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# A Biologist's Guide to Mathematical Modeling in Ecology and EvolutionSarah P. Otto & Troy Day

 TABLE OF CONTENTS:Preface ix Chapter 1: Mathematical Modeling in Biology 11.1 Introduction 11.2 HIV 21.3 Models of HIV/AIDS 51.4 Concluding Message 14Chapter 2: How to Construct a Model 172.1 Introduction 172.2 Formulate the Question 192.3 Determine the Basic Ingredients 192.4 Qualitatively Describe the Biological System 262.5 Quantitatively Describe the Biological System 332.6 Analyze the Equations 392.7 Checks and Balances 472.8 Relate the Results Back to the Question 502.9 Concluding Message 51Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology 543.1 Introduction 543.2 Exponential and Logistic Models of Population Growth 543.3 Haploid and Diploid Models of Natural Selection 623.4 Models of Interactions among Species 723.5 Epidemiological Models of Disease Spread 773.6 Working Backward--Interpreting Equations in Terms of the Biology 793.7 Concluding Message 82Primer 1: Functions and Approximations 89P1.1 Functions and Their Forms 89P1.2 Linear Approximations 96P1.3 The Taylor Series 100Chapter 4: Numerical and Graphical Techniques--Developing a Feeling for Your Model 1104.1 Introduction 1104.2 Plots of Variables Over Time 1114.3 Plots of Variables as a Function of the Variables Themselves 1244.4 Multiple Variables and Phase-Plane Diagrams 1334.5 Concluding Message 145Chapter 5: Equilibria and Stability Analyses--One-Variable Models 1515.1 Introduction 1515.2 Finding an Equilibrium 1525.3 Determining Stability 1635.4 Approximations 1765.5 Concluding Message 184Chapter 6: General Solutions and Transformations--One-Variable Models 1916.1 Introduction 1916.2 Transformations 1926.3 Linear Models in Discrete Time 1936.4 Nonlinear Models in Discrete Time 1956.5 Linear Models in Continuous Time 1986.6 Nonlinear Models in Continuous Time 2026.7 Concluding Message 207Primer 2: Linear Algebra 214P2.1 An Introduction to Vectors and Matrices 214P2.2 Vector and Matrix Addition 219P2.3 Multiplication by a Scalar 222P2.4 Multiplication of Vectors and Matrices 224P2.5 The Trace and Determinant of a Square Matrix 228P2.6 The Inverse 233P2.7 Solving Systems of Equations 235P2.8 The Eigenvalues of a Matrix 237P2.9 The Eigenvectors of a Matrix 243Chapter 7: Equilibria and Stability Analyses--Linear Models with Multiple Variables 2547.1 Introduction 2547.2 Models with More than One Dynamic Variable 2557.3 Linear Multivariable Models 2607.4 Equilibria and Stability for Linear Discrete-Time Models 2797.5 Concluding Message 289Chapter 8: Equilibria and Stability Analyses--Nonlinear Models with Multiple Variables 2948.1 Introduction 2948.2 Nonlinear Multiple-Variable Models 2948.3 Equilibria and Stability for Nonlinear Discrete-Time Models 3168.4 Perturbation Techniques for Approximating Eigenvalues 3308.5 Concluding Message 337Chapter 9: General Solutions and Tranformations--Models with Multiple Variables 3479.1 Introduction 3479.2 Linear Models Involving Multiple Variables 3479.3 Nonlinear Models Involving Multiple Variables 3659.4 Concluding Message 381Chapter 10: Dynamics of Class-Structured Populations 38610.1 Introduction 38610.2 Constructing Class-Structured Models 38810.3 Analyzing Class-Structured Models 39310.4 Reproductive Value and Left Eigenvectors 39810.5 The Effect of Parameters on the Long-Term Growth Rate 40010.6 Age-Structured Models--The Leslie Matrix 40310.7 Concluding Message 418Chapter 11: Techniques for Analyzing Models with Periodic Behavior 42311.1 Introduction 42311.2 What Are Periodic Dynamics? 42311.3 Composite Mappings 42511.4 Hopf Bifurcations 42811.5 Constants of Motion 43611.6 Concluding Message 449Chapter 12: Evolutionary Invasion Analysis 45412.1 Introduction 45412.2 Two Introductory Examples 45512.3 The General Technique of Evolutionary Invasion Analysis 46512.4 Determining How the ESS Changes as a Function of Parameters 47812.5 Evolutionary Invasion Analyses in Class-Structured Populations 48512.6 Concluding Message 502Primer 3: Probability Theory 513P3.1 An Introduction to Probability 513P3.2 Conditional Probabilities and Bayes' Theorem 518P3.3 Discrete Probability Distributions 521P3.4 Continuous Probability Distributions 536P3.5 The (Insert Your Name Here) Distribution 553Chapter 13: Probabilistic Models 56713.1 Introduction 56713.2 Models of Population Growth 56813.3 Birth-Death Models 57313.4 Wright-Fisher Model of Allele Frequency Change 57613.5 Moran Model of Allele Frequency Change 58113.6 Cancer Development 58413.7 Cellular Automata--A Model of Extinction and Recolonization 59113.8 Looking Backward in Time--Coalescent Theory 59413.9 Concluding Message 602Chapter 14: Analyzing Discrete Stochastic Models 60814.1 Introduction 60814.2 Two-State Markov Models 60814.3 Multistate Markov Models 61414.4 Birth-Death Models 63114.5 Branching Processes 63914.6 Concluding Message 644Chapter 15: Analyzing Continuous Stochastic Models--Diffusion in Time and Space 64915.1 Introduction 64915.2 Constructing Diffusion Models 64915.3 Analyzing the Diffusion Equation with Drift 66415.4 Modeling Populations in Space Using the Diffusion Equation 68415.5 Concluding Message 687Epilogue: The Art of Mathematical Modeling in Biology 692Appendix 1: Commonly Used Mathematical Rules 695A1.1 Rules for Algebraic Functions 695A1.2 Rules for Logarithmic and Exponential Functions 695A1.3 Some Important Sums 696A1.4 Some Important Products 696A1.5 Inequalities 697Appendix 2: Some Important Rules from Calculus 699A2.1 Concepts 699A2.2 Derivatives 701A2.3 Integrals 703A2.4 Limits 704Appendix 3: The Perron-Frobenius Theorem 709A3.1: Definitions 709A3.2: The Perron-Frobenius Theorem 710Appendix 4: Finding Maxima and Minima of Functions 713A4.1 Functions with One Variable 713A4.2 Functions with Multiple Variables 714Appendix 5: Moment-Generating Functions 717Index of Definitions, Recipes, and Rules 725General Index 727Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press