Mathematics in Population Biology

Description

The formulation, analysis, and re-evaluation of mathematical models in population biology has become a valuable source of insight to mathematicians and biologists alike. This book presents an overview and selected sample of these results and ideas, organized by biological theme rather than mathematical concept, with an emphasis on helping the reader develop appropriate modeling skills through use of well-chosen and varied examples. Part I starts with unstructured single species population models, particularly in the framework of continuous time models, then adding the most rudimentary stage structure with variable stage duration. The theme of stage structure in an age-dependent context is developed in Part II, covering demographic concepts, such as life expectation and variance of life length, and their dynamic consequences. In Part III, the author considers the dynamic interplay of host and parasite populations, i.e., the epidemics and endemics of infectious diseases. The theme of stage structure continues here in the analysis of different stages of infection and of age-structure that is instrumental in optimizing vaccination strategies. Each section concludes with exercises, some with solutions, and suggestions for further study. The level of mathematics is relatively modest; a "toolbox" provides a summary of required results in differential equations, integration, and integral equations. In addition, a selection of Maple worksheets is provided. The book provides an authoritative tour through a dazzling ensemble of topics and is both an ideal introduction to the subject and reference for researchers. Cover Title Page Copyright Page Contents Preface Chapter 1. Some General Remarks on Mathematical Modeling Bibliographic Remarks PART 1. BASIC POPULATION GROWTH MODELS Chapter 2. Birth, Death, and Migration 2.1 The Fundamental Balance Equation of Population Dynamics 2.2 Birth Date Dependent Life Expectancies 2.3 The Probability of Lifetime Emigration Chapter 3. Unconstrained Population Growth for Single Species 3.1 Closed Populations 3.1.1 The Average Intrinsic Growth Rate for Periodic Environments 3.1.2 The Average Intrinsic Growth Rate for Nonperiodic Environments 3.2 Open Populations 3.2.1 Nonzero Average Intrinsic Growth Rate 3.2.2 Zero Average Intrinsic Growth Rate Chapter 4. Von Bertalanffy Growth of Body Size Chapter 5. Classic Models of Density-Dependent Population Growth for 37 Single Species 5.1 The Bernoulli and the Verhulst Equations 5.2 The Beverton–Holt and Smith Differential Equation 5.2.1 Derivation from a Resource–Consumer Model 5.2.2 Derivation from Cannibalism of Juveniles by Adults 5.3 The Ricker Differential Equation 5.4 The Gompertz Equation 5.5 A First Comparison of the Various Equations Chapter 6. Sigmoid Growth 6.1 General Conditions for Sigmoid Growth 6.2 Fitting Sigmoid Population Data Chapter 7. The Allee Effect 7.1 First Model Derivation: Search for a Mate 7.2 Second Model Derivation: Impact of a Satiating Generalist Predator 7.3 Model Analysis Chapter 8. Nonautonomous Population Growth:Asymptotic Equality of 75 Population Sizes Chapter 9. Discrete-Time Single-Species Models 9.1 The Discrete Analog of the Verhulst (Logistic) and the Bernoulli Equation: the Beverton–Holt Difference Equation and Its Generalization 9.2 The Ricker Difference Equation 9.3 Some Analytic Results for Scalar Difference Equations 9.4 Some Remarks Concerning the Quadratic Difference Equation Bibliographic Remarks Chapter 10. Dynamics of an Aquatic Population Interacting with a Polluted Environment 10.1 Modeling Toxicant and Population Dynamics 10.2 Open Loop Toxicant Input 10.3 Feedback Loop Toxicant Input 10.4 Extinction and Persistence Equilibria and a Threshold Condition for Population Extinction 10.5 Stability of Equilibria and Global Behavior of Solutions 10.6 Multiple Extinction Equilibria, Bistability and Periodic Oscillations 10.7 Linear Dose Response Bibliographic Remarks Chapter 11. Population Growth Under Basic Stage Structure 11.1 A Most Basic Stage-Structured Model 11.2 Well-Posedness and Dissipativity 11.3 Equilibria and Reproduction Ratios 11.4 Basic Reproduction Ratios and Threshold Conditions for Extinction versus Persistence 11.5 Weakly Density-Dependent Stage-Transition Rates and Global Stability of Nontrivial Equilibria 11.6 The Number and Nature of Possible Multiple Nontrivial Equilibria 11.7 Strongly Density-Dependent Stage-Transition Rates and Periodic Oscillations 11.8 An Example for Multiple Periodic Orbits and Both Supercritical and Subcritical Hopf Bifurcation 11.9 Multiple Interior Equilibria, Bistability, and Many Bifurcations for Pure Intrastage Competition Bibliographic Remarks PART 2. STAGETRANSITIONS AND DEMOGRAPHICS Chapter 12. TheTransitionThrough a Stage 12.1 The Sojourn Function 12.2 Mean Sojourn Time, Expected Exit Age, and Expectation of Life 12.3 The Variance of the Sojourn Time, Moments and Central Moments 12.4 Remaining Sojourn