malnutrition. and higher levels of stress. which causes the cycles. 3. Levin and Cornejo] for instructions to build your own low budget “people’s” chemostat and [Smith and Waltman.3. once the nutrient is depleted. A chemostat is a continuously stirred tank that receives fresh nutrients through an input tube and outputs residual nutrients and bacterial cells through an output tube.1 Single species growth model We first present a simple model of the growth of a single species of bacteria in a chemostat. Then the system of ODEs is . and these behavioral tradeoffs define the dynamics of the decline.” Figure 3. 1995] for a comprehensive discussion of the mathematical theory of population dynamics in a chemostat. and may predispose hares to predation. increased parasite loads. quite different from the population dynamics of bacteria growing in spatially structured habitats such as petri dishes or biofilms. One assumes that the continuous stirring produces a mass-action interaction between the bacteria and nutrient. which reduce fertility. Chemostats are widely used devices in microbiology labs to maintain bacterial populations over long periods of time. and that the bacteria growth depends on the nutient availability according to a Hollings Type II response. We caution the reader that the population dynamics of bacteria swimming in well-mixed liquid culture provided by chemostats is. But food quality and quantity affect body condition. See [Levin and Cornejo. The immediate cause of death is virtually always predation. The authors claim that “hares in peak and declining populations must trade off safety and food. The impact of food is felt largely in winter and it is mostly indirect.11: Canadian lynx and snowshoe hare pelt trading records of the Hudson’s Bay Company [from http://www. The input and output pump rates coincide so the volume remains constant. Their research shows that predation by other mammals is the main driver. a priori.3.edu/education/webfeatsII/Word2HTML/ HTML%20Sample/pred1. but food plays a non-trivial role. The result is a time lag in both the indirect effects and the direct effects of predation. |{z} N +γ | {z } output (3. 1973] dN dt αN β B N +γ | {z } = f N0 − f N − |{z} |{z} input output (3. In Problem (27) we will show that this system coincides with a type of predator-prey system.12: Canadian lynx and snowshoe hare pelt trading records of the Hudson’s Bay Company from 1875-1904 [Gilpin. and β measures the appetite of the bacteria.57) depletion during growth dB dt αN = B − fB .64 CHAPTER 3.58) growth where N is the nutrient density. One can nondimensionalize the system by measuring nutrient concentrations in terms of N0 and bacterial concentrations in terms of flow rate to obtain the equivalent system of ODEs dN dt dB dt cN B N +a = 1−N − = cN B − B. Adding (3. N0 = N (0). B is the density of bacteria.60) we obtain the single linear . f is the flow rate per unit volume.59) and (3.60) It is easy to see that all solutions are uniformly bounded. MODELS OF COMMUNITIES Figure 3.59) (3. N +a (3. it only exists for c > 1 and c > a + 1.g. 3.61) N (t) + B(t) = 1 + c1 exp(−t)..first quarter preserved . (c − a − 1)/(c − 1)). where every individual receives a fitness benefit. in lichens.2 Competiton: (write this) 3. interperate biologically. rock) and holds in water needed by the alga. a The eigenvalues of J( c−1 .4.3. (3. • Lichens are composed of a fungus and either an alga or a cyanobacterium (and sometimes both). increased survivorship from the other.4 Mutualism Mutualism is an symbiosis between individuals of two different species. e. c−a−1 c−1 ) are −1 and 0. one eigenvalue is always negative and the other is negative for a > c − 1 and positive for a < c − 1. The Jacobian matrices are c −1 − a+1 J(1.3 Predation: (write this) Lenski checked this model in his lab with bacteria and phage. The fungus attaches the whole lichen to its substrate (tree. This the extinction equilibrium point is locally attracting for a > c − 1 and a saddle for a < c − 1.3. and globally attracting competitive exclusion 3. It follows that extinction occurs if either c < 1 or c > 1 and a/(c − 1) > 1. It follows that if c > 1 and a/(c − 1) < 1. The first equilibrium point corresponds to bacterial extinction. The eigenvalues of J(1. then the second equilibrium point is locally attracting. and/or odors. 0) = c 0 −1 + a+1 and J a c−a−1 . 0) are the diagonal entires. 0) and (a/(c − 1).3. nectar. The existence of the second equilibrium point depends on the parameters. while the animal receives food. Interestingly. MUTUALISM 65 ODE d(N + B) = −(N + B) + 1. c−1 c−1 = −1 −1−a+c c−1 −1 0 . the fungus and alga are so closely intertwined that the combination is classified as species . problem . It follows that all orbits must approach an equilibrium point. The alga does photosynthesis and produces sugars to provide energy for both. The plant expends less energy on pollen production and instead produces showy flowers. Simple algebra yields two equilibrium points: (1. dt which can be easily solved using an integrating factor (3.62) Thus limt→∞ N (t) + B(t) = 1.like previous example of Rosengarden. Well known examples include: • Many plants depend on animals to spread their pollen. (3. cows.63) = r2 N2 (1 − N2 /K2 ) + α21 N1 N2 . (1 + α ¯ 12 )/(1 − α ¯ 12 α ¯ 21 )). MODELS OF COMMUNITIES • Tropical reef-building corals rely on a photosynthesizing algae (zooxanthellae) to provide energy while the coral with shelter and inorganic nutrients. while the microorganisms benefit from having a stable supply of nutrients in the host environment. these protozoans may account for 60% of the termite’s weight. u2 ) = u1 (1 − u1 + α ¯ 12 u2 ) (3. This leads to expulsion of the zooxanthellae by the coral. u02 ) plane where u01 > 0 and u02 > 0). the interior equilibrium point (u∗1 . u02 (t0 )) ∈ Q1 (the first quadrant in the (u01 . Show that (0. u∗2 ) = ((1 + α ¯ 21 )/(1 − α ¯ 12 α ¯ 21 ). where each species has logistic growth in the absence of the other: dN1 dt dN2 dt = r1 N1 (1 − N1 /K1 ) + α12 N1 N2 (3. the orbit of every interior initial condition converges to (u∗1 . green algae and fungi (lichens). 0). Sometimes this mutualism is upset by environmental stresses associated with unusually warm or cool water temperatures. pollination of plants by insects. The herbivorous animal benefits from access to a large source of fixed energy. When α ¯ 12 α ¯ 21 > 1. If not. We nondimensionalize and rewrite the system as du1 dτ du2 dτ = f (u1 . (1. u2 ) = ρu2 (1 − u2 + α ¯ 21 u1 ). and when it exists. a phenomenon known as ”bleaching. Such a system of ODEs is a called a monotone system.66) Simple algebra yields three equilibrium points: (0. the orbit must intersect the positive u01 axis or the . Proof. Theorem 4. a change in salinity. (3. u∗2 ) is an attracting node. (0. Another common mutualism occurs in the guts of animals (e. and coral and zooxanthellae algae. and the analysis of this system is similar to the competition model. 0). such as cellulases. 2. We consider the following simple two species model of mutualism with intraspecies competition.64) where the cooperation coefficients αij > 0. 1) and (1. Problem 35. the orbit of every interior initial condition diverges to infinity. which digest cellulose. termites) that eat plants.. (0. The following theorem is an immediate consequence of the fact that for sufficiently large τ . these animals live in a symbiosis with microorganisms.65) = g(u1 . a fourth equilibrium point.66 CHAPTER 3. they need protozoans in their gut to digest the cellulose.g. We first claim that if (u01 (τ0 ). and when α ¯ 12 α ¯ 21 < 1. u∗2 ). Although termites eat wood. many of which are not very effective at digesting cellulose. i 6= j measure the per capita effect of species j on the population growth of species i. Often. (Global behavior of solutions) 1. (u∗1 . When α ¯ 12 α ¯ 21 < 1. 0) is an unstable node. then (u01 (τ ). 1). or excessive exposure to sunlight or shading. which inhabit part of the gut and secrete enzymes. 0) are saddles with stable manifolds contained in the coordinate axes. u02 (τ )) ∈ Q1 for all τ ≥ τ0 . The time reversal of this system is a Lotka-Volterra competition system.” which may lead to death of the coral unless it can re-establish another algal mutualism. the solutions u1 (τ ) and u2 (τ ) are monotone functions of τ . amebiasis (Entamoeba histolytica). but they ultimately kill their host after they reproduce. 1995]. Approximately three billion people globally are infected with helminths. fungi. But an application of the chain rule shows that (d/dτ )u01 = (∂f /∂u1 )u01 + (∂f /∂u2 )u02 = (∂f /∂u2 )u02 > 0 at the point of intersection. especially wasps and flies. and helminths (worms). May. PARASITOIDISM 67 positive u02 axis. at least in the short term. Parasitoids are a desirable natural pest control. then (u01 (τ ). (3. and unsegmented flatworms. is introduced. an adult female parasitoid lays eggs in or on the larvae of the developing host (see Figure 3. 3. The majority of parasites have a very narrow ecological niche using only one or two host species. Mills and Getz. Either way. mostly of young children in sub-Saharan Africa.” Problem 36. The same argument shows that if (u01 (τ0 ). often a parasitoid. If u01 (t) hits the positive u02 axis. and it is estimated that 10% of insects are parasitoids. where one replaces Q1 and Q3 by Q2 and Q4 . The larvae become free-living insects and the host dies. 1996]. Examples include viruses. Human diseases caused by protozoa include malaria (Plasmodium spp. Parasitoids are similar to parasites. Thus unlike competition. u02 (0)) ∈ Qi for i = 2 or 4. u02 (τ )) enters Q1 or Q3 . α21 > 0.. When a natural enemy. and Tonkyn. bacteria. The eggs develop within the host while taking nutrients from the host. and sometimes to an “orgy of mutual benefaction. giardiasis (Giardia lamblia). or within. A similar proof can be used to prove Theorem (3d).67) = r2 N2 + α21 N1 N2 . a single host organism from which they derive metabolic advantage. u01 (τ ) and u02 (τ ) are ultimately monotone functions of τ and converge to a limit or to infinity. Parasitism is observed mostly in insects. Malaria is estimated to infect 300 − 500 million people every year and cause over a million deaths.5 Parasitoidism Parasites are organisms that spend a significant part of their life history attached to. and eventually emerge from the host. u02 (τ )) ∈ Qi for all τ ≥ 0 or (u01 (τ ). For insects.68) with α12 . african sleeping sickness (Trypanosoma brucei). 1973. where it remains for all larger τ . Anderson. Suppose that (u01 (0). u02 (τ )) ∈ Q3 for all τ ≥ τ0 . Hassell. 1981. References on modeling parasitoids include [Hassell and May. and the diseases they carry include Dracunculiasis (Dracunculus medinensis). so such an intersection can not occur. 1996]. Parasites usually do not kill their host.5. Then either (u01 (τ ). Parasites are classified according to whether or not they are transmitted by a vector (usually through the bite of an arthropod) [Despommier et al. protozoa. u02 (τ0 )) ∈ Q3 .). cooperation always leads to coexistence. and leishmaniasis (Leishmania donovani).13). Analyze the dynamics of the simpler model of mutualism without intraspecies competition dN1 dt dN2 dt = r1 N1 + α12 N1 N2 (3.3. filariasis (filarial nematodes). . onchocerciasis (Onchocerca volvulus). Few insect species escape parasitism. then clearly (d/dt)u01 (τ ) < 0 at the point of intersection. segmented flatworms. Viruses are called the ultimate parasites because they can not reproduce outside of their host cells. and trichuriasis (Trichuris trichiura). it is able to suppress the abundance of an insect pest to a level at which it no longer causes economic damage [Mills and Getz. There are three basic types of helminths: roundworms. The wasp’s larval stage is a small. In the absence of parasitoids. The eggs from additional parasitoidizations are not viable. Pn ) denote the fraction of non-parasitoidized hosts in generation n. and let c be the average number of viable eggs by a parasitoid on or in the host. MODELS OF COMMUNITIES Figure 3. The adult wasps continue to search for other hornworms to parasitize. 4. the host population will grow with exponential rate λ > 1. i. Then Hn+1 = λHn f (Hn . Then the mean number of encounters per host per generation is µ = Ne /Hn = aPn . The larvae continue to develop into adults and leave through an opening in the end of the cocoon.13: A tobacco hornworm parasitized by a parasitic wasp. Let f (Hn . (3. A host may be parasitized by many parasitoids. Once the hornworm wasp larvae fully develop they spin white oblong cocoons on the surface of the hornworm. Let Hn denote the population (or density) of hosts and Pn the population (or density) of parasitoids in generation n. Parasitoids can lay unlimited number of eggs and thus can parasitize any number of hosts. but only the first encounter matters. The hosts and parasitoids live one generation (thus the models are discrete dynamical systems) 2. The tiny black wasp lays its eggs in hornworms. white legless grub that develops inside the hornworm. The parasitoidized hosts give rise to the next generation of parasitoids. Parasitoids search randomly for hosts and that the number of encounters Ne per generaton between parasitoids and hosts is proportional to the product of their populations. 3. The basic models of parasitoidism make the following assumptions: 1. 4.. Ne = aPn Hn . . Hosts that have not been parasitoidized will give rise to the next generation of hosts. The number of parasitizations of a host per generation is given by a Poisson random variable with mean µ. This is well-mixing again. Pn ) (3. Pn )). Thus the probability that a host escapes parasitization during a generation is given by the zero term of the Poisson density function exp (−µ) for the number of mobile parasitoids encountering the spot where the host is located. 2. 3.70) Nicholson and Baily made the following assumptions: 1.68 CHAPTER 3.69) Pn+1 = cHn (1 − f (Hn .e. ¯ P¯ ) = (λ log λ/ac(λ − 1). ¯ P¯ ) = 1 + log λ/(λ − 1).14 for a typical solution. parasitoid levels become so low that the model predicts the eventual extinction of the parasitoid. Moreover. Liu. c(1 − exp (−aP )) acH The matrix J(0.71) Pn+1 = cHn (1 − exp (−aPn )).14: A solution of the Nicholson and Baily model compared with actual data [from home.net/˜sharov/PopEcol/lec10/gnichdyn. during some generations. (3. Li. and Malkin. log λ/a). See Figure 3. P ) = exp (−aP ) .comcast. det J(H. the sequence of angles corresponding to the iterates is strictly decreasing. An easy calculation shows ¯ P¯ ) = λ log λ/(λ − 1) and trace J(H. ¯ P¯ ) − 4 det J(H. 0) and (0. There exists experimental data to test the model and the model does a reasonable job of fitting some data sets for a dozen or so generations [Hassell and May. but it is unknown whether the sequences are unbounded. However. Computer simulations indicate that most forward orbits exhibit global oscillations. 1). host-parasitoid interactions to be stable over many generations. Little is also known about the existence of periodic points except for λ close to one. . and thus the fixed point (0. The authors [Hsu. This is hard to prove. P ) is an unstable spiral. 0) and (H. λ −aλH J(H.3. with eigenvectors (1. ¯ ¯ Jury criterion (see Section 5) to conclude that the fixed point (H.72) Recall that a fixed point is stable if and only if both eigenvalues of the Jacobian matrix have modulus less than one. PARASITOIDISM 69 The Nicholson-Baily model is the discrete dynamical system Hn+1 = λHn exp (−aPn ) (3. ¯ P¯ ) > 1 and trace2 J(H. Figure 3. and is unstable if at least one eigenvalue has modulus greater than one. 0) has eigenvalues 0 and λ > 1. after a few generations. But under natural conditions. the solution’s fluctuations in the populations of both the host and the parasitoid can get unrealistically large. 2003] prove that in polar coordinates centered at the non-zero fixed point. Use the Problem 37. and the existence of chaotic dynamics has not be ruled out. that det J(H.gif ].5. 0) is a degenerate saddle with the coordinate axes as stable and unstable manifolds. The Jacobian matrix is There are two fixed points: (0. Show that for all λ > 1. ¯ P¯ ) < 0. 1973]. P∗ ) that is an attracting spiral.76) (3. 1998. Costantino. is given by: c cel ea A(k) − L(k) bA(k) exp − V V P (k + 1) = (1 − µl )L(k) c pa A(k + 1) = exp − A(k) P (k) + (1 − µa )A(k). 1975] replaced the exponential growth of the hosts by a Ricker growth term to account for intraspecific competition of the hosts. and Costantino. Although there is no closed form expression for the interior fixed point.6 Flour beetle model and chaos The Beetle Team [Costantino. roughly the durations of the egg. 2001] conducted extensive experiments on laboratory populations of flour beetles. density dependent. simulations indicate that resonances cause the closed orbit to degenerate into periodic and quasi-periodic orbits. The time step in their discrete model is two weeks. [Beddington. larval. Cushing. The team also created a stochastic version of their model to account for the deviations of the data from the deterministic model and which allowed extremely rigorous statistical validation. The LPA model. Find all fixed points of this system. 2006. 2002. We have all eaten them. (3. Dennis. However. Eventually. The density-dependence is due to cannibalism of immobile stages with rates based on random Poisson encounters (as in the Nicholson-Bailey model).75) c −r + (c + r)H∗ Show that (H∗ . 1997. P∗ ) has simple form: 1 − rH∗ −H∗ J(H∗ . Cushing. The adults and larvae eat eggs and the adults eat pupae. Free. The authors modeled the beetle populations using a discrete.73) Pn+1 = cHn (1 − exp (−aPn )). P∗ ) can undergo a Hopf bifurcation. they observed that numbers of flour beetles underwent transitions from steady to periodic to quasiperiodic to chaos. Desharnais. and Lawton.74) Problem 38. After a Hopf bifurcation. three developmental stage structured model (a non-linear Leslie-type model). Cushing. The resulting discrete dynamical system is Hn+1 = Hn exp (r(1 − Hn /K) − aPn ) (3. and in 1997 they reported the first experimental evidence of chaos in an actual laboratory population.70 CHAPTER 3. Neubert and Kot.77) (3. MODELS OF COMMUNITIES Later authors searched for modifications of the basic model to stabilize the equilibrium point. 3. Cushing.78) . Desharnais. Beetles exhibit rather boring dynamics under natural conditions. Desharnais. when team members created extremely high adult mortality (about 96%) and varied the adult cannibalism rate of pupae (Cpa below). V L(k + 1) 2 Flour = beetles are one of the main pests of grains. 1992]. and pupal stages. and Dennis. 2 The large number of beetles raised in a tightly controlled environment (to minimize environmental stochasticity). the closed orbit loses its smoothness and breaks up into a strange attractor [Kon. and Henson. Dennis. (3. In the 1970s. the Jacobian matrix at (H∗ . Henson. and the relative ease of counting them resulted large and reliable data sets. These transitions correlated well with bifurcations in their mathematical models. (3. named for the stages (with egg and larvae grouped together). P∗ ) = . and show that in some parameter regions (which are actually biologically realistic) there is an interior fixed point (H∗ . Note that the nonlinear terms in (3. The exponential expressions represent the fractions of individuals surviving cannibalism in one unit of time. Problem 39. the LPA model reduces to a linear Leslie model. much of their analysis requires computer simulations. Figure 3. 3. and they effect a local and bifurcation analysis at the fixed points. various methods have been developed to estimate the Lyapunov exponents from the time series.7 Do real populations exhibit chaos? The hallmarks of chaotic dynamical systems are aperiodicity (orbits do not settle down to equilibrium points or limit cycles) along with exponential sensitivity to initial conditions. However.15 shows a bifurcation diagram exhibiting many local and global bifurcations [Cushing. The authors prove that the LPA system is dissipative (has a trapping region). and a useful definition of chaos is that the system has a positive Laypunov exponent on a set of positive volume in the phase space. cal /V . with cannibalism coefficients cel /V. P (k) is the population of non-feeding larvae. If one assumes that all cannibalism rates are zero. and the parameters µl and µa are the larval and adult probabilities of dying from causes other than cannibalism in one time unit.78) are all due to cannibalism. Analyze the log term behavior of the linear model. The sensitivity to initial conditions is frequently measured using Lyapunov exponents. The LPA model also has an invariant closed curve that bifurcates into a strange attractor [Ugarcovici and Weiss. population time series tend to . pupae and callow at time k. DO REAL POPULATIONS EXHIBIT CHAOS? 