SARAH P. OTTO and TROY DAY

Transcription

1 A Biologist's Guide to Mathematical Modeling in Ecology and Evolution SARAH P. OTTO and TROY DAY Univsr?.ltats- und Lender bibliolhek Darmstadt Bibliothek Biotogi Princeton University Press Princeton and Oxford

2 Contents Preface ix Chapter 1: Mathematical Modeling in Biology Introduction HIV Models of HIV/AIDS Concluding Message 14 Chapter 2: How to Construct a Model Introduction Formulate the Question Determine the Basic Ingredients Qualitatively Describe the Biological System Quantitatively Describe the Biological System Analyze the Equations Checks and Balances Relate the Results Back to the Question Concluding Message 51 Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology Introduction Exponential and Logistic Models of Population Growth Haploid and Diploid Models of Natural Selection Models of Interactions among Species Epidemiological Models of Disease Spread Working Backward Interpreting Equations in Terms of the Biology Concluding Message 82 Primer 1: Functions and Approximations 89 Pl.l Functions and Their Forms 89 PI.2 Linear Approximations 96 PI.3 The Taylor Series 100 Chapter 4: Numerical and Graphical Techniques Developing a Feeling for Your Model Introduction Plots of Variables Over Time ' Plots of Variables as a Function of the Variables Themselves 124

3 vl Contents 4.4 Multiple Variables and Phase-Plane Diagrams Concluding Message 145 Chapter 5: Equilibria and Stability Analyses- One-Variable Models Introduction Finding an Equilibrium Determining Stability Approximations Concluding Message 184 Chapter 6: General Solutions and Transformations One-Variable Models Introduction Transformations Linear Models in Discrete Time Nonlinear Models in Discrete Time Linear Models in Continuous Time Nonlinear Models in Continuous Time Concluding Message 207 Primer 2: Linear Algebra 214 P2.1 An Introduction to Vectors and Matrices 214 P2.2 Vector and Matrix Addition 219 P2.3 Multiplication by a Scalar 222 P2.4 Multiplication of Vectors and Matrices 224 P2.5 The Trace and Determinant of a Square Matrix 228 P2.6 The Inverse 233 P2.7 Solving Systems of Equations 235 P2.8 The Eigenvalues of a Matrix 237 P2.9 The Eigenvectors of a Matrix 243 Chapter 7: Equilibria and Stability Analyses Linear Models with Multiple Variables Introduction Models with More than One Dynamic Variable Linear Multivariable Models Equilibria and Stability for Linear Discrete-Time Models 279, 7.5 Concluding Message 289 Chapter 8: Equilibria and Stability Analyses- Nonlinear Models with Multiple Variables Introduction Nonlinear Multiple-Variable Models Equilibria and Stability for Nonlinear Discrete-Time Models Perturbation Techniques for Approximating Eigenvalues Concluding Message 337

4 Contents vii Chapter 9: General Solutions and Tranformations Models with Multiple Variables Introduction Linear Models Involving Multiple Variables Nonlinear Models Involving Multiple Variables Concluding Message 381 Chapter 10: Dynamics of Class-Structured Populations Introduction Constructing Class-Structured Models Analyzing Class-Structured Models Reproductive Value and Left Eigenvectors The Effect of Parameters on the Long-Term Growth Rate Age-Structured Models The Leslie Matrix Concluding Message 418 Chapter 11: Techniques for Analyzing Models with Periodic Behavior Introduction What Are Periodic Dynamics? Composite Mappings Hopf Bifurcations Constants of Motion Concluding Message 449 Chapter 12: Evolutionary Invasion Analysis Introduction Two Introductory Examples The General Technique of Evolutionary Invasion Analysis Determining How the ESS Changes as a Function of Parameters Evolutionary Invasion Analyses in Class-Structured Populations Concluding Message 502 Primer 3: Probability Theory 513 P3.1 An Introduction to Probability 513 P3.2 Conditional Probabilities and Bayes' Theorem 518 P3.3 Discrete Probability Distributions 521 P3.4 Continuous Probability Distributions 536 P3.5 The (Insert Your Name Here) Distribution 553 Chapter 13: Probabilistic Models Introduction " Models of Population Growth Birth-Death Models Wright-Fisher Model of Allele Frequency Change Moran Model of Allele Frequency Change Cancer Development 584

5 viii Contents 13.7 Cellular Automata A Model of Extinction and Recolonization Looking Backward in Time Coalescent Theory Concluding Message 602 Chapter 14: Analyzing Discrete Stochastic Models Introduction Two-State Markov Models Multistate Markov Models Birth-Death Models Branching Processes Concluding Message 644 Chapter 15: Analyzing Continuous Stochastic Models Diffusion in Time and Space Introduction Constructing Diffusion Models Analyzing the Diffusion Equation with Drift Modeling Populations in Space Using the Diffusion Equation Concluding Message 687 Epilogue: The Art of Mathematical Modeling in Biology 692 Appendix 1: Commonly Used Mathematical Rules 695 Al.l Rules for Algebraic Functions 695 A1.2 Rules for Logarithmic and Exponential Functions 695 A1.3 Some Important Sums 696 A1.4 Some Important Products 696 A1.5 Inequalities 697 Appendix 2: Some Important Rules from Calculus 699 A2.1 Concepts A2.2 Derivatives 701 A2.3 Integrals A2.4 Limits 704 Appendix 3: The Perron-Frobenius Theorem 709 A3.1: Definitions 709 A3.2: The Perron-Frobenius Theorem 710 Appendix 4: Finding Maxima and Minima of Functions 713 A4.1 Functions with One Variable 713 A4.2 Functions with Multiple Variables 714 Appendix 5: Moment-Generating Functions 717 Index of Definitions, Recipes, and Rules 725 General Index 727