Preface ix

Chapter 1: Mathematical Modeling in Biology 1

1.1 Introduction 1

1.2 HIV 2

1.3 Models of HIV/AIDS 5

1.4 Concluding Message 14

Chapter 2: How to Construct a Model 17

2.1 Introduction 17

2.2 Formulate the Question 19

2.3 Determine the Basic Ingredients 19

2.4 Qualitatively Describe the Biological System 26

2.5 Quantitatively Describe the Biological System 33

2.6 Analyze the Equations 39

2.7 Checks and Balances 47

2.8 Relate the Results Back to the Question 50

2.9 Concluding Message 51

Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology 54

3.1 Introduction 54

3.2 Exponential and Logistic Models of Population Growth 54

3.3 Haploid and Diploid Models of Natural Selection 62

3.4 Models of Interactions among Species 72

3.5 Epidemiological Models of Disease Spread 77

3.6 Working Backward--Interpreting Equations in Terms of the Biology 79

3.7 Concluding Message 82

Primer 1: Functions and Approximations 89

P1.1 Functions and Their Forms 89

P1.2 Linear Approximations 96

P1.3 The Taylor Series 100

Chapter 4: Numerical and Graphical Techniques--Developing a Feeling for Your Model 110

4.1 Introduction 110

4.2 Plots of Variables Over Time 111

4.3 Plots of Variables as a Function of the Variables Themselves 124

4.4 Multiple Variables and Phase-Plane Diagrams 133

4.5 Concluding Message 145

Chapter 5: Equilibria and Stability Analyses--One-Variable Models 151

5.1 Introduction 151

5.2 Finding an Equilibrium 152

5.3 Determining Stability 163

5.4 Approximations 176

5.5 Concluding Message 184

Chapter 6: General Solutions and Transformations--One-Variable Models 191

6.1 Introduction 191

6.2 Transformations 192

6.3 Linear Models in Discrete Time 193

6.4 Nonlinear Models in Discrete Time 195

6.5 Linear Models in Continuous Time 198

6.6 Nonlinear Models in Continuous Time 202

6.7 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 254

7.1 Introduction 254

7.2 Models with More than One Dynamic Variable 255

7.3 Linear Multivariable Models 260

7.4 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 294

8.1 Introduction 294

8.2 Nonlinear Multiple-Variable Models 294

8.3 Equilibria and Stability for Nonlinear Discrete-Time Models 316

8.4 Perturbation Techniques for Approximating Eigenvalues 330

8.5 Concluding Message 337

Chapter 9: General Solutions and Tranformations--Models with Multiple Variables 347

9.1 Introduction 347

9.2 Linear Models Involving Multiple Variables 347

9.3 Nonlinear Models Involving Multiple Variables 365

9.4 Concluding Message 381

Chapter 10: Dynamics of Class-Structured Populations 386

10.1 Introduction 386

10.2 Constructing Class-Structured Models 388

10.3 Analyzing Class-Structured Models 393

10.4 Reproductive Value and Left Eigenvectors 398

10.5 The Effect of Parameters on the Long-Term Growth Rate 400

10.6 Age-Structured Models--The Leslie Matrix 403

10.7 Concluding Message 418

Chapter 11: Techniques for Analyzing Models with Periodic Behavior 423

11.1 Introduction 423

11.2 What Are Periodic Dynamics? 423

11.3 Composite Mappings 425

11.4 Hopf Bifurcations 428

11.5 Constants of Motion 436

11.6 Concluding Message 449

Chapter 12: Evolutionary Invasion Analysis 454

12.1 Introduction 454

12.2 Two Introductory Examples 455

12.3 The General Technique of Evolutionary Invasion Analysis 465

12.4 Determining How the ESS Changes as a Function of Parameters 478

12.5 Evolutionary Invasion Analyses in Class-Structured Populations 485

12.6 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 567

13.1 Introduction 567

13.2 Models of Population Growth 568

13.3 Birth-Death Models 573

13.4 Wright-Fisher Model of Allele Frequency Change 576

13.5 Moran Model of Allele Frequency Change 581

13.6 Cancer Development 584

13.7 Cellular Automata--A Model of Extinction and Recolonization 591

13.8 Looking Backward in Time--Coalescent Theory 594

13.9 Concluding Message 602

Chapter 14: Analyzing Discrete Stochastic Models 608

14.1 Introduction 608

14.2 Two-State Markov Models 608

14.3 Multistate Markov Models 614

14.4 Birth-Death Models 631

14.5 Branching Processes 639

14.6 Concluding Message 644

Chapter 15: Analyzing Continuous Stochastic Models--Diffusion in Time and Space 649

15.1 Introduction 649

15.2 Constructing Diffusion Models 649

15.3 Analyzing the Diffusion Equation with Drift 664

15.4 Modeling Populations in Space Using the Diffusion Equation 684

15.5 Concluding Message 687

Epilogue: The Art of Mathematical Modeling in Biology 692

Appendix 1: Commonly Used Mathematical Rules 695

A1.1 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 699

A2.2 Derivatives 701

A2.3 Integrals 703

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