71 where L(k) is the population of feeding larvae at time k. Figure 3.76)-(3.3.7. Together. Assuming that a time series arises as an orbit of a low dimensional dynamical system. However.15: Bifurcation diagram of LPA model . they make the dynamics of deterministic systems appear random. where V denotes the habitat size. 2004]. 2002]. and A(k) is the population of sexually mature adults at time k The parameter b is the average number of larvae recruited per adult per unit time in the absence of canibalism. cpa /V. and Godfray. I am unaware of any other populations where chaotic behavior has been convincingly demonstrated. Ellner. the Beetle Team combined experiments.72 CHAPTER 3. [Tilman and Wedin. Turchin. Problem 40. MODELS OF COMMUNITIES be short and noisy. and mathematical models to make a compelling case for the existence of chaos in laboratory flour beetle populations. time-series analysis. Combining the same three methods. which makes the reliable estimation of Lyapunov exponents extremely difficult and unreliable [Hastings. As mentioned in the previous section. Why might chaotic behavior be common in real-life populations? Why might it be uncommon? . Hom. 1991] make a strong case for chaos in the dynamics of a perennial grass. 1993]. it is not necessary to kill the entire mosquito population . The letter S denotes the compartment of susceptible individuals. the duration of the epidemic. and are credited with formulating what is currently called the susceptible-Infected (SI) compartment model as a system of two ODEs. and R denotes the compartment of recovered or removed individuals which all have permanent immunity. I denotes the compartment of infected individuals. Ross also proved that bird malaria is transmitted by mosquitoes and he was awarded the second Nobel prize for Medicine. McKendrick and Kermack [Kermack and McKendrick. such as the initial conditions that lead to an epidemic. 1933] extended and elaborated these models. the number of cases at the peak of the epidemic. people believed that diseases such as cholera were caused by a miasma. We model both both types of disease. Before then.4). a noxious form of bad air. Infectious disease can be endemic (always present) or epidemic (significant outbreaks. He developed transmission models for malaria [Ross. The first two models are the susceptible-Infected-recovered (SIR) compartment model and the SIR model with demography (with non-disease related births and deaths). while for macroparasites. We remind the reader that formally. Most are extracellular. such ODE models 73 . the shape of the epidemic curve. 1927. Macroparasites like fungi and helminth worms. 1932. etc. In these section we introduce and analyze the most common epidemiological models of the transmission of directly transmitted infectious diseases. For microparasites it is usually sufficient to model just the hosts. This means that to eliminate malaria. one usually must model the life stages of the parasite along with the host. are usually multicellular and grow and multiply outside the host. About 60 years latter. We concentrate on diseases caused by microparasites. [Anderson. Ross devised ODE models to understand the mechanisms of how disease spread. The mass-action hypothesis requires members of the S compartment to mix homogeneously with members of the I compartment. Infection usually induces lifetime immunity or death. Kermack and McKendrick applied this model to describe the progression of the bubonic plague in India during 1905-1906 (see Figure 4.as was widely believed. Most are intracellular and elicit an immune response. These models are frequently called compartment models since the population is partitioned into compartments and they progress from the S compartment to the I compartment to the R compartment. 1991] contains a lucid history of transmission models. We study the key aspects of epidemics. in a background of no disease or endemic disease). usually single celled bacteria or viruses which multiply inside the host. 1910] and derived the first threshold theorem that identified a critical mosquito density required for malaria epidemics.Chapter 4 Modeling the Spread of Infectious Diseases Germ theory was developed in the 1850s. which is frequently.1 SIR models Basic SIR model We begin with the most basic SIR model.2) = νI. MODELING THE SPREAD OF INFECTIOUS DISEASES assume that the population in each compartment is either infinite or individuals are divisible.3) is equivalent to ds dt di dt dr dt ˆ = −βsi (4. In a study of the spread of gonorrhoea in the US. and r(t) = R(t)/N . If the probability of transmission during each contact (the transmissibility) is τ . The parameter ν denotes the removal rate. [1995]). But the mass-action assumption does not take this. [Hethcote and Yorke. each infected individual gives a constant percentage of susceptible individuals a big kiss.4) ˆ − νi = βsi (4. Some authors prefer to present this system using densities instead of numbers. 4. host populations. Then κI/N of these contacts are with infected individuals. into account. Removed individuals are neither infected nor infectable.1 4. and R(t) the number of removed individuals at time t.1). (4.1).5) = νi. We denote by S(t) the number of susceptible individuals at time t.6) . Having twice as many susceptable individuals results in twice as many new infections. Thus 1/ν is the period of infectivity. heterogeneities in pathogens. One of my students explained this form of mixing as every unit of time.1) = βSI − νI (4.2). and the interactions between them enormously affect the dynamics of infection. Suppose that each susceptible has κ adequate contacts per unit time. (4. Defining the densities s(t) = S(t)/N. (4. Infected individuals recover and move from the infected class to the recovered class at rate ν. In reality. called the McKendrick-Kermack model (see Figure 4. I(t) the number of infected individuals at time t. then the expected number of adequate contacts per susceptible per unit time is κτ I/N . 1984a] showed that 60% of all infections were caused by only 2% of the entire population. i(t) = I(t)/N . then s(t) + i(t) + r(t) = 1 and (4. Thus β = κτ /N . The parameter β is called the transmission rate of the infectious disease and the time dependent quantity F = βI is called the force of infection or transmission rate. where an adequate contact is a contact that is sufficient for transmission. although some authors argue mistakenly (Diekmann et al.3) The term βSI for the number of new infectives corresponds to mass-action mixing of the infected and susceptible classes. The system of ODEs reflecting mass-action mixing can be written as dS dt dI dt dR dt = −βSI (4.74 CHAPTER 4. (4.1. or many other heterogeneities. 4. 2.1.g.. Note also that i. The usual “Poincar´e” method to analyze systems of ODEs is to first find and classify the equilibria. In particular. 7. no natural births or natural deaths occur. and smooth dependence of solutions on parameters apply. Problem 42. uniqueness. Thus all the usual theorems from ODEs about local existence. the same level of infectiousness once infected. SIR MODELS 75 where βˆ = κτ . Also. hence smooth and analytic. This is a polynomial system of ODEs. Recovered individuals have permanent immunity. if the model is taken literally.. you are much more likely to get the seasonal flu during the winter than the summer. For flu it may be standing near an infected individual that just coughed.e. 3. the following problem shows that there are infinitely many equilibria and they are all degenerate. s. and r are real valued functions. . e. S βS I ν R Figure 4. an individual becomes infectious as soon as they become infected. Disease transmission is direct with no vectors. R) are degenerate equilibria. the transmission coefficient of many infectious diseases are seasonally dependent. The disease is short lived. Observe that in this form the system only involves proportions and is independent of the total population size N . How is β related to the age of onset of the disease? Implicit in the SIR model are the following additional assumptions: 1. The infection has zero latent period. Thus most of the usual dynamical systems methods do not apply.. The recent paper [Mossong et al. and they are the only individuals with any form of immunity. public health interventions during an outbreak will significantly change one or both of these parameters. i. Problem 41. (Degenerate equilibria) Show that all points of the form (S. everybody. 6. All individuals have the same immune system. the population must be infinite. For HIV. and thus.4. including older people who rarely leave their house and children who attend large schools. and the same period of infectiousness.1: SIR component model The form of a contact depends on the disease. 0. The parameters β and ν remain constant. 5. a contact must be a sexual contact or a blood transfusion. However.. 2008] contains the state-of-the-art understanding about the rates of contacts between humans. Frequently. Thus everybody has the same susceptibility to infection. has the same probability of contracting the disease. No age-structure. the term epidemic seems very poorly defined. If R0 < 1. See the reivew [Heffernan et al. 1989]). and we can focus on S and I. R∞ Problem 44. Theorem 5. There are many references containing proofs of the following easy facts (see [Anderson and May. I 0 (0) > 0.76 CHAPTER 4. Since S 0 (t) ≤ 0 and R0 (t) ≥ 0.2) and Property 1 imply I 0 = (βS − ν)I ≤ (βS(0) − ν)I < 0 for R0 < 1.. MODELING THE SPREAD OF INFECTIOUS DISEASES Adding equations (4. Property 1.e. i.3) yields S 0 + I 0 + R0 = 0. However. Proof. Sometimes it refers the situation when at some later time the number of infectives is increasing. Equation (4. S(t) is an decreasing function and R(t) is an increasing function for t ≥ 0. R(0) ≥ 0. . together with Property 2 imply the second statement. then I(t) decreases to zero as t → ∞. then I(t) decreases monotonically to zero as t → ∞ (at an exponential rate). and thus I(∞) = N − S(∞) − R(∞) exist (are finite). Thus 0 ≤ S(t) ≤ S(0) ≤ N and R(0) ≤ R(t) ≤ N .2) implies I 0 (0) = (βS(0) − ν)I(0) > 0 for R0 > 1. Integrate both sides and exponentiate.9) has several epidemics. This gives an alternate proof that I(∞) = 0. Equation (4. i. However. (5) implies that for t sufficiently large. It usually refers to the situation when the number of infectives immediately increases from the initial value. we now show that R0 is a type of threshold value which determines whether an infectious disease will quickly die out or whether it will become epidemic.1). then solutions S(t). and this implies that R(∞) = 0. In the infectious disease modeling literature. an easy qualitative analysis yields the salient features of the solutions. (4.. then I(t) starts increasing. R(t) ≥ 0 for all t ≥ 0. we must define the basic reproductive number R0 = S(0)β/ν. 2005] for many useful perspectives on R0 . (Invariance of positivity) Prove that if S(0). 3. These observations. Equation (4. Hence S(∞). If not.1) is equivalent to d log S/dt = βI. Thus I(t) is increasing at t = 0. R(∞). and then decreases to zero as t → ∞.e. Show that 0 I(t)dt < ∞. To state the main theorem.. Equation (4. R0 (t) > νI(∞)/2. Endemic disease that waxes and wanes can be called epidemic under the latter definition. reaches its maximum. This observation together with Property 2 proves the first statement. it follows that 0 ≤ I(t) ≤ N . i. 2. all initial conditions lead to I(∞) = 0. since I(t) = N − S(t) − R(t). We discuss the biological significance of the dimensionless R0 in the next subsection.2) also implies that I(t) has only one non-zero critical point.e. The disease always dies out. 1991.2) implies I 0 = (βS − ν)I ≤ (βS(0) − ν)I ≤ 0 for R0 = 1. An infectious disease is called endemic if I(∞) > 0.. Thus R = N −S −I. Theorem (5) implies that the SIR model has no endemic disease and the number of infectives does not oscillate. (Dynamics of SIR model) 1. I 0 (t0 ) > 0. Also. This nonlinear system has no closed form solution. Property 2. Hethcote. Hint: One easy proof uses that equation (4. If R0 > 1. If R0 = 1. Problem 43.2). I(t). I(0). and thus the total population size N (t) = S(t)+I(t)+R(t) is constant over time. This observation together with Property 2 proves the third statement. The infectious disease illustrated in Figure (4. a contradiction. and (4. (4.1) by (4.66 and ν = 0.1) by (4.3) which yields the ODE dS −βIS −βS = = . Combining this fact with Property 3 yields that maximum number of infectives Imax satisfies Imax = −ν/β + I(0) + S(0) − ν/β log S(0) + ν/β log ν/β. for every t ≥ 0. Property 5. To obtain this we divide (4. since for S 6= 0 it can be rewritten as 1 −β dS = dR.2(b).7) dI βSI − νI This ODE is separable. It follows from (4. This property allows us to make parametric plots of the number of susceptibles verses the number of infectives for all time. −βS (4. (4.4.44 β = 1.1. To obtain this we divide (4. The system of ODEs has a first integral of motion.9) How do epidemics end? Do they end because there are no longer susceptibles in the population? If so. Property 4. I(t) + S(t) − ν/β log S(t) = I(0) + S(0) − ν/β log S(0).2) that the maximum number of infectives occurs when S = ν/β. We now show this is not true. then it would be the case that S(∞) = 0.10) dR νI ν This ODE is separable.2: (a) Solutions of SIR system of ODEs with β = 1.2) which yields the ODE dS −βSI = .11) . In other words.44 (b) Plots of I verses S with Property 3. since for I 6= 0 it can be rewritten as βS − ν dS = dI. We now show that S(∞) ≥ S(0) exp(−βN/ν) > 0. S ν (4. See Figure 4. (4. SIR MODELS 77 Figure 4.66 and ν = 0.8) Integrating both sides yields −I − S + ν/β log S = C. 2 0. the number of susceptibles decreases and so the rate at which new infections arise decreases. We saw in Theorem 5 that an epidemic occurs if R0 > 1 and the disease quickly dies out if R0 < 1. ν (4.0 3. S drops below ν/β. Thus for a given infectious disease with fixed β and ν.5 Figure 4.0 0. The epidemic ends because of the lack of new infectives and not because of the lack of susceptibles. Letting x = S(∞)/N be the proportion of susceptibles at the end of the epidemic. It follows that as an epidemic proceeds. then equation (4. Equation (4.3.0 4. SH¥LN verses R0 1.12) and since R(t) ≤ N .5 4. R).5 3.0 1. and the rate at which individuals recover exceeds the rate at which new infections occur.5 2. Thus. (4. there is a minimum susceptible population size necessary for a epidemic to occur. then log x = R0 (x − 1). and the duration of infection. Eventually. S + R = N ? 4.1. the property follows. The number R0 number plays several other key roles in modeling.8 0. We plot this relation in Figure 4.78 CHAPTER 4. the average number of contacts an infective has with susceptibles per unit time. (4.14) which is the product of the transmissibility.6 0. Find the basin of attraction for each equilibrium point (S.0 1. 0.3: Plot of S(∞)/N verses R0 .0 2. This threshold value is called the critical community size. . Problem 45.4 0. predicting. and controlling the spread of an infectious disease. I(t) starts decreasing. MODELING THE SPREAD OF INFECTIOUS DISEASES Integrating both sides yields S(t) = S(0) exp(−βR(t)/ν).12) immediately yields a transcendental equation for S(∞) since S(∞) = S(0) exp(−βR(∞)/ν) = S(0) exp(−β(N − S(∞))/ν).13) has a simple form in terms of R0 . Thus R0 is the number of new infections caused by each infective per unit time.13) If we assume that S(0) = N .2 The basic reproductive rate R0 We defined the basic reproductive rate as β R0 = S(0) = τ ν S(0) κ N 1 . The definition of R0 provides strategies to prevent an epidemic by reducing it to less than one. in 1967 the World Health Organization (WHO) mounted a successful worldwide smallpox . and not everybody can take the vaccine. while during the same period HIV/AIDS among homosexual men in England had R0 ≈ 2 or 5.. This is the theoretical underpinning of public health policy. Thus to prevent an outbreak of smallpox. but has a much shorter infectiousness period. may suffer serious morbidity or mortality from the vaccine. they have roughly the same R0 . One can reduce the contact rate α by improving sanitation. To prevent an epidemic.1 provides a rough estimate for R0 for well-known infectious diseases. 1991] We note that an R0 of 38 has been estimated for an outbreak of foot and mouth disease that occurred during unusual environmental conditions [Haydon et al. vaccinations. Some individuals. Different outbreaks of the same disease can have different R0 ’s. Let ρ denote the fraction of the susceptible class that gets vaccinated (we assume that the vaccine is 100% effective).g. SIR MODELS 79 Table 4.1: Representative values of R0 [Anderson and May. Even if a vaccine is 100% effective. the 1968-69 Hong Kong flu caused 34. ♦ Which parameter in R0 is probably responsible for this? During 1981-85. and herd immunity Vaccinating susceptible individuals removes them from the susceptible class. R0 . along with the belief that humans are the only natural hosts of the smallpox virus. the 1957-58 Asian flu caused about 70. we require R0 < 1. One can reduce S(0) with vaccines (and culling for animal epidemics). the 1918-19 Spanish flu caused about 500.1. and the 2003 SARS epidemic caused 800 deaths worldwide. To prevent an epidemic. a 1962 measles outbreak in Senegal had R0 ≈ 1 or 2.000 deaths. This will occur when ρ ≥ ρc = 1 − 1/R0 . and the phenomenon is called herd immunity. HIV/AIDS among female prostitutes in Nairobi had R0 ≈ 11 or 12.4. Malaria Measles Pertussis Chicken pox Polio Smallpox Dengue HIV/AIDS SARS Flu Ebola >100 12-18 12-17 9 5-7 3-5 3 2-5 2-5 2-3 1-2 Table 4. and one can reduce the infectious period 1/ν with antibiotics or antivirals. vaccinating an entire population is very expensive. while a 1955-58 measles outbreak in the US had R0 ≈ 5 or 6. or by isolation or quarantine. e. Although HIV/AIDS and smallpox are very different diseases. We ask the question..000 deaths in the US. 1997]. we require that (1 − ρ)S(0)β/ν < 1. Smallpox is much more transmissible than HIV/AIDS. assuming R0 = 5. Finally although they had roughly the same R0 . those with compromised immune systems or severe allergies. vector control. it is only necessary to vaccinate 80% of the population. requesting people to stay home. Based in part on this finding. For example.000 deaths. One can reduce the transmissibility τ by using protective barriers (face masks for SARS and barrier contraceptives for HIV/AIDs). can an epidemic of an infectious disease be prevented by vaccinating only a fraction of the susceptible class? It easily follows from our analysis of the SIR model that the answer is yes. • The initial growth rate of an epidemic is (R0 − 1)ν. experts believe that it will be impossible to irradiate HIV/ AIDS from the human population. R0 and initial exponential growth of infectives It immediately follows from equation (4. Smallpox is one of only a very small number of human infectious diseases that has been completely irradiated around the world. Integrating we obtain that for small t the number of infectives I(t) ≈ I(0) exp (ν(R0 − 1)t). 1. 1. 5. Most vaccines are not 100% effective. 1993]. 0. 0. To briefly recap.e. it would be necessary to vaccinate 99% of the population to prevent outbreaks.) R0 and pathogen evolution Pathogens are thought to evolve to increase their R0 (fitness). Thus the initial exponential growth rate of infectives can be expressed in terms of R0 and the mean duration of infection ν. 0. 1. patients produce about 109 virons per day and there is typically one mutation in each HIV nucleotide per day. Some infectious diseases that may be relatively minor for children may result in serious complications for adults. 7. 3.1) that I 0 (0) = I(0)ν(R0 − 1). The HIV retrovirus genome gets permanently inserted into the human genome. One of the difficulties with a possible malaria vaccine is that. . 0. 