Time and Its Expectation 12.5 Fixed Stage Durations 12.6 Per Capita Exit Rates (Mortality Rates) 12.7 Exponentially Distributed Stage Durations 12.8 Log-Normally Distributed Stage Durations 12.9 A Stochastic Interpretation of Stage Transition Bibliographic Remarks Chapter 13. Stage Dynamics with Given Input 13.1 Input and Stage-Age Density 13.2 The Partial Differential Equation Formulation 13.3 Stage Content and Average Stage Duration 13.4 Average Stage Age 13.5 Stage Exit Rates 13.5.1 The Fundamental Balance Equation of Stage Dynamics 13.5.2 Average Age at Stage Exit 13.6 Stage Outputs 13.7 Which Recruitment Curves Can Be Explained by Cannibalism of Newborns by Adults? Bibliographic Remarks Chapter 14. Demographics in an Unlimiting Constant Environment 14.1 The Renewal Equation 14.2 Balanced Exponential Growth 14.3 The Renewal Theorem: Approach to Balanced Exponential Growth Chapter 15. Some Demographic Lessons from Balanced Exponential Growth 15.1 Inequalities and Estimates for the Malthusian Parameter 15.2 Average Age and Average Age at Death in a Population at Balanced Exponential Growth. Average Per Capita Death Rate 15.3 Ratio of Population Size and Birth Rate 15.4 Consequences of an Abrupt Shift in Maternity: Momentum of Population Growth Bibliographic Remarks Chapter 16. Some Nonlinear Demographics 16.1 A Demographic Model with a Juvenile and an Adult Stage 16.2 A Differential Delay Equation Bibliographic Remarks PART 3. HOST–PARASITE POPULATION GROWTH: EPIDEMIOLOGY OF INFECTIOUS DISEASES Chapter 17. Background 17.1 Impact of Infectious Diseases in Past and Present Time 17.2 Epidemiological Terms and Principles Bibliographic Remarks Chapter 18. The Simplifed Kermack–McKendrick Epidemic Model 18.1 A Model with Mass-Action Incidence 18.2 Phase-Plane Analysis of the Model Equations. The Epidemic Threshold Theorem 18.3 The Final Size of the Epidemic. Alternative Formulation of the Threshold Theorem Chapter 19. Generalization of the Mass-Action Law of Infection 19.1 Population-Size Dependent Contact Rates 19.2 Model Modi?cation 19.3 The Generalized Epidemic Threshold Theorem Chapter 20. The Kermack–McKendrick Epidemic Model with Variable Infectivity 20.1 A Stage-Age Structured Model 20.2 Reduction to a Scalar Integral Equation Bibliographic Remarks Chapter 21. SEIR (-> S)Type Endemic Models for“Childhood Diseases” 21.1 The Model and Its Well-Posedness 21.2 Equilibrium States and the Basic Replacement Ratio 21.3 The Disease Dynamics in the Vicinities of the Disease-Free and the Endemic Equilibrium: Local Stability and the Interepidemic Period 21.4 Some Global Results: Extinction, Persistence of the Disease; Conditions for Attraction to the Endemic Equilibrium Bibliographic Remarks Chapter 22. Age-Structured Models for Endemic Diseases and 341 OptimalVaccination Strategies 22.1 A Model with Chronological Age-Structure 22.2 Disease-Free and Endemic Equilibrium: the Replacement Ratio 22.3 The Net Replacement Ratio, and Disease Extinction and Persistence 22.4 Cost of Vaccinations and Optimal Age Schedules 22.5 Estimating the Net Replacement Ratio: Average Duration of Susceptibility and Average Age at Infection. Optimal Vaccination Schedules Revisited Bibliographic Remarks Chapter 23. Endemic Models with Multiple Groups or Populations 23.1 The Model 23.2 Equilibrium Solutions 23.3 Local Asymptotic Stability of Strongly Endemic Equilibria 23.4 Extinction or Persistence of the Disease? 23.5 The Basic Replacement Matrix, Alias Next-Generation Matrix 23.6 The Basic Replacement Ratio as Spectral Radius of the Basic Replacement Matrix 23.7 Some Special Cases of Mixing Bibliographic Remarks PART 4. TOOLBOX Appendix A Ordinary Differential Equations A.1 Conservation of Positivity and Boundedness A.2 Planar Ordinary Differential Equation Systems A.3 The Method of Fluctuations A.4 Behavior in the Vicinity of an Equilibrium A.5 Elements of Persistence Theory Bibliographic Remarks A.6 Global Stability of a Compact Minimal Set A.7 Hopf Bifurcation A.8 Perron–Frobenius Theory of Positive Matrices andAssociated Linear Dynamical Systems Bibliographic Remarks Appendix B Integration, Integral Equations, and Some Convex Analysis B.1 The Stieltjes Integral of Regulated Functions B.2 Some Elements from Measure Theory B.3 Some Elements from Convex Analysis B.4 Lebesgue–Stieltjes Integration B.5 Jensen’s Inequality and Related Material B.6 Volterra Integral Equations B.7 Critical and Regular Values of a Function Bibliographic Remarks Appendix C Some MAPLE Worksheets with Comments for Part 1 C.1 Fitting the Growth of the World Population (Figure 3.1) C.2 Periodic Modulation of Exponential Growth in Closed Populations (Figures 3.2 and 3.3) C.3 Fitting Sigmoid Population-Growth Curves (Figures 6.1 and 6.2) C.4 Fitting Bernoulli’s Equation to Population Data of Sweden (Figure 6.3) C.5 Illustrating the Allee Effect (Figures 7.2–7.4) C.6 Dynamics of an Aquatic Population Interacting with a Polluted Environment (Figure 10.3E) References Index