1. (I thank Igor Belych for sharing this problem. There is a current worldwide campaign to eliminate polio from the planet. with R0 > 100. The time (in days) between newly reported cases is given in the following sequence: 13. Overall. 1. An outbreak of smallpox occurred in the town of Abakaliki in southeastern Nigeria in 1967. 2. 1971] techniques.80 CHAPTER 4. Assume that R0 of measles is 15 and that the measles vaccine is 95% effective. How does this influence the above calculation? Problem 47. What can you conclude about the prospects of eradicating the measles virus with a single dose of the vaccine? Problem 48. This equality approximately holds for small t. MODELING THE SPREAD OF INFECTIOUS DISEASES irradiation program. 0. Since there are about 1016 HIV genomes in the world today. vaccination against a disease can be completely effective without making everyone immune. I 0 (t) ≈ I(0)ν(R0 − 1). 1. and Skalka. 5. Enquist. Prove that vaccinations raise the average age of first acquiring a disease. Even though the R0 of HIV/AIDS is not greater than that for smallpox.. 2. 0. 5. 0. Racaniello. Let denote the effectiveness of a vaccine. i. 3. 4. or ever will have [Flint. R0 plays the following four key epidemiological roles: • R0 is a threshold value: an epidemic will occur if R0 > 1. By herd immunity. Data Problem 1. 2. 3. 5. This also shows that R0 is a measure of the fitness of the pathogen [May. 4. Also. Fit this data using an SIR model using both the least squares and maximum likelihood [Bailey and Thomas. then the pathogen will also evolve to decrease their virulence.]. thus to become more easily transmitted (larger β) and make individuals sick longer (smaller ν). there were 30 cases of infection in a population of 120 individuals. it is highly probable that HIV genomes exist that are resistant to every antiviral drugs that we have now. Flint et al. 2. Problem (??) shows that if the infection causes mortality at rate δ (virulence). 5. People living there belong to a religious group that is quite isolated and declines vaccination. Problem 46. Are your estimated values of β and ν realistic? Estimate R0 . .4. dt (4.1) yields the separable ODE dR/R = −(1/ρ)dS/S.16) and thus Equation (4.19) to obtain dR γα2 N 2 αγt 2 ≈ sech −φ .21) Kermack and McKendrick confronted this model with data from a plague outbreak in Bombay during 1905 − 6. • The percentage of susceptibles S(∞)/N at the end of an epidemic is the root of the transcendental equation log x = R0 (x − 1).22) dt 2S(0)R02 2 They combined parameters and viewed the formula as involving three parameters.15) dR = νI = ν(N − R − S) = ν(N − R − S(0) exp(−R/ν)). SIR MODELS 81 • The critical vaccination threshold for herd immunity is 1 − 1/R0 . Integrating we obtain S(t) = S(0) exp(−R(t)/ν) (4. 4. (4. ν 2ν (4.4(a)). Since almost everyone who became infected died.1.17) (4.18) The right hand side is a quadratic polynomial in R and the approximate ODE is separable with closed form solution N2 ανt S(0)R0 R(t) ≈ − 1 + α tanh −φ . (Kermack-McKendrick’s “small epidemic analysis”) Dividing Equation (4. They differentiated (4.3 Examples Example 3.1. which they (somehow) estimated by fitting the model to the data (see Figure 4.3) by Equation (4.3) becomes Kermack and McKendrick assume that R(t) << ν.19) S(0)R02 N 2 where α= !1/2 2 S(0)R0 2S(0)(N − S(0))R02 −1 + N N2 and φ = tanh−1 1 α S(0)R0 −1 . N (4. the removal rate R0 was the same as the death rate or the number of deaths per week. Expanding the exponential term in a Taylor series yields the approximate ODE dR dt = ≈ ν(N − R − S(0) exp(−R/ν)) R R2 ν N − R − S(0) 1 − − 2 . (4.20) (4. particularly in young animals.82 CHAPTER 4. [Murray. Central and South America. and ν = .kvl.5 shows the number of infected swine predicted by a simple SIR model during an outbreak in the Netherlands during 1997-1998. The disease is endemic in much of Asia.4: (a) Predictons from Kermack and McKendrick SIR model for weekly death rate from Bombay plague outbreak.dk/ hoehle/teaching/SundSmit/talk.44 per day. 2003] Problem 49. Could this outbreak have occurred if the school contained 100 susceptible students? Example 4. rash. Figure 4. It was believed to have been eradicated in the United Kingdom forty years ago.4(b)). Swine fever usually causes death within 15 days.5: SIR model prediction of classical swine fever virus outbreak in the Netherlands [from www. The presence (the ”dengue triad”) of fever. MODELING THE SPREAD OF INFECTIOUS DISEASES Figure 4. (b) Predictons from SIR model for weekly number of boys infected during an influenza outbreak in a boarding school in England.dina. β = .00218 per day. and parts of Europe and Africa. Assume that N = 763.pdf] Example 5. (Dengue Fever outbreaks in the Americas) Dengue fever is a disease caused by a family of viruses that are transmitted by mosquitoes. but an outbreak occurred in East Anglia in 2000. Figure 4. (Influenza outbreak in a boarding school in England) Murray describes a simple SIR model of an influenza outbreak in a boarding school in England during 1978 (see Figure 4. and headache (and other . Compute R0 . (Classical swine fever outbreak in the Netherlands) Classical swine fever is a highly contagious viral disease of pigs and wild boar. 1. We remind the reader not to be overly impressed with the data fitting capabilities of the SIR model in these examples. The displayed data was used to parametrize the model. SIR MODELS 83 pains) is particularly characteristic of dengue. 2009b].6: SIR model predictions of Dengue fever outbreaks in (a) Venezuela (b) Santiago de Cuba (1997) [C´ aceres. Figure 4.6 shows the number of infected individuals predicted by a simple SIR model during Dengue outbreaks in Venezuela and Cuba Figure 4.The disease commonly frequently found in the tropics and is endemic in about 100 countries. Dengue has four serotypes. Recall the admonishment in Section 1.4.7: SIR model prediction of 2001 Dengue fever outbreaks in Havana [C´aceres. C´ aceres] .7 illustrates an attempt to model a 2001 Dengue outbreak in Havana.1. C´aceres] for a discussion. and Weiss.6 about this type of model validation: the parametrized model may be just regenerating the data which was used to parametrize it. Figure 4. with interesting infection dynamics between them. See also [Pollicott. see [C´aceres. Wang. Notice the terrible fit to the data. Figure 4. There are approximately 50 million cases of Dengue fever virus worldwide. . C´aceres] . Thus system of ODEs is again redundant in the sense that R is determined by S and I. This strong assumption keeps the population size constant at N . Thus the natural definition of the basic reproductive number with all individuals susceptible is R0 = N β/(µ + ν). which accounts for the fact that some infected individuals die a natural death before recovering. (Dynamics of SIR model with vital rates) .23) = βSI − νI − µI (4. Nμ S βS μ I ν R μ μ Figure 4. I(t)) do not leave the square formed by the axes and the lines S = N and I = N . We assume that births occur at rate N µ and natural deaths in all three compartments occur at rate µ.8: SIR model with vital rates The extended system of ODEs is dS dt dI dt dR dt = N µ − µS − βSI (4.1. The mean duration of infection is 1/(µ + ν). 4. We illustrate this extended model in Figure 4. (4. I(0).4 MODELING THE SPREAD OF INFECTIOUS DISEASES SIR model with vital rates We now add vital rates to the standard SIR model to account for births and natural deaths. Problem 50. and (4. Actually.25) Adding equations (4.25) yields S 0 + I 0 + R0 = 0. Theorem 6. Newborns enter the susceptible class with no immunity.23). I(t). and thus the total population size N (t) = S(t) + I(t) + R(t) is constant over time. R(0) ≥ 0. The following theorem shows that R0 is a threshold which separates I(∞) = 0 from endemic disease I(∞) > 0.8. solutions (S(t).24) = νI − µR. then the solutions S(t). R(t) ≥ 0 for all t ≥ 0. (4.24).84 CHAPTER 4. since S ≤ N and I ≤ N . Recall that the latter is not possible in the out vital rates. Prove that if S(0). which reduces their mean duration of infection from 1/ν. since the function L(S.29) Problem 52. and thus (N. (ν/β)(R0 − 1)). Rohani.27) The eigenvalues of the first matrix are −µ < 0 and (R0 − 1)(µ + ν). where a1 = µR0 and a2 = µ(µ + ν)(R0 − 1). Proof.9). 0. 0) is globally attracting. there are two equilibrium points (N. the equilibrium point (N. Thus I(t) decreases to zero as t → ∞. Thus I(t) decreases to zero as t → ∞. I ∗ . a2 > 0. and thus (S ∗ . 0. The period of oscillation is 2π/ r µ µ+ν (R0 − 1) − R02 . There are no limit cycles since any limit cycle must contain (N. R∗ ) = ((µ + ν)/β. If R0 < 1. 0. the equilibrium point (N. SIR MODELS 85 1. the number of infectives I(t) approaches (µ/β)(R0 − 1) as t → ∞. If R0 = 1. 0. Problem 51. (4. I ∗ ) is ∆ = µ2 R02 − 4(R0 − 1)µ(µ + ν). [Earn. If R0 > 1. λ2 = −µR0 /2 ± i µ2 µ+ν µ (R0 − 1) − R0 . Bolker. 0. . The first is a saddle and the second is attracting.26) (4. R∗ ). 0) is attracting. Thus if I(0) > 0. 0) and thus attain negative values. The condition to have oscillatory solutions is R02 < 4(µ + ν)/µ.1.4. Thus it follows from the Poincar´e-Bendixson theorem that (N. I) = S − S ∗ log S + I − I ∗ log I is a global Lyapunov function (♦ verify this. I ∗ ) = = −µ −βN 0 βN − (µ + ν) −µR0 −µ − ν µ(R0 − 1) 0 and . 0) is globally attracting. There are no heteroclinic or homoclinic connections since (N. it follows from the Routh-Hurwitz theorem that both eigenvalues have negative real parts provided R0 > 1. 2 µ (4. 0) J(S ∗ . what is the ρ threshold vaccination rate to prevent an outbreak of the disease? Oscillations The discriminant of J(S ∗ . and Grenfell. 0) is a saddle for R0 > 1. Assuming perfect and life-long immunity. Suppose that ρ percent of the newborns get vaccinated. the equilibrium point (N. From above. 0) is attracting for R0 < 1. 0. and solutions will oscillate to the endemic value (see Figure 4. We first compute the two Jacobian matrices J(N. I ∗ . I ∗ . Since a1 . Model measles using an SIR model with vital rates and compute the period of the oscillation.). If ∆ < 0 then the attracting equilibrium point is a spiral. 2000] contains realistic epidemiological parameters for measles. 2. (4. 0. (µ/β)(R0 − 1). 3. The characteristic equation for the second matrix is x2 + a1 x + a2 = 0. 0) is globally attracting. 0. All solutions with I(0) > 0 approach (S ∗ .28) R0 − 1 q 2 and λ1 . 0) and J(S ∗ . I ∗ ): J(N. R∗ ) is an attracting equilibrium point. 0) and (S ∗ . It ) with discrete state space Z+ ×Z+ [Ball and Clancy.86 CHAPTER 4. and assume that recovered individuals loose their immunity at rate η The system of ODEs is given by dS dt dI dt dR dt = N µ − µS − βSI (4. The basic stochastic SIR model defines a natural continuous time Markov process (St .1. 4.35) Prove an analog of Theorem 6. (4. (SIR model with vital rates and temporary immunity) Consider the SIR model with vital rates. (4.5 Stochastic SIR model Both deterministic and stochastic models are used for epidemiological modelling.9: Oscillations in number of infectives Problem 53. As we have seen. while deterministic models only deal with densities (see (4. MODELING THE SPREAD OF INFECTIOUS DISEASES Infected 6 5 4 3 2 1 0 10 20 30 40 50 Figure 4.34) (4. 1979]. Stochastic models only deal with a finite populations. I +1) and occurs with transition rate βSI and removal corresponds to the transition (S. A simple transmission model for such a disease which includes vital rates and disease caused mortality is given by dS dt dI dt = N µ − µS − βSI (4.31) = νI − µR − ηR. .5). I) → (S−1.33) = βSI − µI − δI. I − 1) and occurs with transition rate νI. Becker. deterministic models often exhibit threshold behaviour. (SI with vital rates) Consider an infectious disease for which there is no recovery.4).6)). such as untreated TB or many plant diseases. I) → (S. (4. Problem 54. 1993. They are also mathematically easier to analyze than stochastic models. Infection corresponds to the transition (S.32) Prove an analog of Theorem 6. (4.30) = βSI − νI − µI (4. 1971] and Barbour [Barbour.i (t) = P (S(t) = s. although it is not easy too obtain a closed form solution.1). It follows from the basic theory of Markov chains that in the basic stochastic SIR model the infectivity period is an exponentially distributed random variable D with parameter γ. I(0))]. . otherwise the contact has no efect. 1974] (see [Becker. SIR MODELS 87 We let ps. because nothing will happen during most time steps. How does one simulate this stochastic model on a computer? I recommend the short description [Bolker. Kurtz [Kurtz.3) can be recovered from this system by dividing the original state variables by the population size and letting taking the limit as the population size becomes infinite. But this is extremely inefficient. 1979] for a nice survey) made the approximation and convergence rigorous.4.10: Epidemic in stochastic SIR model . This algorithm generates an exponentially distributed random number that determines the time to the next event (infection or recovery) and then selects among the two transitions with probabilities proportional to their individual rates.10 shows a Gillespie simulation of the stochastic SIR model with β = . The Gillespie algorithm [Gillespie. During their infectious period. but their hypotheses require that the sizes of the susceptible and infected subpopulations be large. as well requiring a large number of contacts between infectives and susceptibles. Then the Kolmogorov forward equations for p are dps.i (t) = dt (4.36) KEEP GOING Formally the basic deterministic SIR system (4. an infected individual has contacts with other individuals at times determined by a Poisson process with rate β. Always extinction. It is easy to write down a one step difference equation for P [T = t | (S(0). 1976] is much faster. The obvious idea is to use very small time steps to be sure that only a single infection or recovery occurs during the step. Each contact is with an individual chosen uniformly at random from the N available (allowing self-contacts). Bolker]. no thresholds Figure 4. (4.2). it becomes an Infective. One says that the epidemic lasts for time T = inf{t ≥ 0 : I(t) = 0} with final outbreak size F = N − I(0) − S(T ). If the individual contacted is Susceptible. Bolker] which includes computer code in R.1 and γ = 1 [Bolker.1. These assumptions often do not hold in applications to disease transmission in households or for a disease imported by a small number of individuals into a community previously free from the disease. (4. I(t) = i). 1970. Figure 4. 1/R0 is the fraction of susceptible individiuals at equilibrium.37) λk+1 = βSk Ikα (4. In principle. The system of ODEs is . One can relate R0 to the average time Ts spent as a susceptible before becoming infected (at equilibrium). See [??. 2. In other words. Suppose it is known that I(0) ≈ 0 (and thus S(0) ≈ N ). I 0 (t) ≈ I(t)ν(R0 − 1). We present four indirect methods to estimate R0 from epidemiological and seroprevalence data [Dietz. some investigators use the Time Series SIR model. It immediately follows from (4. denoted SIS and due to Ross. Samore.88 CHAPTER 4. Tan. This model assumes no births or natural deaths. and so R0 = 1 + 1/(µTs ).. The authors in [Lipsitch. et al. bj] for a historical discussion along with an applications to modeling UK measles. Gonorrhea is an example of a non-fatal disease that does not confer protective immunity. 4.13) implies that R0 ≈ log S(∞) − log S(0) S0 . Robins. The parameter α has no biological significance but helps with the data fitting.39) where X is a binomial or negative geometric random variable and λk+1 is the expected number of infectives in the k + 1 generation. R0 is determined from the knowledge of the effective contact rate and the removal rate.40) The initial and final numbers of susceptibles S(0) and S(∞) can sometimes be estimated with seroprevalence data. has all infected individuals recover and immediately reenter the susceptible class. 4. Thus Ts = 1/(βI ∗ ) = 1/(µ(R0 − 1))). in an SIR model.Then equation (4. 4. In other words. and thus I(t) ≈ I(0) exp (ν(R0 − 1)t). S(∞) − S(0) (4. 4.38) Ik+1 = X(λk+1 ) or X(λk+1 ). 1993]. 1. (4. Cooper. The latter is difficult to estimate directly.1. The number of susceptibles can sometimes be estimated with seroprevalence data.1. 2003] estimate R0 for SARS. Gopalakrishna.7 Estimating R0 from epidemiological data Except for sexually transmitted diseases with have clearly defined contacts.2 SIS model A two compartment model. Ma. Cohen. Thus the initial exponential growth rate of infectives is ν(R0 − 1). This discrete phenomenological model is described by Sk+1 = Sk + Bk − Ik+1 (4.2) and the definition of R0 that I 0 (0) = I(0)ν(R0 − 1). it is usually impossible to directly estimate the number of secondary cases generated for each primary case. James. 3. R0 − 1 is the ratio of the average lifetime and the average age when the infection is acquired. For small time. From knowledge of the number of susceptibles S ∗ in the endemic state it easily follows that R0 = N/S ∗ . Chew. No immunity is conferred.6 MODELING THE SPREAD OF INFECTIOUS DISEASES Time Series Stochastic SIR model If fitting infection data well is the primary objective of a transmission model. (4. then I(t) = I(0). (4. the total population size is not constant. The system of ODEs is given by dS dt dI dt = −βSI + νI (4.43) dt βN − ν If one defines the basic reproductive ratio R0 = βN/ν. 3. (SIS model with disease mortality) Consider the SIS model and assume that some infected individuals are killed by the disease. N .44) = βSI − νI − µI.45) Problem 56. (Dynamics of SIS model without vital rates) 1.41) = βSI − νI.2. If R0 < 1. then the number of infectives I(t) decreases monotonically to zero as t → ∞.42) yields the logistic ODE βI dI = (βN − ν)I 1 − . Theorem 7. and thus the total population size is constant. The system of ODEs is given by dS dt dI dt = N µ − βSI + νI − µS (4.49) . but there are no births and natural deaths. As a consequence. (4. The system of ODEs is given by dS dt dI dt = N µ − βSI + νI − µS (4.47) where δ is the death rate of infected individuals (often called the virulence of the disease). SIS MODEL 89 dS dt dI dt = −βSI + νI (4. over time. (4. and the SIS model cannot be reduced to a single ODE. (SIS model with vital rates) Prove an analog of Theorem 7 for the SIS model with vital rates. Substituting S = N − I into (4. Problem 57. 2.46) = βSI − νI − δI. If R0 = 1.42) Adding these equations yields S 0 + I 0 = 0.48) = βSI − νI − δI − µI. Find the equilibria of this model and establish their stability. then I(t) approaches I ∗ = β/(ν(R0 − 1)) as t → ∞. then the following theorem is an immediate consequence of our analysis of the logistic ODE. Problem 55. If R0 > 1.4. (4. (SIS model with vital rates and disease mortality) Combine the previous two models. 55) = rF SF IM − αF IF . and include a problem on using the model to study malaria. (4. 2003] and explain the model to study the spread of gonorrhea. Show that the change of variables x(k) = (αI(k))/(N (1 − γ + α)) and a = 1 − γ + α transforms this difference equation to x(k + 1) = ax(k)(1 − x(k)) as in (2. IM . 1910] to study malaria. Since there is little or no acquired immunity following a gonorrhea infection. Gonorrhea is transmitted back and forth between males and females. IF denote the number of sexually active females. αM . Show that the population size is constant N . the number of susceptible males. the number of susceptible females.56) where the parameters rM . MODELING THE SPREAD OF INFECTIOUS DISEASES Problem 58. SM . IF ) = rF IM (NF − IF ) − αF IF . Shistosomiasis is transmitted back and forth between humans and a trematode worm. We assume only heterosexual contacts.38). an SIS model seems appropriate.53) = rM SM IF − αM IM (4. invented by [Ross. and SM . We let NM .51) N 1. N (4.3 SIS criss-cross models This class of models. We follow [Murray. Since NM = SM + IM and NF = SF + IF this system of four ODEs reduces to a system of two ODEs dIM dt dIF dt = f (IM . are useful to model diseases that are transmitted back and forth between two interacting but distinct groups.54) = −rF SF IM + αF IF (4. Malaria/Dengue is transmitted back and forth between humans and female Anopheles/Aedes mosquitoes. IM denote the number of sexually active males. and the incubation period is short compared to the infectious period. The system of ODEs is dSM dt dIM dt dSF dt dIF dt = −rM SM IF + αM IM (4. IF ) = rM IF (NM − IM ) − αM IM (4.57) = g(IM .52) 2. (4.50) N α I(k + 1) = I(k) 1 − γ + S(k) . SF . Use this fact to show that α I(k + 1) = I(k) 1 − γ + α − I(k) . (Discrete time SIS model without vital rates) The discrete time SIS model without vital rates is given by the system of difference equations α S(k + 1) = S(k) 1 − I(k) + γI(k) (4. rF . and the number of infected males. and consider only sexually active individuals. and the number of infected females. (4.90 CHAPTER 4. 4. αF are positive.58) . 60) The average of a female is 1/αF . McKenzie. 2004] for a state of the art mathematical model of malaria transmission. this malaria model formally coincides with the above gonorrhea model (4. since O · (f. 2003] for an attempt to parameterize this model. ρ M + NF ρ F + NM where ρM = αM /rM and ρF = αF /rF . Macdonald. 2000]. 1952. IF∗ ) is globally attracting. See [Murray.59) (4. 1978]. All orbits are bounded since both components of the vector field are negative for IM and IF sufficiently large. 1976a] extend this basic model to account for the heterogeneous nature of the population at risk. and whether an individual is symptomatic or asymptomatic while infected. (4. and Nold.58) and (4. 1988. SIS CRISS-CROSS MODELS 91 There are at most two equilibrium points. . Dietz. Find an analogous expression for R0 and interpret it biologically. Similarly. is attracting. The Ross-MacDonald malaria model captures the basic features of the interaction between the human host population and the female mosquito vector population. 0) is globally attracting. 1982. We recommend [McQueen and McKenzie. Problem 60. (NF rF )/αM is the infectious contact number for the female population. which only exists R0 > 1. ρM ρF αF αM (4. 0. 0) is attracting for R0 < 1 and a saddle for ∗ . Hethcote. Problem 59. and for R0 > 0 the equilibrium ∗ . point (IM [Lajmanovich and Yorke. Their extension takes into account the facts that many infected females are asymptomatic and that a symptomatic individual is unlikely to find a sex partner. show that (0. 0) and NM NF − ρM ρF NM NF − ρM ρF ∗ ∗ (IM . then either the disease free state or an endemic state is globally attracting.62) where Repeat the analysis of the above gonorrhea model for the malaria model.A. Show that the equilibrium point (IM for R0 > 1. (0.58): dx dt dy dt = (abM/n)y(1 − y) − rx = ax(1 − y) − µy. Thus the quantity (NM rM )/αF is the infectious contact number for the male population. [Aron and May. . This implies that for R0 < 0 the equilibrium point (0. level of sexual activity (active or very active). See also [J.4. They also use the model to test the efficacy of public health measures to control the spread of gonorrhea. Their main mathematical result is that if the population is irreducible (it can not be decomposed into two subpopulations having no contact). Identify the type of local bifurcation. and rM /αF is the fraction of susceptible males infected by the female during this time. Define the parameter NM NF NM rM NF rF R0 = = . The authors partition the population into eight groups by sex (male or female). IS ) = . Using linear stability analysis.3. Bendixson’s criterion implies there are no limit cycles. this equilibrium point corresponds to endemic disease. IS∗ ). g) = (−rM IF − αM ) + (−rF IM − αF ) < 0. Up to a simple rescaling.. Biologically.61) (4. 2: Model parameters 4. Parameter x y N M m = M/N a b r µ MODELING THE SPREAD OF INFECTIOUS DISEASES Description Proportion of human population infected Proportion of the female mosquito population infected Size of human population Size of female mosquito population The number of female mosquitos per human host Number of bites on man per unit time by a single mosquito Proportion if bites on man that produce an infection Per capita recovery rate for humans Per capita mortality rate for mosquitos. It is illustrated schematically in Figure (4. called the SEIR model.11) and is described by the system of ODEs Nμ S μ βE E μ α I μ ν R μ Figure 4. but are not yet infectious. is frequently used by disease modelers and is commonly found in the literature. This model.11: SEIR model with vital rates .4.4 4.1 SEIR models Basic SEIR model with vital rates We now extend the SIR model with vital rates to allow for individuals who are exposed (E) to infection.92 CHAPTER 4. Table 4. . However. Thus if I(0) > 0. 0. 2. ν(R0 − 1)/β).67) (µ + α)(µ + ν) The following theorem is an extension of Theorem 6 and has a similar proof. The asymptotic number of infectives for the SEIR and SIR models with vital rates coincide. 0. I ∗ .69) = pβSI + νL − µI − δI. 0.63) = βSI − αE − µE (4. SEIR MODELS 93 dS dt dE dt dI dt dR dt = N µ − βSI − µS (4. If R0 < 1. µ(R0 − 1)/β. 0.4. Susceptibles are infected at rate SI and move either into the exposed (or latent) class E or directly into the infectious class I (fast progression). In the infectious state. Theorem 8. Problem 61. 0. Prove an analog of Theorem 8. Exposed individuals progress to active disease when they are infectious at a constant rate (slow progression). If R0 = 1. the equilibrium point (N. the number of infectives I(t) approaches (µ/β)(R0 − 1) as t → ∞. (A simple model of tuberculosis transmission) [Blower et al. R∗ ) = (1/R0 . individuals suffer an increased death rate due to disease. 3. one can prove that the growth of the infective population is slower for the SEIR model. 0) is globally attracting. This basic model assumes that nobody recovers from the disease. the equilibrium point (N.68) = (1 − p)βSI − µE − νE (4.64) = αE − νI − µI (4. 1995] This model assumes that susceptibles are born into the population at rate N . 0) and (S ∗ . (4. which makes sense because individuals need to spend time passing through the exposed class. µ(R0 − 1)((µ + ν)/(αβ)). (Dynamics of SEIR model with vital rates) 1. 0. If R0 > 1.65) = νI − µR.66) where 1/α is the mean latent period (incubation period) for the disease and has a similar proof. (4..70) where δ is the death rate due to TB and p is the proportion of “fast progressors”. 0) is globally attracting. (4. The first is a saddle and the second is attracting. Thus I(t) decreases to zero as t → ∞. The basic reproduction ratio for this model is N βα R0 = . dS dt dE dt dI dt = N µ − βSI − µS (4.4. Thus I(t) decreases to zero as t → ∞. E ∗ . there are two equilibrium points (N. where β0 is the mean transmission coefficient. including multiple stable subharmonics of any period (resulting from period doubling and saddle node bifurcations). Regular periodic outbreaks also occurred for other childhood viral diseases such as mumps. 1990] devised a clever method to compute R0 as the spectral radius of the next generation operator.. this system exhibits harmonic resonance (if the forcing frequency is near the natural freqency). The transmission coefficient β for these diseases changes throughout the year. MODELING THE SPREAD OF INFECTIOUS DISEASES As models get more complicated. Several authors have modeled this transmission coefficient using a periodic function β(t) = β0 (1 + α cos(2πt)). collected before the routine use of the measles vaccine in these cities. chickenpox. As to be expected from a nonlinear harmonic oscillator. 1989] for an exposition of periodicity in epidemiological models.94 CHAPTER 4. it becomes increasingly difficult to compute R0 . 2002] for many examples illustrating this method. rubella. multiple subharmonic resonances. 1994] (see bifurcation diagram in Figure 4.2 Seasonally forced SIR and SEIR models The measles incidence data in Figure 4. and lower during the summer months. 4. 1976] initiated the rigorous analysis of the periodically forced SIR model with vital rates. Figure 4. and period doubling cascades that produce strange attractors [Kuznetsov and Piccardi. The model produces recurrent undamped epidemics of all frequencies observed in measles time series. [Dietz. None of the epidemiological models that we have considered so far can explain the periodic nature of these epidemics.4. and a zoo of local and global bifurcations. See also [Castillo-Chavez et al. 1995] .13).. We recommend [Hethcote and Levin.12: Measles time series from [Mollison.12. The same is true using a piecewise constant transmission coefficient. The authors [Diekmann et al. shows a regular biennial pattern of epidemics between 1948 and 1966. it is higher when school is in session. and provides useful insights into these questions. and is usually very difficult to estimate reliably. The transmission rate of an infectious disease is defined to be the rate of new infections per unit time. and the vast majority of investigators use an SIR-type model. Dowell [2001]. pneumococcus.edu/current/documents/pcmi earn lecture2.ias.74) β(k) = S(k)I(k) [Fine and Clarkson.13: Measles bifurcation diagram cmi. 1991]. and polio peaks in the summer.4. In most cases the underlying mechanisms are uncertain. The data determine I(k) and I(k + 1). We consider the simplest SIR model with time varying transmission coefficient dS dt dI dt dR dt = −β(t)SI (4. to compute or estimate this quantity requires a mathematical model. . 1982]. They may include prevalence and virulence of the pathogen. S(k) frequently depends on the birth rate.3 as function of α [from Transmission models with time dependent transmission coefficient There is strong seasonality of the infection rate of many acute infectious diseases: influenza. There are also biennial cycles of measles. etc. The transmission rate has been thought impossible to measure directly [Anderson and May. while respiratory syncytial virus peaks in the spring. How can one estimate β(t) from infection data? The most widely used method is based on defining the transmission coefficient is I(k + 1) (4. but not S(k). The causes of these cycles is one of epidemiology’s challenge problems. contact rate. seasonal changes in the environment..73) νI. etc. immune system response. SEIR MODELS 95 Figure 4.pdf] 4. This is independent of any SIR or mathematical model. However.72) = (4. In reality.4. and rotavirus peak in the winter.71) = β(t)SI − νI (4.4. does there exist a non-negative transmission function β(t) such that the solution I(t) of the SIR model coincides with f (t) on a large time interval? They prove that this is always possible. MODELING THE SPREAD OF INFECTIOUS DISEASES One can obtain (4. Our algorithm is derived from the complete solution of a mathematical inverse problem for SIR-type transmission models. In the next section we derive a new formula for recovering the transmission coefficient that uses only the infection data. All previous measles transmission models assume that the effective contact rate has one-year period driven solely by the mixing of children in school. Proof. (4.74) from (4. Wang. and then equate with (4. Given a smooth positive function f (t).76) = (4. We rewrite (4.76) implies that f 0 (t) + νf (t) = β(t)S(t)f (t). The idea is to use the full system of ODEs instead of the crude discrete approximation of one ODE. Given an arbitrary smooth positive function f (t) and recovery rate ν > 0.96 CHAPTER 4. The authors apply this algorithm to historic UK measles data and observe that for most cities.75) = β(t)SI − νI (4. In [Pollicott.72) by discretizing using time steps of length 1/ν = 1.77) νI. The proofs is very elementary. the effective contact rate has strong biennial and three times per year signatures. From this theorem we see that there are no restrictions on how f (t) increases. This clearly illustrates the danger of overfitting a transmission model with time-varying functions. Consider again the simplest SIR model with time-dependent transmission coefficient dS dt dI dt dR dt = −β(t)SI (4.79) . The algorithm yields that almost any “smooth” infection profile can be perfectly fitted by an SIR model with a variable transmission coefficient.5 Recovering the transmission coefficient from data We present a new algorithm to recover this time-varying transmission coefficient directly from infection data. We now show this condition is also sufficient. there exists β0 > 0 such that if β(0) < β0 there is a solution β(t) with this initial condition such that I(t) = f (t) for 0 ≤ t ≤ T if and only if f 0 (t)/f (t) > −ν for 0 ≤ t ≤ T . Theorem 9. in the sense that its logarithmic derivative is always bounded below by −ν. but to find β(t) the function f (t) cannot decrease too fast. It is easy to see that f 0 (t)/f (t) > −ν is a necessary condition. 2009a] the authors pose the following inverse question. and T > 0. 4. and Weiss.76) as f 0 + νf S= . ν > 0. which must be positive for 0 ≤ t ≤ T .78) βf then compute (d/dt)S.75) to obtain 0 d f 0 + rf f + rf = −β(t) f (t) dt βf βf (4. since Equation (4. (4. With the restriction of ν.85) = αSA − βAI A. The authors show that one can robustly estimate β(t) by smoothly interpolating the data with a spline or trigonometric function and then applying the formula for continuous data. (4.88) .e. The initial transmission coefficient must also satisfy Z T eP (s) f (s)ds < 1/β(0). The goal is to obtain parameter regions for a network which correspond to globally attracting disease-free equilibrium points. Assuming that the computer network does not change during a virus attack.6. Navarro.87) = δI − σRS R (4. (4. where p= x0 − px − f = 0.84) 0 Of course epidemiological data is discrete. A of non-infected computers equipped with anti-virus.80) (4. We use the transformation x = 1/β to obtain the linear ODE β 0 − pβ − f β 2 = 0.. Using the method of integrating factors we obtain an explicit solution of the form: Z t 1 −P (t) −P (t) eP (s) f (s)ds. 4. I of infected computers.82) t p(τ )dτ.6 Modeling the spread of a computer virus There is now a substantial literature on modeling the spread of computer viruses using epidemiological models.81) (4. The total population N is divided into four compartments: S of non-infected computers subjected to possible infection. f (f 0 + rf ) This is a Bernoulli equation. and Monteiro. In the case of no restriction on ν. MODELING THE SPREAD OF A COMPUTER VIRUS and this 97 2 f 00 f − f 0 . the basic SAIR model is given by the the following system of ODEs [Piqueira. i.86) = βSI SI + βAI AI − σIS I − δI (4. and R of removed ones due to the infection or not. not continuous. f 0 + νf > 0. = x(t) = x(0)e −e β(t) 0 where Z P (t) = (4. this can be prevented by choosing ν sufficiently large such that ν · min{f (t) : 0 ≤ t ≤ T } > − max{f 0 (t) : 0 ≤ t ≤ T }.83) 0 The only problem that could arrive with this procedure is for the denominator of p(t) be be zero. 2005] dS dt dA dt dI dt dR dt = −αSA − βSI SI + σIS I + σRS R (4. and the hope is that this will provide strategies about how to build networks that are hard to infect. the zero in the demominator is prevented by requiring that the denomimator is always positive.4. As I write these notes. Parameter α βSI σIS σRS βAI δ MODELING THE SPREAD OF INFECTIOUS DISEASES Description conversion rate of susceptible computers into antidotal ones Infection rate of new computers Recovery rate of infected computers Recovery rate of removed computers with system administrator intervention Infection rate of antidotal computers due to the onset of new virus Recovery rate of infected computers Table 4. .7. σIS + δ (4.7 4. Show that E1 = (0. 3.” 4.1 Evolution and transmission of infectious diseases Infection with multiple strains We now present a simple model of the transmission dynamics of an infectious disease caused by a pathogen having multiple strains. According to [Richards.2). When all evolutionary paths result in death.3: Model parameters where Problem 62. N. “Software instructions are very brittle. and thus unstable. The zero eigenvalue for E1 is innocuous because the A-axis is the center manifold. You will notice that both contain a zero eigenvalue. The model also describes the competition between a wild strain and mutant strains of a pathogen (see Section 4. 0. Show that E1 is locally attracting (forgetting about the zero eigenvalue) provided that R0 = βAI N < 1. 0) are the two disease-free equilibrium points.98 CHAPTER 4.7. Can you interpret this condition in terms of computer networks? What can a systems administrator do to prevent a virus epidemic on her system? An interesting question is whether computer viruses can evolve like ordinary viruses.89) 4. Compute the associated Jacobian matrices and compute the eigenvalues. 0. 2010]. 0. 2. and those instructions are so densely coded that almost all mutations or recombinations would yield completely non-functional programs. It’s not even a question of reduced fitness . evolution just doesn’t have much to work with. 0) and E2 = (N. 1.they simply wouldn’t work. Since E2 has both positive and negative eigenvalues. researchers are studying the competition and transmission dynamics of the seasonal and pandemic strains of H1N1 influenza. They cary out a specific set of instructions. it is a saddle. In this case we prove a competitive exclusion principle showing that asymptotically a single strain will evolve or the disease will die out (typically). where N = S(0) + A(0) + I(0) + R(0). 1/βj 1/β1 I1 (t) Ij (t) exp((σj − σ1 )t) = . .91). .93) to obtain r X dN =− dk Ik (t) ≤ 0.g.90). .94) k=1 Thus N (t) ≤ N (0) for all t ≥ 0. . EVOLUTION AND TRANSMISSION OF INFECTIOUS DISEASES 99 Similar models are used to study the spread of infectious diseases with multiple infected subclasses.90) j=1 = βj SIj − νj Ij − µIj − dj Ij . We denote Ik the number of individuals infected by strain k. r. 2. σ2 .4. = r X j = 1. . (4.93) implies that u˙ j = S for j = 1. Lajmanovich and Yorke.93) that if σj /S(0) > 1. and thus the number of infections caused by strain j starts increasing. dt (4.95) Ij (0) I1 (0) Assume. Observe that the total population size N (t) is a decreasing function of time. . . Using the fact that S(t) ≤ N (t) − I1 (t) − · · · − Ir (t) − R(t) ≤ N (0) − Ij (t) . r. . . 2. classes contain individuals having different contact rates with other individuals.91) as dIj = βj Ij (S − σj ). . then Ij0 (0) > 0. 1976b]. . βj and observe that (4. Examples include sexually transmitted diseases [Hethcote and Yorke. If R0 (i) = N (0)/σi denotes the basic reproduction rate of the i − th strain. that σ1 = min{σ1 . 1984b. and thus the exponentials of the u0j s coincide up to multiplicative constants. σr }. (4. .91) (4. . 2. without loss of generality. It also immediately follows from (4.95) implies that the limit limt→∞ Ik (t) = 0 exponentially for all σi > σ1 . and can include a super-spreader class.. (4.7. the system of ODEs is dS dt dIj dt dR dt = N (t)µ − µS − S r X βj Ij (4. . 2. r. dt j = 1. compute (d/dt)N (t) be adding (4.92) j=1 Define σj = (µ + νj + dj )/βj and rewrite (4.93) Pr where N (t) = S(t) + k=1 Ik (t) + R(t). (4. . . We now prove that if the basic reproduction rate of all strains is less than one then the disease die out. where the subclasses are based on the frequency of sexual activity. We now follow [Bremermann and Thieme. . (4. Then assuming that superinfection with more than one strain can not occur and that an individual infected with any strain obtains permanent immunity to subsequent infection from all strains. Then since I1 (t) ≤ N (0) for all t ≥ 0. This implies that for j = 1. Thus the u0j s all coincide up to multiplicative constants. . . To see this. . 1989] and define uj = 1 log (Ij /Ij (0)) + σj t. and (4. e. this calculation shows that strains that have lower basic reproduction rate of other strains die out. r νj Ij − µR. . λ4 = 2 Simple algebra yields that real(λ3 ).100 CHAPTER 4. 0 0 β2 (σ1 − σ2 ) 0 0 −d1 −d2 0 A routine calculation yields that the eigenvalues are λ1 = 0. . I2∗ = 0. and that for σ1 < σ2 every equilibrium point is a saddle. . and if R0 (i) = N (0)/σi < 1 for all i. I2∗ = 0.101) (4. There is a one parameter family of equilibrium points (S ∗ = σ1 . . N ∗ ) parameterized by N ∗ with Jacobian matrices −β1 I1∗ − µ β1 σ1 −β2 σ1 0 β1 I1∗ 0 0 0 . (4. λ2 = β2 (σ1 − σ2 ). Finally. λ3 . and assume I1∗ > 0. q 1 ∗ ∗ 2 ∗ 2 −β1 I1 − µ ± (β1 I1 + µ) − 4β1 σ1 I1 . MODELING THE SPREAD OF INFECTIOUS DISEASES we crudely estimate dIj dt = βj Ij (S − σj ) (4. I1∗ = µ(N ∗ − σ1 )/(σ1 β1 ). . . real(λ4 ) < 0 and that λ2 < 0 if σ1 > σ2 and λ2 > 0 if σ1 < σ2 . r.102) j=1 To simplify the exposition we prove the claim for r = 2. we show that if σ1 < σj . and the disease dies off. a one dimensional local unstable manifold. . . It follows from the local stable and center manifold theorems that for σ1 > σ2 every equilibrium point has a three dimensional local stable manifold and a one dimensional center manifold. 2.98) (4.99) Thus IjM AX = 0 or IjM AX ≤ N (0) − σj . j = 2. We consider the equivalent system of ODEs dS dt dIj dt dN dt = N (t)µ − µS − S r X βj Ij (4. The zero eigenvalue is innocuous because the N ∗ -axis is the center manifold. then there is a unique endemic non-zero equilibrium with I2 = I3 = · · · = Ir = 0 that is locally attracting. while for σ1 < σ2 every equilibrium point has a two dimensional local stable manifold. then IjM AX = 0 for all i. r = − r X dj Ij .97) Let IjM AX = lim supt→∞ I(t) and choose a sequence of times {tk } such that limk→∞ I(tk ) = IjM AX and I (tk ) ≥ 0. . and a one dimensional center manifold. It follows that for σ1 > σ2 every equilibrium point is effectively locally attracting.100) j=1 = βj Ij (S − σj ) j = 1. We have that 0 0 dIj (tk ) dt ≤ βj IjM AX (N (0) − IjM AX − σj ). (4. ≤ lim tk →∞ (4. and R0 (1) > 1.96) ≤ βj Ij (N (0) − Ij (t)) − σj )). We can use the above analysis to explore conditions under which the mutant pathogen will replace the wild type pathogen in the population. which is the SIR model with disease caused mortality and vital rates.7. but not yet infectious). latency (some individuals are infected. with d(0) = 0. Thus no more than two strains which have the same R0 persist in the population. Thus one would expect a pathogen to evolve to increase its R0 . p > 1 and c > 0. schools). In [Bremermann and Pickering.g. and the development of antibiotic resistant strains presents one of the major health challenges to medical science.103) can take the same value only twice.7. If we assume convexity.7. . to model different strains of the disease or infected individuals with very different contact rates). 4. transmission coefficients. We have not excluded the long-term coexistence of different strains with the same σk . vancomycin-resistant enterococci. We assume that a population is initially in equilibrium with an endemic disease caused by a wild type virus. 2002]. A common source of selective pressure on pathogenic bacteria is antibiotics.1 that if the R0 of the mutant strain is less than the R0 of the wild strain then the mutant stain dies out. social structures (e. Mathematically. and d0 (β) → ∞ as β → ∞.7. families. multiple infected classes (e. Now introduce into this susceptible population a small number of individuals infected with the mutant strain. EVOLUTION AND TRANSMISSION OF INFECTIOUS DISEASES 101 Problem 63. their claim is that d = d(β) is a strictly convex function. 1991. and multi-resistant Mycobacterium tuberculosis.2 Invasion by a mutant Bacteria and viruses continuously evolve. the equilibrium number of susceptibles S ∗ = σ1 . seasonal patterns in risks of infection. drugs. to make individuals sick longer (smaller ν). while if the R0 of the mutant strain is greater than the R0 of the wild strain then the wild strain dies out. (SIR model with vital rates and disease virulence) Analyze the above system with only one infected class. Assuming that only the wild strain exists. We have presented the most basic SIR and SIS type models.g. More realistic models incorporate heterogeneous mixing.4. 1998] for a discussion of the global dynamics. and to be less deadly (smaller d).3 Modeling the spread of antibiotic resistance Antibiotics have substantially reduced the threat posed by infectious diseases. and in general be more easily transmitted (larger β). The CDC estimates that 44. and this can result in the evolution of virulence. d0 (0) = 0.. monotone increasing function of β.000 people in North America die annually of infections from drug-resistant germs. An example of such dependence is d = cβ p . Important examples include penicillin-resistant Streptococcus pneumoniae. We refer the interested reader to [Anderson and May. Castillo-Chavez et al. etc. We proved in Section 4. 4. Now. and in cases many. carriers (asymptomatic but infectious).. then σ(β) = µ + ν + d(β) β (4. Viruses also compete directly with each other for reproductive success. age-dependent contact rates. and that different strains produce the same immunologIcal response (identical ν) but have different transmission coefficients βj and disease mortality rates dj = dj (β)... See [Graef et al. suppose the virus spontaneously mutates. maternal immunity. These gains are now seriously jeopardized by the emergence and spread of bacterial strains that are resistant to some. 1983] the authors argue that the disease induced mortality d for some diseases rises “disproportionately” as β increases. methicillin-resistant Staphylococcus aureus (MSRA). multi-resistant salmonellae. In other words.105) = βSIsen − cIsen − rsen Isen − f hIsen . β cβ β . and . Individuals are infected with either a drug sensitive strain or a drug resistant strain. 0.92) are being used to study the spread of antibiotic resistant strains in hospitals and to search for strategies to contain this public health crisis [Bonhoeffer et al. The system of ODEs is dS dt dIsen dt dIres dt Parameter λ µ c β rsen rres f h = λ − µS − βS(Isen + Ires ) + rsen Isen (4. This process is called acquired resistance and we assume this only happens in treated hosts..4: Model parameters where the model assumes: 1. 4. (4. 1997.104) + rres Ires + f h(1 − s)Isen (4.. Lipsitch et al. (4.107) Description birth rate death rate of susceptible individuals death rate of infected individuals (natural and disease-associated mortality) infection transmission coefficient recovery rate of untreated sensitive infections recovery rate of untreated resistant infections scaling parameter reflecting the fraction of patients treated 0 ≤ f ≤ 1 maximum rate when all patients are treated Table 4. . there is a preexisting. Patients being treated for the sensitive strain will either recover (at rate f h) or become infected by a resistant strain (with rate sf h). 0). When these patients receive drugs. There are three two equilibrium points: c + rres βλ − µ(c + rres ) c + f h + rsen (λ/µ. 0.106) = βSIres − cIres − rres Ires + f hsIsen . We present the simplest mass-action model in the case of resistance to a single drug [Bonhoeffer et al. the cost of resistance ∆r = rres − rsen > 0. (4. The fitness cost associated with resistance is reflected by a higher rate of clearance (recover) of the infection on patients infected with resistant bacteria than those infected with sensitive bacteria (rsen < rres ). In a fraction s of sensitive strain infected patients.93).102 CHAPTER 4. small subpopulation of resistant bacteria. 2000]. (4. 1997]. Patients who are treated and cured become immediately susceptible again. 2. No co-infection 3.90).. . MODELING THE SPREAD OF INFECTIOUS DISEASES Mathematical compartment models similar to (4. the resistant population will grow and will quickly dominate the infection. The third equilibrium point corresponds to endemic infection with both strains. Schuster]. for example by presenting an epitope that the immune system fails to recognize. viruses are not independent entities in the quasispecies. have a very high frequency of copying errors. Schuster. it has been estimated that every HIV patient produces all possible single and double point virus mutations and a large fraction of triple point mutations every day [Perelson et al.. and let xi denote the density or frequency of genotypes having sequence i. The matrix Q = {Qij } has no-negative entries and is column stochastic (the sum of entries in every column is one) and is called the mutation matrix. Quasi-species models describe the evolution and selection of virus genotypes. The next problem provides useful insights into the global dynamics. Consider the proportion of resistant infections ρ = Ires /(Ires + Isen ). The equilibrium point (λ/µ. cβ(∆r + f h(s − 1)) cβ(∆r + f h(s − 1)) 103 . A critical element of the model is that the frequency of any individual virus is a function of both its own replication rate and the probability that it will arise by the erroneous replication of other members of the population. 0. The second equilibrium point corresponds to endemic infection with the resistant strain.. Novak.7. called a quasispecies [Bull et al. Problem 64. Most of the mutants have impaired fitness. fi the fitness of sequence i. which are considered to form a single genotype. Also. 1992. EVOLUTION AND TRANSMISSION OF INFECTIOUS DISEASES (f h + ∆r)((c + f h + rsen )µ − βλ) f hs(βλ − (c + f h + rsen )) . dt and has closed form solution ρ(t) = exp((f h − ∆r)t) − 1 . 0) corresponds to no long-term infection with either strain. 4. Show that ρ satisfies the Bernouli ODE dρ = (f h − ∆r)ρ(1 − ρ) + f hs(1 − ρ)2 . 2005. and −c − f h − rsen + βλ/µ. With some probability. such as the flu virus and HIV virus.4. Thus a necessary condition that this resistant strain endemic equilibrium point be attracting is that the cost of resistance is less than the selective pressure against the sensitive bacteria. so that the entire population forms a cooperative structure that evolves as a single unit. These viruses produce an extraordinarily high rate of mutants. The basic . 1997]. sometimes close to one base-pair substitution per genome per generation. −c − rsen + βλ/µ. These virus populations consist of a “cloud’ of mutant genotypes.7. this mutant becomes the ancestor of a new quasispecies which completely replaces the currently existing one. and the Jacobian has eigenvalues −µ. 1971] in which he studied the error-prone self-replication of biological macromolecules. h−∆r exp((f h − ∆r)t) − 1 + f(f hs) Deduce that limt→∞ ρ(t) = 1 if ∆r < f h and limt→∞ ρ(t) = 1 h−∆r 1− f(f hs) if ∆r > f h. and the Jacobian has an eigenvalue −f h + ∆r.4 Quasi-species models Many RNA viruses. find the time required for the proportion of resistant infections to reach a given level 0 < σ < 1. with long formulae for the other two eigenvalues. Their origin is a seminal paper by Eigen [Eigen. The sizable genetric variation of RNA viruses is believed to enhance the evolution of drug resistance and immune escape. This is in contrast to a species. Consider a population of infinitely many genotypes consisting of N nucleic acid sequences. with the goal of understanding the origin of life. For example. but occasionally a new mutant will out-compete all currently existing virions. and Qij ≥ 0 the per unit time probability that sequence j will mutate to sequence i. but are linked by mutual couplings. Consequently. Problem 65. N. . (4. find limt→∞ xi (t). A critical element of the model is that the frequency of any individual virus in the quasispecies is a function of both its own replication rate and the probability that it will arise by the erroneous replication of other members of the population. . i.. Prove that if j xj (0) = 1. Prove that in the case of no mutations. . 1. Assuming that QF = {Qij fj } is diagonalizable over R. 1.110) where the eigenvector vk = (l1k .108) Pn where fˆ(~x) = x = (x1 . . Under the assumption in 2. viruses are not independent entities in the quasispecies but are linked by mutual couplings. ~ removed at this time-dependent rate to ensure selection and to keep the total density constant. Thus all three ingredients for Dawinian evolution are present: reproduction. and are given in terms of the components of the dominant eigenvalue (l1k . . Q is the identity matrix. Show that −fˆ(~x) is a Lyapunov function. whose coordinates represent the limiting or stationary frequencies of genotypes. show that the explicit solution of (4.e. The matrix QF is called primitive if there exists m ∈ N such that all entries of (QF )m are positive.108) is equivalent to the linear system N X dzi Qij fj zj . vn of QF . . and selection. dt j=1 i = 1. then j xj (t) = 1 for all t ≥ 0. Show that system (4. · · · lnk ). . P P 2. PN j=1 k=1 ljk ck (0) exp(λk t) (4. then if fi > fj for all j 6= i. . . . xN ) denotes the mean fitness of the population and genotypes are i=1 fi xi . asymptotically the fittest genotype takes over the population.108) can be written as PN lik ck (0) exp(λk t) xi (t) = PN k=1 . . Although the system is nonlinear. Define zk (t) = xk (t) exp 0 fˆ(~x(s))ds and verify that (4. (4. 3. · · · lnk ). it is a Bernoulii system. but is more general. there is a single attracting equilibrium point xi = 1 and xj = 0 for all j 6= i. MODELING THE SPREAD OF INFECTIOUS DISEASES quasi-species model is the system of ODEs N X dxi = Qij fj xj − fˆ(~x)xi . .109) = dt j=1 Note that all of the entries of the matrix QF are non-negative.. This equilibrium represents a balance between mutation and selection. . mutation. . . . and a change of variables converts it to a linear system having an explicit solution. 4. so that the entire population forms a cooperative structure that evolves as a single unit.108) possesses a globally attracting equilibrium point. 3.. Consequently. In this case the PerronFrobenius (1) theorem guarantees a unique maximal positive eigenvalue λ1 > |λ2 | ≥ |λ3 | ≥ · · · ≥ |λn | having an eigenvector v1 with all positive entries. This of course includes the case when all the entries of QF are positive. λn and eigenvectors v1 . 2. Under the assumption in 2. Thus with no mutations.109) in terms of the eigenvalues λ1 . find the explicit solution of (4. R t Problem 66. Hence this is a conserved system.104 CHAPTER 4. The extra term adds a quadratic nonlinearity and ensures selection. . On the other hand. B. to an upper triangular block matrix with each block a square reducible or zero matrix [Varga. We illustrate this in the simplest case in the following problem. and D. 1) and thus does not have all positive entries. ~x(0)) = xi (t) log2 xi (0) i and showed that if all fi > 0 and Q is symmetric. a wild strain and a mutant strain. In view of the simple dynamics of this model. As µ approaches 1 − fm /fw from below. Maximal size of RNA virus genome . Is the mean fitness increasing over time? The following problem provides an interesting simple example.108) possesses a globally attracting equilibrium point. Observe that for µ > 1 − fm /fw the eigenvector of QF corresponding to the dominent eigenvalue fm is (0. Suppose that A never mutates and always produces a single offspring. the point (0. both equlibrium points merge and coalesce. where the wild strain is fitter than the mutant strain. C. (4. there are two equilibrium points: (0. Thus for large time. x∗m ) with positive coordinates. 6. 1 − k replicas of themselves. if most of the virus copies of a favored strain have errors. there would be no evolution. Consider a quasispecies consisting of four genomes A. The degeneracy of the eigenvalues of the linearized systems complicates the analysis. This bifurcation is called an error catastrophe. where 0 < k < 1. Problems (67) and (68) show that some stationary frequencies may be zero. and k of each of the other two types. on average. 2010]. Problem 67. 1) is repelling and there is a globally attracting equilibrium point (x∗w . In general.108).112) Verify (at least numerically) that if µ < 1 − fm /fw . The error threshold is the maximum copying error rate that will ensure survival of the favored strain.4. genome A will become extinct and the other three genomes will be present in equal numbers. the matrix QF has non-negative entries and thus is similar. both stains are present in nearly these proportions. 1) is globally attracting and thus the mutant strain becomes fixated (only the wild type strain survives). Show that asymptotically. the strain would die out. Weinberger [Weinberger.108) becomes dxw dt dxm dt = fw (1 − µ)xw − xw (fw xw + fm xm ) (4. Does this contradict the Perron-Frobenius theorem? Problem 69. then −I is a global Lyapunov function for the quasispecies system (4.111) = fw µxw + fm xm − xm (fw xw + fm xm ).7. a natural question is whether one can construct a global Lyapunov function. EVOLUTION AND TRANSMISSION OF INFECTIOUS DISEASES 105 5. If the virus makes no copying errors and hence produces no mutants. via a change of variables by a permutation matrix. An important aspect in the theory of quasispecies is the error threshold. Also assume that that the mutant strain makes no copying errors. Let µ be the probability that reproduction of the wild strain results in the mutant strain . Problem 68. Show that system (4. 2002] defined the notion of “pragmatic information” by X xi (t) I(~x(t). At the threshold. the favored strain and a second strain have precisiely the same replacement rate. If µ > 1 − fm /fw . Consider two virus strains. Then (4. whose coordinates represent the limiting or stationary frequencies of genotypes. Suppose that the other three genomes all produce. and is thus a conserved quanity. whereas B is the top of an almost flat mountain. and that B is surrounded by mutants with high (but lower) fitness. quasispecies wander over the fitness landscape searching for peaks. N can be thought of as the average number of virions produced over a lifetime of an infected cell.108) and our previous analysis to the case where the mean fitness of the population is a conserved quantity. Different viruses will attach to different host cells: HIV to lymphocytes. z∗ ) is . then the second equilibrium point (x∗ . and malaria to red blood cells. the mean fitness term was subtracted from the right hand side to insure that the total density of all genotypes add up to one. then x0 < 0. with probability p > 0 the mutant strain mutates to the wild strain. Imagine two genomes. We note that asuch an error catastrophe has not yet been empirally obsered. Problem 70.113) = kxz − βy (4. Let x∗ = γ/(kN ). Problem 71. Some viruses such as polio or hepatitis C operate close to the error threshold. In system (4.106 CHAPTER 4.115) where x denotes the density of the uninfected cells. natural selection does not simply choose the fittest sequence. whereas in the case of budding viruses. The carrying capacity for the healthy cell population is λ/l.111) one allows “back-mutations”.e. which represent regions of high fitness values. y∗ . The immune system removes infected cells at a rate of βy. MODELING THE SPREAD OF INFECTIOUS DISEASES To recapitulate. If λ > (lγ)/(kN ). where the viral replication is limited by the availability of uninfected cells. Assume that A has a higher fitness than B. Problem (68) shows that B could be the winner and force A into extinction. The system of ODEs is dx dt dy dt dz dt = λ − lx − kxz (4.. fitness can be viewed as a function on the set of all genomes and its graph is called the fitness landscape. that A is surrounded by mutants with very low fitness. The lifespan of a free virus particle is 1/γ. and z is the number of free virus particle. The error threshold is the critical mutation rate above which A is forced to extinction and less fit genomes take over the population. but the fittest quasispecies. Both A and B are local maxima. the idea being to increase the mutation rate of a pathogenic virus to push it over the error threshold so that it becomes extinct. If x > λ/l. in system (4.8 In-host model of viral infection We follow [Nowak and May. 0).108). The rate of viral production is proportional to the removal rate of infected cells. Suppose. but A is a sharp peak in the fitness landscape. y the density of infected (virus producing) cells. One equilibrium point is (λ/l. Under the evolution of system (4. A will be selected and B will disappear. In particular this prevents extinction of the entire population. and present a simple model of a general viral infection in a host. A and B. The authors introduce this model to study the primary phase of HIV and SIV in the beginning of their book in virus dynamics.114) = N βy − γz. i. How does this effect the conclusions in Problem (68)? 4. 0. 2001]. In the case of lytic viruses. uninfected cells die at rate kx and become infected at rate kxz. quasispecies climb the mountains in the fitness landscape. (4. Drugs are being developed to exploit the error catastrophe. N represents the average burst size of a single infected cell. Uninfected cells are produced at rate λ. With mutation. Modify system (4. It follows from Problem (65) that the absence of mutation.108). However. Under the guidance of natural selection. y∗ . The basic idea is to use express the transmission coefficient. (4. from transmission models in terms of variables like x. y. the virus is cleared. z∗ ). Analyze the system (4. z∗ ) is a globally attracting equilibrium point.115) assuming that x is constant. y. An HIV infected patient’s T-cell count changes. y∗ . y. and whose derivative dV (x. the virus causes endemic disease. Thus V is a Lyapunov function and (x∗ . See [Antia et al. 2002]. 2.116) In other words. It also shows that the equilibrium point (x∗ . z) 6= (x∗ .4. but on a time scale of years. y∗ . Biologically.113)-(4. Linear stability analysis shows that the equilibrium point (λ/l.117) V (x. (Long term behavior) 1. Remark 2. z∗ ) is attracting for R0 > 1. 2007] prove that (x∗ . etc. y. Gilchrist and Sasaki. 2007] extends this analysis to multiple virus strains with mutations. death rate.119) (4. z) = (x − x∗ ) − x∗ log x∗ z β y (z − z∗ ) − z∗ log . . One of the exciting systems biology challenges in infectious disease modeling is to combine in-host models such as the above viral infection model with between-host transmission SIR-type models. z) dt ∂V dx ∂V dy ∂V dz + + ∂x dt ∂y dt ∂z dt l xy∗ z yz∗ x∗ = − (x − x∗ )2 − βy∗ + + −3 x x x∗ yz∗ y∗ z = (4. z∗ ). 0) is attracting for R0 < 1 and unstable for R0 > 1. and (4.118) − y∗ log y∗ N β z∗ which vanishes at (x∗ . Biologically. 0. (4. z) with y(0) > 0 or z(0) > 0 converges to (x∗ . [De Leenheer and Pilyugin. the second equilibrium point exists if R0 = (λ/l)/x∗ > 1. β kx∗ 107 . then the orbit of every positive initial condition (x.114). y.115). y. y∗ . If R0 > 1. Coombs et al.113). z) converges to (λ/l. The recent paper [De Leenheer and Pilyugin. y∗ . IN-HOST MODEL OF VIRAL INFECTION λ − lx∗ λ − lx∗ x∗ . 0. 1994.. They define the non-negative function x + (y − y∗ ) − (4. z∗ ) is globally attracting using a Lyapunov function.8.120) is negative for (x. (4. removal rate. z in (4. 0). 2007. y∗ . .. z∗ ). Theorem 10. then the orbit of every positive initial condition (x. Proof. If R0 < 1. Problem 72. 9 CHAPTER 4. α(B). susceptible individuals contact more pathogens than the infectious dose and become infected. If a healthy person ingests 100 V. and malaria and leishmaniasis. and R be the numbers of the susceptible. he will not become clinically ill with cholera. 1979.123) dt dB = π(B) + ξI (4. and Weitz. Disease occurs through transmission via direct contact with reservoirs containing human pathogens. indirectly increasing the transmittability of the pathogen to the susceptible. and infected individuals shed pathogens back into the reservoir. Nalin. 1981]. but non-differentiable terms in the differential equation model.5. If the in-reservoir pathogen density is above the MID. Wang. and the recovered. respectively. MODELING THE SPREAD OF INFECTIOUS DISEASES Case Study 3: iSIR model with immunological threshold We now discuss our recent epidemiological model of infectious diseases for which the primary mode of transmission is indirect [Joh.125) α(B) = a(B−c)n . Below. I. which are transmitted through contact with a contaminated reservoir. the immunological threshold is about 105 to 107 bacteria [Levine. and not via direct person-to-person contact. For obvious reasons. but we assume all population members possess the same ”average immunity. so the transmittability of the disease. Since S 0 +I 0 +R0 = 0. Let S. 2006]. the infected. Such diseases include cholera and schistosomiasis. The basis for explicitly modeling the MID is that the innate human immune system is capable of eliminating low levels of pathogens and staving off disease. We consider the natural family of transmittability responses ( 0. The value c reflects a combination of immunological and ecological factors.122) dt dR = δI − µR (4. Our model is described by the set of ODEs dS = −α(B)S − µS + µN (4. B<c (4. and Finkelstein. which complicates the analysis of the model. is an increasing function of B. like many infectious diseases. Cholera. Codeco. α(B) and π(B). we introduce a family of reservoir mediated SIR models with a threshold pathogen density for infection. and Levin. We define the threshold via the threshold pathogen density c by requiring that α(B) = 0 for B ≤ c. We denote B as pathogen density in a reservoir. n n (B−c) +H . B ≥ c. a higher pathogen density increases the chance that a susceptible individual becomes infected. Black. Faruque. the minimum infectious dose can be re-scaled as a threshold pathogen density for infection. Building upon prior epidemiological models of cholera [Capasso and Paveri-Fontana. Clements. we describe the functional terms. Mekalanos.108 4. We assume there is a minimum infectious dose (MID) of pathogens necessary to cause infection. The innate immunity of individuals varies. The disease gets amplified in the body. A key difference between this iSIR model and other SIR or indirect disease models is the explicit incorporation of a MID.” Assuming the contact rate to the reservoir is identical for every individual.124) dt The definitions of all parameters are explained in Table 4. We name these models iSIR models. Weiss. respectively. For most strains of cholera. 2008].121) dt dI = α(B)S − µI − δI (4. Jensen. cholerae bacteria. which are transmitted through contact with an infected insect vector. 2001. corresponding to human-pathogen contact and in-reservoir pathogen dynamics. Cisneros. The threshold results in continuous. the total human population N is conserved. has an immunological threshold. The term rK is the in-reservoir pathogen birth rate in the absence of shedding and 1/µ is the average life-span of a susceptible individual. ξN is the pathogen shed rate when all individuals are infected and 1/(µ + δ) is the average duration of infection. . respectively.9.4. The prevalence of pathogens in reservoirs suggests that there are stable steady states with positive pathogen densities but no infected individuals. The growth rate of the pathogen density. CASE STUDY 3: ISIR MODEL WITH IMMUNOLOGICAL THRESHOLD Parameter S I R N B α(B) π(B) µ δ ξ a c r K H Description Number of the susceptible Number of the infected Number of the recovered Total population Pathogen density in a reservoir Transmittability Pathogen growth rate Per capita human birth or death rate Recovery rate Pathogen shed rate Maximum rate of infection Threshold pathogen density for infection Maximum per capita pathogen growth efficiency Pathogen carrying capacity Half-saturation pathogen density 109 Dimension cell liter−1 day−1 cell liter−1 day−1 day−1 day−1 cell liter−1 day−1 day−1 cell liter−1 day−1 cell liter−1 cell liter−1 Table 4.5: Model parameters where n is a positive integer. we interpret this dimensionless number as the ratio of two factors: (i) the average number of pathogens shed over the time course of infection if all individuals were infected. We assume that pathogens exhibit logistic growth. which plays a role analogous to R0 for models of directly transmitted infectious diseases ξN µ ζ= . We call this the pathogen enhancement ratio. We introduce an important threshold parameter ζ. π(B) = rB(1 − B/K). ξN/(µ + δ). π(B). The first term. The Holling’s Type II and III functional responses [Holling. (4. Hence.126) µ+δ rK From (4. Here we analyze the threshold model with Holling’s Type II functional response. Without human hosts. is the natural in-reservoir growth rate of pathogens in the absence of human hosts. We now state our main technical result. and (ii) the average number of pathogens reproduced in the reservoir over the time course of an uninfected individual. Pathogens might be free-living or exist on a variety of zoonotic hosts. 1959a] correspond to cases of n = 1 and n = 2.124).. represents the total number of pathogens shed into the reservoir during the average period of infectiousness assuming all individuals were infected and there was no feedback. the pathogen density will reach its carrying capacity. It follows that for almost all initial conditions. B0 ) is attracting with I0 > 0. Corollary 1. If c > K. and there exist up to two additional equilibrium points. For larger ζ. 2. MODELING THE SPREAD OF INFECTIOUS DISEASES Theorem 11. there are two equilibrium points: (N. B). (Equilibrium points and asymptotic behavior of solutions) 1. the equilibrium point (N. I(∞) = 0. If c < K. 0. K) is attracting. I0 . there are two additional equilibrium points. 0) is a saddle and (S0 . I1 . and (S2 .110 CHAPTER 4. B1 ) is a saddle. I2 . the disease becomes endemic. For sufficiently small ζ. Hence. In this case. the disease becomes endemic. and the solution for almost every initial condition converges to (N. 0) is a saddle. there are no additional equilibrium points. B2 ) is attracting with I2 > I1 > 0 . the solution for almost every initial condition converges to one of the two attracting equilibrium points. for almost all initial conditions. (Zero threshold case) The special case where there is no immunological threshold occurs when c = 0. Thus. and asymptotically the population becomes either disease free or the disease becomes endemic (depending on initial conditions). B0 ). I0 . 0. (S1 . after a saddle node bifurcation. (N. and the population becomes asymptotically disease free. 0. 0. It follows that the solution for almost every initial condition converges to (S0 . . Human activities have been increasingly fragmenting natural habitats and creating metapopulations from previously continuous populations. There is growing interest in creating population and transmission models that relax the well-mixing assumptions. [Levins.1) occupy patches of high quality habitat and use the intervening habitat only for movement from one patch to another. Good references include [Hanski. network models. Although this is almost never the case.Chapter 5 Spatial Population Models Until now. 1969] introduced the following simple model of a metapopulation. and neglect local population dynamics. van Nouhuys]. For example. etc. t + ∆t]. this assumption allows us to model populations using ODEs or difference equations. There is no consideration of fitness. Let e denote the local extinction rate and c the colonization(migration) rate. the probability that an unoccupied patch becomes colonized is proportional to the fraction of occupied patches p(t).” More precisely. 1991. reaction-diffusion PDE models. There is limited migration from patch to patch. An occupied site can have one individual or 1010 individuals. we have been assuming that all interacting populations are spatially homogeneous and well-mixed. it does not matter.. The model assumes that during this time period the probability that a colonized patch goes extinct is e∆t. During the time interval [t. selection. The use of p rather than population size N is a dramatic departure from classical population dynamics. Movement in space is only implicit.1 Metapopulation models A metapopulation is “a population of populations that go extinct locally and recolonize. van Nouhuys. 1999. local populations connected by occasional dispersal between local populations. Thus the metapopulation concept has now become one of conservation biology’s predominant paradigms. and thus we expect that p(t)e∆t colonized patches go extinct. The model also assumes that during this time period. 2001] is considered the starting point for thinking about population dynamics with implicit spatial structure. some patches that were inhabited at time t go extinct and some patches that were vacant become colonized. 5. 111 . it is a set of relatively isolated. spatially distributed. Let p(t) denote the fraction of occupied patches at time t. We briefly introduce four main classes of popular models: metapopulation models. Hanski and Gilpin. and agent based models. The theory of island biogeography [MacArthur and Wilson. with finite life spans. Metapopulation models consider local populations as individuals. 1991. mountain sheep in Southern California (see Figure 5. Thus p(t + ∆t) − p(t) = (1 − p(t))p(t)c∆t − ep(t)∆t. Some investigators have started using island biogeography and metapopulation models to study microbial populations [Crump et al. Problem 73. The metapopulation model is spatially implicit and easy to analyze rigorously. the dotted lines show fenced highways [Bleich. and thus for many types of predictions about patterns of species abundance. Thus the metapopulation will persist. a mainland. and Holl. Setting rp(1 − p/K) = 0 and solving for p yields the equilibrium proportion of colonized patches p = 0 and p = 1 − (e/c). Heidi.. the metapopulation will go extinct. For this reason it is a popular modeling tool to study spatial population problems. 2004.1) dt Equation (5. Kimura argues that genetic drift is the main determinant of evolution and not Darwinian selection. In other words. Island biogeography and metapopulation models. and taking the limit. Wehausen. if e < c and will go extinct if c ≤ e. Dolan. If dispersal is too infrequent. where r = c − e and K = 1 − (e/c). and thus the probability is p(t)c∆t. arrows indicate documented intermountain movements. ´ selection can be neglected. despite frequent local extinctions. immune to extinction. Heidi et al. Suppose that d% of the patches become uninhabitable: if species attempt to colonize an uninhabitable patch they die. Thus this model shows that migration between local populations promotes the persistence of a regional metapopulation. the metapopulation will persist provided the relatively long distance dispersal events occur sufficiently often to provide for recolonization of patches that have suffered local extinction. assuming each tree is equally likely to reproduce whatever its species. we obtain Levins’ ODE dp = cp(1 − p) − ep. 1985]). 1990]. each gene is equally likely to enter the next generation whatever its allelic type. is the sole source of colonists to the surrounding patches. . (5. Use this model to test the consequences of habitat destruction. In this theory of evolution. Benjamin. We would then expect that (1 − p(t))p(t)c∆t unoccupied patches become occupied in this time interval. 2005. for which fitness and selection play no role. meaning p > 0.1) can be rewritten as p0 = rp(1 − p/K). 2004] devised a similar neutral theory for forest ecology.1: Patches of mountain sheep. In island biogeography. and Sam. 2007]. SPATIAL POPULATION MODELS Figure 5.. Warning: many population papers with “spatial” in the title use metapopulations. Hubbell [Hubbell and Borda-de Agua. How does this effect the equilibrium proportion of colonized patches? Problem 74.112 CHAPTER 5. Shaded areas indicate mountain ranges with resident populations. Modify Levin’s model to this case with a single species. This is a logistic ODE. are precursors to the neutral theory of evolution (Kimura [Kimura. Ramette and Tiedje. (5. dt (5. Do populations.6) ∆t (∆x)2 = . standing waves.g.3) pn+1 (k) − pn (k) = pn (k − 1)(1/2) + pn (k + 1)(1/2) − pn (k) (5.. 1994] for details. e. Then pn (k) satisfies the difference equation pn+1 (k) = pn (k − 1)(1/2) + pn (k + 1)(1/2). which says that if u and v are solutions of equation (5.5. and spiral waves. x) denotes the population density at point x at time t.2) is stable if the spectrum of the linear operator 4 + df (u∗ ) lies in the left-half plane. The choices of right or left steps are independent at each step.2 113 Reaction-diffusion PDE equation models There is a growing literature that spatially extends the population models presented in previous sections by adding a diffusion term to account for animals moving in space. tend to steady states like for many of the ODE models? Not necessarily. The main tool for analyzing reaction-diffusion equations is the comparison principle. since random diffusion was believed to homogenize any inhomogeneities and lead to homogeneous solutions. an infinite dimensional version of Poincar´e’s method of linear stability analysis from ODEs. These were unexpected.2). then u ≤ v on D.2. f (u) models the local population growth. 1972] discovered stable. 1993].2.1 The diffusion equation We begin by deriving the diffusion equation as a special scaling limit of a simple symmetric random walk. (5. See [Smoller. [Segel and Jackson.4) We can rewrite this expression as 1 (pn (k + 1) − 2pn (k) + pn (k − 1)) .5) 2 Dividing both sides by ∆t. These special solutions of reaction-diffusion equations were discovered by Turing (see [Allen. Imagine a (drunk) animal walking along the x-axis. However. Do animals really move by diffusion with uncorrelated steps? Of course not. We warn the reader that the analysis of PDEs is significantly more sophisticated than the analysis of ODEs. imposing the diffusive scaling (limit) condition (∆x)2 = 2D∆t.2) can be traveling waves. REACTION-DIFFUSION PDE EQUATION MODELS 5. 2007] for an elementary mathematical presentation). The single species models are of the form du = f (u) + D4u. We then present a couple of spatial population models where movement by diffusion coupled with exponential or logistic population growth provides a reasonable fit to data.2) where u(t. spatially inhomogeneous patterns of standing waves in population models of plankton. Several authors have studied spatial population models with more sophisticated movement terms. and if u ≤ v on the boundary ∂D of a domain D. We denote pn (k) the probability that after n∆t steps the animal is at site k∆x. Another useful tool for proving stability of solutions is the method of linearization. and 4u denotes the Laplacian of u. and recalling the second difference limit definition of second derivative yields pn+1 (k) − pn (k) pn (k + 1) − 2pn (k) + pn (k − 1) =D . Qualitatively and quantitatively the results are not very different [Holmes. During every time interval ∆t the animal moves distance ∆x to the right with probability p = 1/2 and moves distance ∆x to the left with probability q = 1/2. (5. 5. now moving randomly in space. Solutions of equation (5. starting from correlated random walks. which says that a time-independent solution u∗ of equation (5. this first approach to modeling animal movement is not as foolish as it may first appear. (5. If we start with a collection of animals on the real line.7) for t > 0 and n(0. and John. this corresponds to the Dirac delta initial condition u(0.7) with respect to x [Tikhonov. x. Suppose an animal is at the origin at time 0.10) ∂t ∂x geometrically as saying that the rate of population density change is proportional to the “curvature” of population density. (5.13) hdi = s u(t. SPATIAL POPULATION MODELS The constant D is called the diffusion coefficient. s2 = 2 x + y 2 . then at time t the expected density of animals at position x is given by Z ∞ (x − v)2 1 exp − n0 (v)dv. ∆x → 0 such that (∆x)2 = 2D∆t leads to the diffusion equation ∂u ∂2u (5. s)2πsds = πDt 0 and hd2 i = Z ∞ s2 u(t. 0 The variance of the distance is hd2 i − hdi2 = 4Dt − πDt. The PDE is 2 ∂n ∂ n ∂2n =D + = D4n. Mathematically. y) is a radial function since it depends on s2 = x2 + y 2 . If it undergoes diffusion. x) = √ 4Dt 4πDt −∞ One can check that this function solves (5. what is its expected distance. x.114 CHAPTER 5. x) = exp − 4Dt 4πDt Note that u(t.8) u(t. x) = δ(x). Arsenin. Taking the special scaling limit as ∆t. Note also that u(t. y) = 4πDt 4Dt Note that u(t. The fundamental solution is 1 x2 √ . y) is the joint normal probability density function with mean 0 and variance 2Dt. One can interpret the diffusion PDE ∂n ∂2n =D 2 (5. from the origin at time t? Straightforward calculations yield that the mean distance and mean distance squared are Z ∞ √ (5. x) is the probability density function for a normal random variable with mean 0 and variance 2Dt. There are several ways to obtain the fundamental solution. given that at time t = 0 is was at the origin. One can derive the two-dimensional diffusion equation in a completely analogous way from a random walk on Z2 .7) = D 2. including taking the Laplace or Fourier transform of (5. x. x) = n0 (x). which at time t = 0 has density n0 (x). (5. ∂t ∂x where u(t. 1977]. s)2πsds = 4Dt. The population density increases where the curvature is positive and decreases where it is negative.14) .9) n(t. x) denotes the population density at point x at time t.12) u(t. A fundamental solution of the diffusion equation gives the probability density of finding the animal at position x at time t > 0. (5. (5.11) ∂t ∂x2 ∂y 2 which has the closed form fundamental solution 2 x + y2 1 exp − . where in 1905 a landowner released several muskrats in the wilderness of Bohemia. there exist careful records of the spread of the muskrats (see Figure 5.12) and the fundamental solution is x2 + y 2 1 exp at − . then at time t the expected density of animals at position (x. The muskrat population grew rapidly and dispersed widely over central Europe. in time. If we start with a collection of animals on R2 which at time t = 0 have density n0 (x. Fortunately. (5. was brought to Central Europe. However.17) to the threshold nT and obtain r 4D r(t) = 4aD − log (4πnT Dt/n0 ). Solutions of this PDE are not waves in the strict mathematical sense.2 Skellam’s model and the European invasion of muskrats We now add a smooth local population growth term f (n) to our diffusion model and obtain a reaction-diffusion equation ∂n = D4n + f (n).17) u(t. a native rodent of North America. x) is given by (5. the population density expands out to infinity. Skellam defined an “invasionfront” and considered the asymptotic propagation speed of the front. (5. (5. and allows a closed form solution (♦ Verify this!). then at time t the expected density of animals at position x is given by n(t.15) ∂t We first consider the diffusion equation with exponential local population growth ∂n = D4n + an.16) ∂t The simple change of coordinates u0 = u exp (−at) transforms (5.16) into (5. Skellam] attempted to fit this data using the above model. y) = n(0) exp at − . about 50 km southwest of Prague. In the early 1950s.20) t t .2. Show that an invading population eventually becomes extinct.19) exp at − 4πDt −∞ −∞ 4Dt Modeling a muskrat invasion The muskrat. REACTION-DIFFUSION PDE EQUATION MODELS 115 Problem 75. Thus u0 (t. below which the population density is so small it can not be measured. Set the population density in (5. (5. y) = 4πDt 4Dt If we start with n(0) animals at time t = 0 all located at the origin.2(a)).2. and he considered the speed of how this level set propagates through space. He let nT be the threshold population density. (5. 5.18) 4πDt 4Dt Notice the radial dependence of n = n(r).7). w)dvdw. y). y) is given by 1 x2 + y 2 n(t.5. (5. x. [Skellam. r = x2 + y 2 and that. x. x. y) = Z ∞Z ∞ 1 (x − v)2 + (y − w)2 n0 (v. which yields a diffusion coefficient of about 12.22) for > 0. y) −→ t→∞ ∞ √ if r > (2 Da + )t √ if r < (2 Da + )t. For the muskrat invasion. In 1957. were accidently released near Rio Claro. Use the Skellam model to estimate the speed of the invasion front.17) to show that ( 0 n(t. [Okubo. Africanized honey bees. Problem 77. Thus this model predicts that for large time the invasion front expands radially at speed 2 aD. 1980] estimates the exponential growth rate as 2. . Africanized honey bees had spread from Brazil south to northern Argentina and north to Texas. and their descendants have since spread throughout the Americas.116 CHAPTER 5.2: (a) Invasion region by year (b) Invasion speed [from [Skellam. or killer bees. Hence if one travels in a straight line away from the origin with a velocity larger (smaller) than √ 2Da. Arizona. x. (5. as predicted by the model. Following the release. S˜ ao Paulo State. a and D. Florida. Although the actual invasion fronts from the muskrat invasion were not true circles. there is no reason such √ a limit needs to exist. SPATIAL POPULATION MODELS Figure 5. then over long time one will observe zero (infinitely many) animals. Problem 76.65 per year. As of 2002.2 km2 /yr.21) Note that a priori. Since the area of a cirlce A = πr2 . Use Equation (5. the model predicts that the area of invasion (from the p origin up√to the threshold contour) A(t) = πr2 ≈ 4aDπt2 . t (5.2(b)) does convincingly show that the area of invasion grows linearly in time. This implies that the speed of the front is 2 aD = 11. the African queens eventually mated with local drones. Both parameters. regardless of the threshold of detection.4 km/yr. Compare this speed with the invasion speed of muskrats. can be estimated with independent experiments.16 and thus 2 aD = 32. the data (see Figure 5. New Mexico. and southern California. √ For the slope of the best fit line through the invasion data √ we estimate that 2 πaD ≈ 20. and thus the square root of the area on invasion A(t) ≈ 2 πaDt grows linearly in time. Skellam]] Then the asymptotic spead of the invasion front is lim t→∞ √ r(t) = 2 aD.2 (km/yr) . 5.24) The solution connects the constant solutions.23). Problem 78.2. x) = 0 and n(t. collared turtle dove. Lubina and Levin. Show that the unstable eigenvector of the Jacobian matrix at (K. and Okubo. 0) and (K. and which connects two constant solutions at plus and minus infinity. Clearly n(t. 1974. cereal leaf beetle. house finch. one branch of the unstable manifold of the saddle emanates into the third quadrant. Thus a traveling wave can be viewed as a bounded solution representing the progressive replacement of one steady state (ahead of the front) by another (behind the front). 1937] produced a rigorous proof.23) and showed heuristically the existence in one spatial dimension of a traveling wave solution that describes the spread of the allele in a population. Kareiva. The traveling wave solution is very different than fundamental solutions of the diffusion and Skellam equations where the population density peaks at the point of introduction and decreases exponentially in time. Petrovsy. Humphrey.25) = −cv − a(1 − ψ/K)ψ. (5. one makes the ansatz that the solution n(t. ∂t (5. to obtain the ODE D d2 ψ dψ +c + a(1 − ψ/K)ψ = 0. and which. He introduced and studied the diffusion equation with logistic local population growth ∂n = D4n + n(1 − n/K)n. Metz. . Unlike the Skellam differential equation. if viewed in a frame traveling at the wave speed. appears constant. and Diekmann. Bosch. Prove that the equilibrium point (K. This second order nonlinear ODE can be rewritten as the first order system dψ dy dv D dy = v (5.2. and thus by the stable manifold theorem. 2 dy dy (5. Himalayan thar. A traveling wave is a wave with constant shape and speed in time. 1. 2. To find the traveling wave solution in one spatial dimension with speed c > 0. 0) is an attracting spiral for c2 < 4aD and an attracting node for c2 > 4aD. grey squirrel. Fisher studied the spread of an advantageous allele in a population. x) = ψ(y). Levin. Later that year [Kolmogorov. and Piskounov. this PDE has no non-trivial closed form solutions. REACTION-DIFFUSION PDE EQUATION MODELS 117 See [Andow. 1988] for further examples (including the cabbage white butterfly. and substitutes this into Equation (5. Prove that the equilibrium point (0. and California sea otter) and more modeling details.5. 0). 1990. 1990. The Fisher model is the cornerstone of many spatial biological models. 0) has both coordinates of the same sign. 0) is an attracting node for c2 < 4aD and a saddle for c2 > 4aD. x) = K are constant solutions. European starling. where y = x − ct.26) There are two equilibrium points: (0. so that ψ(∞) = 0 and ψ(−∞) = K.3 The Fisher model and traveling waves In the 1930s. This traveling wave solution is also stable with respect to local perturbations [Bramson.21). and Brown.27) ∂t where R denotes the Gaussian curvature and r denotes the average curvature. x). and by the Poincar´e-Bendixson theorem must approach (0. but infected foxes become aggressive and loose their sense of direction . Aronson and Weinberger. Problem 80. Thus for each c > 2 aD there is a traveling wave. x) is nonnegative. 0) is an attracting spiral. v(y)) = (K. x) evolves into a traveling wave solution with speed 2 aD. Show that the divergence of the vector field is negative. This viral disease is almost 100% fatal. We note that the Fisher equation is quite similar to the PDE describing how the Ricci flow changes the curvature of metrics on surfaces ∂R = 4R + R(R − r).118 CHAPTER 5. √x) is continuous in x1 < x < x2 .2. 0). 5. u(0. The Ricci flow is the main tool in Perelman’s proof of the Poincar´e Conjecture. 1986] began the study of using reaction-diffusion PDEs to model the spatial spread of infectious diseases. which would correspond to a negative population density. Thus a necessary condition for a traveling wave solution is for the wave speed c2 > 4aD. the existence of such a traveling wave corresponds to the existence of a heteroclinic connection between the two equilibrium points. has compact support. 0). but they are not bounded functions. −1 ≤ v ≤ 0}. [Aronson and Weinberger. Show that there are traveling wave solutions to the diffusion equation and the Skellam equation. Healthy foxes tend not to travel very far. then the solution u(t. and conclude from Dulac’s theorem that there are no closed orbits. This proves the existence of a heteroclinic connection between the two equilibrium points. 1983]. 1978] showed that if n(0. 0). In other words. v). √ We conclude from Problem (79) that wave speed c√> 2 aD is both a necessary and sufficient condition for the existence of a traveling wave. Hint: Assume there is and obtain a linear ODE with constant coefficients for the wave profile. rabies is spread mostly by foxes. When the equilibrium point (0. The requirement that the traveling wave connect the two constant solutions translates into the requirement that limy→−∞ (ψ(y). In England. x) = 1 for x < x1 and u(0. √ Is the asymptotic wave speed equal to 2 aD? The answer depends on the initial condition n(0. SPATIAL POPULATION MODELS 3. Construct a triangular trapping region consisting of the intervals {(u. x) = 0 for x > x2 . Argue that the branch of the unstable manifold of the saddle emanating into the thrid quadrant must stay trapped in this trianglar region for positive time. (5. Stanley. Problem 79. 0) and limy→∞ (ψ(y). any forward asymptotic orbit would be forced to enter the negative u plane. This minimum wave speed coincides with the asymptotic wave speed in (5. where x1 < x2 . v(y)) = (0. Similar results hold for two spatial dimensions with the same asymptotic speed. and u(0. 0 ≤ u ≤ 1} and {(1.4 Modeling the spatial spread of rabies [Murray. We stress that the asymptotic invasion speed in the Skellam and Fisher models coincide. Solve it and show that no solutions are bounded for all time. x) = S(y). bacteria compete as colonies. 1986] for more details. the system becomes dS dt dI dt = −SI (5. (5. After nondimensionalizing (♦ carry out the details!).30) where D is the diffusion coefficient of rabid foxes. y = x − ct.35) The authors then linearize (5. Murray proposed the following model for the spatial spread of rabies dS dt dI dt dR dt = −βSI (5. in physically structured habitats. (5.32) where λ = 1/R0 . the capacity to direct their course towards nutrients or/and away from toxins. (5. dt2 dt (5.34) (5. REACTION-DIFFUSION PDE EQUATION MODELS 119 and appear to travel in random directions.2. The first is chemotaxis. See [Murray. and search for a traveling wave solution of the form S(t. Murray.35) at the leading edge of the front S = 1 and I = 0 to obtain the linear ODE d2 I dI +c + I(1 − 1/R0 ) = 0.5 Why do bacteria move? (ADD MUCH MORE) We have been working with Yan Wei and Bruce Levin at Emory University to study why bacteria have and maintain the flagella. like semi-solids.2.5. I(t. 2 dt dt c (5. The second is to avoid predation by protozoa. Show that a necessary condition for I to be always non-negative is that R0 > 1 which corresponds to wave speeds c > 2(1 − 1/R0 )1/2 . In accord with this hypothesis. and Brown.36) Problem 81.28) = D4I + βSI − νI (5.29) = νI. With the aid of experiments and reaction diffusion models we develop and explore the properties of a third hypothesis for the motility of bacteria: competition for resources in physically structured habitats.31) = D4I + SI − λI. motors and other machinery to be motile? Currently there are two hypotheses with experimental evidence in their support. Stanley. rotifers.33) satisfying the boundary conditions S(−∞) = S(∞) = 1 and I(−∞) = I(∞) = 0. Substituting into (255) and (266) they obtain the system of ODEs dS = SI dt 2 d I dI +c + I(S − 1/R0 ) = 0. Because of . x) = I(y). 5. 2003. At first Murray assumes that space is one dimensional. nematodes and the like. The spatially extended spruce budworm population model [Ludwig. and Murray. (5. colonies of motile bacteria increase the surface area and thus their collective capacity to sequester diffusing resources at a greater rate than non-motile cells.6 Chemotaxis (write this) 5.2.2.1).39) where Bk is the population density of strain k and N is the nutrient density. Our model is the following system dB1 dt dB2 dt dN dt αN B1 K +N αN = D2 4B2 + B2 K +N N = DN 4N − ν (B1 + B2 ).41) = DN2 4N2 + r2 N2 (1 − N2 /K2 ) − α21 N1 N2 .25 for ODE version) dB B βB 2 = D4B + rB 1 − − 2 P. Williamson. SPATIAL POPULATION MODELS the movement of the bacteria within them. .120 CHAPTER 5. The results of our experiments support this hypothesis and are consistent with the predictions of the model.38) (5. ADD MUCH MORE HERE 5.40) 2. Maini. Using motile and non-motile strains of E.42) See [Okubo. 1989] for a discussion of traveling wave solutions and applications to the spread of the invasive grey squirrel in England.37) (5.7 Further examples of spatial population models We now briefly mention a few additional examples of spatially extended models 1. (3. In this way motility provides these bacteria a competitive advantage over less motile cells. The spatially extended Lotka-Volterra competition model (see Equations (3. 1979] (see Equation 2. coli K-12 in we test the hypothesis that motility provides bacteria a competitive advantage in physically structured habitats (soft agar) that they do not have in mass (liquid) culture.2) for ODE version) dN1 dt dN2 dt = DN1 4N1 + r1 N1 (1 − N1 /K1 ) − α12 N1 N2 (5. and Weinberger. K +N = D1 4B1 + (5. dt KS N0 + B 2 (5. Aronson. coolantarctica. However. There has been a great deal of research on identifying the network structures that occur in nature and studying what these imply about the ecosystem dynamics.com/Antarctica%20fact%20file/wildlife/ whales/foodweb. Using the theory of random matrices. but the individual populations less stable [Tilman. etc. and susceptible individuals do not all face the same .3 121 Network models Networks are one of the hottest topics in biology.2. In the real world.3. 1999]. Figure 5. 2006] for a rational assessment of the role of networks in population modeling. In the 1970s. in the sense that those with more species and more complicated network interactions are more robust and better able to withstand shocks. The current thinking on this topic seems to be that more species make the community more stable. NETWORK MODELS 5. A major driving question has been whether more complex ecosystems are more stable. which study complex interactions in biological systems determined by networks.gif] Modeling infectious diseases on networks ODE models of infectious diseases assume the populations are well-mixed. Many universities are creating institutes for systems biology. The role of the network structure is still poorly understood. real ecosystems have evolved over many years and are not randomly constructed networks. The relationships between species in real ecosystems are much more complex than the simple A eats B eats C food chain in Section (3.3) illustrates a simplified Antarctic food web.6). Everybody has heard of small world networks and the six degrees of separation between people. Figure (5. [May.3: A simplified Antarctic food web [from www. predation. he proved that large randomly assembled community models tend to have unstable equilibrium points. I recommend the opinionated survey [May. each individual has contact with only a small fraction of the entire population.) between them. 2001] studied Lotka-Volterra models with many species and random interactions (competition. A food web extends the concept of food chain from a simple linear pathway to a network of interactions between species.5. The interactions can take place at home. and went directly to the hospital. shopping. 2005] provide an easy and convincing heuristic that if pd > p2r (1 − pn ) .43) that an epidemic requires long range interactions of disease transmission. . When this happens. If each node has degree 2. To account for infection through occasional long range contacts. the index case was a woman who shared a home with a large. isolated. The model has discrete time steps ∆t. and with sufficiently many simplifying assumptions about the network structure. During the 2003 SARS outbreak in Canada. A transmission model can be defined on any contact network via (1). are analytically tractable. We note that this network is not scale free and not small world. and Brunham. Such models seem more realistic than SIR models. the node will never transmit the infection to its neighbors. (2). She died at home. the disease progressed very differently in Toronto and Vancouver. an infected node will randomly select a susceptible node anywhere on the network and infect it with probability pd . The basic reproductive number R0 . Figure 5. at work. In Toronto. 2005]. 2pn (5. It is believed that 209 SARS infections resulted from this index case. or removed state. Especially at the beginning stages of a disease. on a bus. and the distribution of degrees is a fundamental characteristic of a network. The contact patterns of the first few infections are believed to play a major role in determining whether an epidemic will occur [Meyers. An investigator studying the spread of a sexually transmitted disease in this high school could create an SIR network model using this network.43) then the number of infected individuals will start to grow exponentially. 2005]. plays a less important role when studying disease transmission on networks. In Vancouver. An infected node will recover during ∆t with probability pr . on an airplane. Pourbohloul. was a man who lived with only one other person.122 CHAPTER 5. The number of edges emanating from a node is called the degree of the node. and treated. We now briefly discuss a simple discrete time SIR model on a small world network [Saram¨aki and Kaski. etc. and (4) above. 3. Each node is in either a susceptible. at school. a long range transmission is likely to start a new infective cluster. The nodes of a contact network are individuals and the edges represent interactions between the individuals. SPATIAL POPULATION MODELS risk of becoming infected. where he was diagnosed. The reason is the effect of a secondary infection caused by a nearest neighbor transmission is quite different than an infection caused by a long range transmission. It is believed that only four others in Vancouver were ultimately infected. It follows from (5. [Saram¨aki and Kaski. Skowronski. undiagnosed. the first infected person (index case). Newman. while a nearest neighbor transmission will only expand an existing cluster. 4. defined as the average number of secondary infectious caused by each primary infection. in a hospital.4 shows the network of romantic and sexual relations at a midwestern US high school. An infected node will independently infect each susceptible neighbor (connected by a single edge) during time interval ∆t with probability pn . infected. 2. 1. An elderly person living alone at home is much less likely to come into contact with an infected person than a young adult who works in a large office building or a child who attends a large school. multi-generational family. Models using social or contact networks are nothing new in the sexually transmitted disease modeling world. We call two nodes neighbors if they are connected by an edge. He interpreted the surprising result of his model as saying that.6) on five different types of networks (see Figure 5. local resources. not mathematical models. individuals (frequently called agents) explicitly interact with each other and with their environment. In the words of one of the leaders in the ABM field. even with relatively mild assumptions on each individual’s nearest neighbor preferences.4 Agent based models In ODE models.” The economist and 2005 Nobel Laureate Thomas Schelling developed an early ABM by moving pennies and dimes on a chessboard according to certain simple rules. Many ABMs have a huge number of parameters and their details are almost impossible to communicate in any reasonable way. health. Using computer simulations.. I have written a few papers on the dynamics of the Schelling model [Gerhold. Pollicott and Weiss. 2008] compare features of epidemics (see Figure 5. Thus ABMs can provide insights on how individual variability and the interactions of individuals with each other and the environment lead to population or community outcomes. social status. even if all individuals prefer integration. Weiss. Gerhold et al. ABMs are discrete dynamical systems. etc. and can move in space. Schneider. you don’t know it. . In principle. Glebsky. Joshua Epstein [Epstein.5. and can be studied using the tools of dynamical systems. individuals are only implicit: subpopulations interact with other subpopulations. etc. age. The extra complexity significantly increases computational requirements and limits the ability to conduct sensitivity analysis. as well as space variations including habitat. assuming massaction mixing. and Zimmermann. roads. “If you can’t grow it. an integrated city would likely unravel to a segregated city. One can simulate (grow) and study virtual populations and ecosystems.4. ABMs can incorporate individual variations in sex. topography. AGENT BASED MODELS 123 Figure 5.5). ABMs are computer programs. 5. The underlying assumptions are frequently so specific that ABMs can never lead to any general understanding on the population level. Notice that the profile of the epidemics on the spatial and scale-free networks are quite similar. as well as for the lattice and small-world networks. This is one way to bridge the gap between two modeling philosophies.” To their detractors. In agent based models (ABMs). [Keeling and Rohani. And in the words of a leading population expert. I don’t have time to study virtual populations.4: Each circle represents a student and lines connecting students represent romantic relations occuring within the 6 months preceding the interview. “I have enough trouble studying real populations.. size. 2007]. 5: Five different types of networks containing 100 individuals. lattice.124 CHAPTER 5. and small-world [Keeling and Rohani. From left to right and from top down: random. SPATIAL POPULATION MODELS Figure 5. 2008] . spatial. scale-free. spatial.6: Epidemics occurring over five different types of networks containing 100 individuals.4. lattice. scale-free. From left to right and from top down: random. and small-world [Keeling and Rohani. AGENT BASED MODELS 125 Figure 5. 2008] .5. Longini Jr. Riley. At each time step a lattice site is randomly selected. The state of a site at time n + 1 depends on the state of the site at time n along with the state of neighboring sites at time n. 2002] construct an agent based model to explore the role of spatial scale in maintaining coexistence in this community. it grows much faster than strain (C) and drives (C) to extinction in a culture containing only the two bacteria.]. A beats B. Strain (S) outcompetes strain (R) because it is better able to absorb nutrients. The system “selforganizes” with no central control. ABM model for competition between strains of E. but C beats A. and Chakravarty. Like them or not. S. I have also been working to take a hydrodynamic limit of the Schelling process to obtain a PDE. Initially. in a similar spirit of obtaining the heat equation as a hydrodynamic limit of a symmetric random walk. 2003]. the number of ecology papers using ABMs has been growing exponentially. and are studying our biosphere (the union of all ecosystems on the earth) as a CAS [Levin. and is also immune to this toxin. 2003]. . R}) is given by fi . 3. and Bohannan.126 CHAPTER 5. and pandemic flu [Germann. strain (R) is immune to the toxin. or E (empty). physical. If an empty lattice point is selected for updating. SPATIAL POPULATION MODELS 2001. and social sciences. such as smallpox [Epstein. coli There have been several recent discoveries of communities exhibiting the rock-paper-scissors game relationships between populations.. In a CAS. According to [Grimm. the fraction of its neighbors having strain i. i. The size of the square neighborhood determines the spatial scale of the interactions. Singh. S. movement. 1999]. and Anderson. The phase space for the agent based model is a 250 × 250 square lattice with periodic boundary conditions. either actually or metaphorically. and Weiss. the use of ABMs is increasing in the biological. Some leading ecologists believe that CASs offer an integrative approach to studying ecology. the probability that it gets replaced with a cell of type i (with i ∈ {C. Production of the toxin requires considerable cell resources. It has been hypothesized that competing species can coexist if ecological processes such as dispersal. The Schelling model is also a prototypical example of a complex adaptive system (CAS). strain (S) is sensitive to the toxin and gets killed when exposed to it. Thus it makes sense to write S > R > C. 2001]. coli bacteria containing the following three strains: 1. and compare their model’s predictions with results from laboratory experiments. Singh et al. this would form another bridge between the two modeling techniques. Feldman. The authors [Kerr. and drives it to extinction in a culture containing only the two bacteria. strain (C) outcompetes strain (S) by killing it with the toxin. every lattice point is randomly and independently assigned the state C. but can not produce it. Donnelly. strain (C) can produce the toxin colicin. 2006]. Cummings. and interaction occur over small spatial scales. R. foot and mouth disease [Ferguson. 1.e. is known as the Gaia hypothesis. However. but C > S. The concept that the biosphere is itself a living organism. Vainchtein. 2. and Macken. and because strain (R) does not need to produce the toxin and is immune to the toxin. If an occupied site in state C is selected. If we are successful. it is killed with probability δC . 2. One of the central aims of ecology is to identify mechanisms that maintain biodiversity. the emergence of macroscale behaviour results from the local interactions of the individual parts. ABMs have provided new insights and strategies to control infectious diseases. B beats C. Kadau. One example occurs in cultures of E. 4. and Bohannan.5. There is no reproduction in this model. If an occupied site in state S is selected. 4. Figure 5. Subfigure (c) shows the total distribution of the three states over the same small neighborhood simulation of 5000 time steps. subfigure (d) shows the total distribution of the three states over the entire lattice neighborhood simulation of 5000 time steps. 2002] illustrates their results. There appears to be no convergence to a limit state. If an occupied site in state R is selected. Feldman.0 + τ fC . Figure 5.7: Agent based simulations of three strains of E. it is killed with probability δR . it is killed with probability δS. By contrast. AGENT BASED MODELS 127 3. where δS0 denotes the probability of death of an S cell without any C neighbors. Riley. Notice that the sensitive strains goes extinct after about 100 steps and the toxin producing strain goes extinct after about 480 steps. The investigators run simulations where the neighborhood size ranges from the smallest eight nearest neighbor sites (where dispersal and interaction are completely local) to the entire lattice (where dispersal and interaction occur over large scales). coli . Subfigures (a) and (b) show the result of simulations with the eight point neighborhood at 3000 and 3200 time steps. and fC is the fraction of its neighbors having strain C. respectively.7 from [Kerr. τ denotes the toxicity of C cells. the authors combine the three strains in a static plate culture where bacteria only interact with their nearby neighbors. whereas all populations coexist when ecological processes are localized. and on a well shaken flask which is close to well-mixing of the three strains. Finally. SPATIAL POPULATION MODELS In the laboratory. The laboratory results are shown Figure 5. we note that there is no explicit movement of agents in this model. which qualitiatively resemble the corresponding results for the agent based simulations in Figure 5. Figure 5.7.8.128 CHAPTER 5.8: Results from laboratory experiment of three strains of E. coli . The authors conclude that diversity is rapidly lost in experimental communities when dispersal and interaction occur over relatively large spatial scales. Suppose that (x∗ . y).3) We call (x∗ . 0). y ∗ ) d x x = J(x∗ . 0) is a 129 . but most of the analogous results hold for larger systems of ODEs. and it follows from Sternberg that if (0.1) = g(x. y ∗ ) is topologically conjugate (i. The Hartman Grobman theorem states that if (x∗ .2) The Jacobian of this system is J(x. and stresses insight over rigor and depth. y ∗ ) have non-zero real parts. (6. the topological conjugacy is C 1 smooth. y) = ∂f ∂x ∂g ∂x ∂f ∂y ∂g ∂y ! . y ∗ ) is hyperbolic. y y dt (6. y ∗ ) hyperbolic if both eigenvalues of J(x∗ . If (x∗ . Consider the smooth planar system dx dt dy dt = f (x. can be gently deformed) into the orbit structure for the linearized system in a small disk around (0.1 Local stability and bifurcations We present the main results for systems of two ODEs. then the equilibrium point (0. y ∗ ) . 0) is a center or degenerate (non-isolated) equilibrium point for the linearized system. Sternberg showed that for planer systems.Chapter 6 Mathematical Methods I recommend Strogatz’s text on Nonlinear Dynamics and Chaos to learn the basic analytical tools to analyze ODEs and dynamical systems. y ∗ ) is an equilibrium point.. then the orbit structure for the nonlinear system in a small disk around (x∗ . 6. This condition excludes imaginary eigenvalues and zero eigenvalues. Strogatz’s textbook is highly readable.e. y ∗ ) is non-hyperbolic. y) (6. and consider the linearized system at (x∗ . The following brief discussions of mathematical tools are meant to complement the Strogatz text. 2) has a hyperbolic equilibrium point (x∗ . etc. Hsu.1 illustrates a simple example.130 CHAPTER 6. if (0. It occurs when J(x∗ . y ∗ ) has a single zero eigenvalue. See [Kuznetsov. Fall et al. hyperbolic fixed points do not bifurcate. especially targeted to population models. so the main value of this method is in establishing global stability. 2005.. However. y ∗ ) is a nonlinear repelling node. Good references. and is one of the main ways of showing the existence of a limit cycle. Sallet. a non-hyperbolic equilibrium point can bifurcate.1: Example of saddle node bifurcation www. The bifurcation can be supercritical or subcritical.. Figure 6. Local stability is usually determined using the method of linearization. y 0 = −y [from A Hopf bifurcation is the birth of a limit cycle from an equilibrium point. Gatto and Rinaldi. y ∗ ) is a nonlinear attracting spiral. 2004] for details. Figure 6. 0) is a linear repelling node. y ∗ ) is a linear attracting spiral.2 Lyapunov functions Lyapunov functions are a tool to determine the local stability. MATHEMATICAL METHODS linear saddle. and Tewa. if (x∗ . Thus hyperbolic fixed points are stable under small C 1 perturbations of the ODE.gif x0 = x2 + β. Figure 6. resulting in an attracting or repelling limit cycle. If the functions f and g are very slightly perturbed (in the C 1 topology). y ∗ ) with the same local phase portrait and stability. In particular. where the additional co-dimension-one period doubling bifurcation can also occur. A Hopf bifurcation creates an oscillation. . A saddle-node bifurcation is a collision and disappearance of two equilibrium points: a saddle and a node.1) and (6. It occurs when J(x∗ . 1977. then (x∗ . then (x∗ . then (x∗ . Suppose the system (6. Gurel and Lapidus. Iggidr. Both local bifurcations can occur for hyperbolic fixed points for discrete dynamical systems. but all the analogous results hold for larger systems of ODEs. 6. Wake. y ∗ ) has a pair of purely imaginary eigenvalues. y ∗ ). and sometimes prove global stability. include [Fall. These are the two co-dimension one local bifurcations for ODEs. 2002]. i. for ODEs two ways are by far the most pervasive: stable node bifurcations and Hopf bifurcations.2 illustrates a simple example.e. then the perturbed systems will also have a hyperbolic equilibrium point near (x∗ . 1968. We present the main results for systems of two autonomous ODEs. y ∗ ) is a nonlinear saddle. they do not disappear or change their stability under sufficiently small perturbations of the ODE.scholarpedia. Although there are several ways this can happen. of an equilibrium point.org/wiki/images/a/a5/2DSaddleNode. LYAPUNOV FUNCTIONS 131 Figure 6. y ∗ ) is a globally attracting. y(0)) = 0 Remark 3.scholarpedia. if D = R2 and V (x.2: Example of supercritical Hopf bifurcation [from www. y ∗ ) is an isolated equilibrium point. Furthermore. y ∗ ) is a locally attracting equilibrium point. i.6) for all t. Trajectories can’t stop before they reach the equilibrium point because dV /dt < 0. 3. since Z V ((x(t). y ∗ ) = 0 2.2. y ∗ ). in the sense that the only equilibrium point contained in a sufficiently small disk around (x∗ . Then (x∗ . If (x(t).7) .. (6. one can find a function with bowl-shaped graph with the equilibrium point at the bottom such that on the bowl. then ∂V 0 ∂V 0 d V (x(t). y(s)) ds < 0. a smooth function V : R2 → R with the following properties: 1. y ∗ ) such at V (x. y) > 0 for (x. Suppose there exists a local Lyapunov function. y ∗ ) is (x∗ . suppose (x∗ .4) = g(x. y ∗ ) denotes a solution of the ODE. y) (6. y).5) where f and g are smooth (C 1 ) functions. y(t)) = x + y <0 dt ∂x ∂y (6. y) ∈ D \ (x∗ . then (x∗ .e.gif] Theorem 12.org/wiki/images/7/7f/SuperHopf. ds (6. The idea of the proof is that around the equilibrium point. (Stability via Lyapunov functions) For the system dx dt dy dt = f (x. V (x∗ . (Facts about Lyapunov functions) t V (x(s). There exists an open disk D centered at (x∗ . y ∗ ). y(t)) 6= (x∗ . trajectories flow down and down to the equilibrium point. y) → ∞ as x2 +y 2 → ∞. y(t)) − V (x(0).6. closed orbits. MATHEMATICAL METHODS 1. y ∗ ) stays inside the disk around (x∗ . Consider theR single ODE x0 = f (x) and suppose x∗ is an isolated equilibrium point. or heteroclinic connections. (Poincar´e-Bendixson theorem): Suppose 1. 6. y > 0. y ∗ ) with x∗ > 0 and y ∗ .9) is smooth on a open set containing R. To construct the set R one usually constructs a trapping region A (see Figure (6. 3. 4. there is a small collection of functions to test whether they are Lyapunov functions. It states that bounded orbits can only accumulate onto equilibrium points. 4. y) ≥ 0 for (x. y ∗ ) for all t > 0. where such trajectories could approach strange attractors. The orbit (x(t). Either way. with chaotic dynamics that are highly sensitive to tiny changes in the initial conditions. A stable fixed point. y) (6. y) ∈ D \ (x∗ . although usually the Lyapunov function can not be expressed in closed form. and yields that (x∗ . 2. Problem 82. Show x that the function V (x) = − x0 f (s)ds is a local Lyapunov function. 1949] showed this is true. the method probably can not be applied. This is a topological annulus that contains no equilibrium point.3 Poincar´ e-Bendixson theorem and closed orbits The Poincar´e-Bendixson theorem severely limits the complexity of limit sets of bounded orbits of a first order system of two autonomous ODEs.8) = g(x. . y(t)) is contained in R for all t ≥ 0. 2. 3. In population dynamics. The first order system of ODEs dx dt dy dt = f (x. and such the vector field corresponding to the ODE is pointing outwards on the inner simple closed curve and pointing inwards on the outer simple closed curve. which could be a center. Theorem 13. the set R contains a closed orbit. R does not contain any equilibrium points. Then the orbit is either a closed orbit or spirals into a closed orbit. There is a weaker version of the local result that only requires that V (x. In practice. y ∗ ) is a stable fixed point. has the property that for all > 0 there exists δ > 0 such that any initial condition starting inside the δ disk around (x∗ . and if these do not work.132 CHAPTER 6. (6. Massera [Massera. the global version of this theorem is frequently applied to an equilibrium point (x∗ . Thus an important consequence is that an autonomous system of two ODEs cannot have a strange attractor. R is a closed and bounded subset of the plane.3)). One can ask whether every locally attracting equilibrium point has a local Lyapunov function. where D is R2 is replaced by the forward invariant region x > 0. y ∗ ). y). The analogous result is generally false in higher dimensions. 6. ROUTH-HURWITZ AND JURY CONDITIONS 133 Figure 6. (Jury criterion) If |a1 | < 1 + a2 < 2. ♦ Prove this. We state these conditions only for two and three dimensional linear systems. which insure that the eigenvalues of the Jacobian matrix at a fixed point has modulus less than one. References include [Anagnost and Desoer. It follows that such a fixed point is locally attracting. the Routh-Hurwitz conditions provide a necessary and sufficient condition for all the eigenvalues of the Jacobian matrix of an equilibrium point to have negative real parts. |a3 < 1|. |λ2 | < 1. (Routh-Hurwitz criterion) If a1 > 0 and a2 > 0. 1. and |a2 −a1 a3 | < |1−a23 |. and a1 a2 > a3 . |λ3 | < 1.4 Routh-Hurwitz and Jury conditions For a system of autonomous ODEs. 1.4. 1977]. then re(λ1 ) < 0. then |λ1 | < 1. 2. (Routh-Hurwitz criterion) If a1 > 0. Lewis. |λ2 | < 1. The Jury conditions are the analog for discrete dynamical systems. The conditions quickly get complicated in higher dimensions. The characteritic equation for A can be written as p(λ) = λ2 + a1 λ + a2 = 0. (n = 3) Let A be a 3 × 3 matrix. Proposition 1. then |λ1 | < 1. (n = 2) Let A be a 2 × 2 matrix. re(λ3 ) < 0. 2. It follows from the method of linearization that such an equilibrium point is locally attracting. re(λ2 ) < 0. The characteritic equation for A can be written as p(λ) = λ3 + a1 λ2 + a2 λ + a3 = 0. re(λ2 ) < 0.3: Trapping region A 6. a3 > 0. (Jury criterion) If |a1 +a3 | < a2 +1. . Proposition 2. 1991. then re(λ1 ) < 0. 1994] contains a highly readable and insightful introduction to ODEs. [Anderson and May. Kuznetsov. 2003] are written by mathematicians who have biology in mind. [Hofbauer and Sigmund. [Nowak and May. 1998] contains a lucid discussion of evolutionary games and population dynamics. written by the “Beetle Team. 1968. 1990. Keeling and Rohani. bifurcation theory. Although first published 35 years ago. 1971. The mathematical biology texts [Edelstein-Keshet. 134 . 5. More rigorous texts include [Alligood. 2001. 1. 2004]. as well as chaos in population models and a complete discussion of the flour beetle model. 2005. May’s book [May. I choose the bibliographic references mostly based on their insights and readability. 1997. III). 3.Chapter 7 Supplementary Material 7. 2008] contain comprehensive accounts of epidemiological models. Murray. 7. 1989] are written by biologists. 1991. Strogatz. 2001] contains a lucid discussion of the mathematical principles of immunology and virology. 4. 2.1 References Below is a short list of books that I recommend for further self study of population dynamics. I have also learned a great deal from May’s collections of expository papers by the world’s top population experts (Theoretical Ecology: Principles and Applications I. I. Kot. [Cushing. Wilson and Bossert. 8. I consider this the main mathematical reference for these lecture notes. and dynamical systems. Sauer.” contains an excellent introduction to the challenges of population modeling and validating models with data. Yodzis. [S. and is still required reading. 2002]. and Yorke. with self study in mind. 2001] was the first to bring a dynamical systems approach to ecology. and are recommended to mathematical biology students. The quantitative biology texts [Smith. Smith’s books 6. Arrowsmith. 12. I also highly recommend Racaniello’s virology textbook [Flint et al. 2006] is a much more authoritative text on immunology. 11. [Shigesada and Kawasaki. For spatial population models and invasions. 1996. 1994] contains a lucid technical discussion of reaction diffusion equations from the dynamical systems point of view. [Begon. Odum and Odum.. 2000] offers many examples of invasions but contains no mathematics. My favorite podcasts are This Week in Virology (TWIV) and This Week in Parasitism (TWIP). REFERENCES 135 9. If one wants to seriously model the spread of infectious diseases. [Allen.7. 2004] and Despommier’s parasitism textbook [Despommier et al... 1982] is readable with little biology background. 2007] contains a self-contained. 1995]. and Townsend. 10. [Kindt et al. it is a good idea to obtain a basic knowledge of mechanisms by which invading microbes cause disease and how the body acts in defense. 2003]. both hosted my Vincent Racaniello and Dickson Despommier. elementary presentation of the relevant partial differential equations. Just listening to these podcasts while driving to work will provide excellent introductions to these fields. The classic text [Elton. I also very much like the gentle introduction to immunology in [Sompayrac.1. Harper. The classic [Mims. 1997] stresses the biology and is very readable. 1971] contain excellent discussions of ecology. The mathematics text [Smoller. . ” American Naturalist (1994): 457–472. R. L.. Ginzburg.A. and R. DG and HF Weinberger. Levin. Ak¸cakaya.R.” Bulletin of mathematical biology 53 (1991): 1–32. R. 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