# Introduction to Mathematical Biology, An, 1st edition

• Linda J.S. Allen

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## Overview

KEY BENEFIT: This reference introduces a variety of mathematical models for biological systems, and presents the mathematical theory and techniques useful in analyzing those models. Material is organized according to the mathematical theory rather than the biological application. KEY TOPICS: Contains applications of mathematical theory to biological examples in each chapter. Focuses on deterministic mathematical models with an emphasis on predicting the qualitative solution behavior over time. Discusses classical mathematical models from population , including the Leslie matrix model, the Nicholson-Bailey model, and the Lotka-Volterra predator-prey model. Also discusses more recent models, such as a model for the Human Immunodeficiency Virus - HIV and a model for flour beetles. KEY MARKET: Readers seeking a solid background in the mathematics behind modeling in biology and exposure to a wide variety of mathematical models in biology.

Preface xi

1 LINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES 1

1.1 Introduction 1

1.2 Basic Definitions and Notation 2

1.3 First-Order Equations 6

1.4 Second-Order and Higher-Order Equations 8

1.5 First-Order Linear Systems 14

1.6 An Example: Leslie’s Age-Structured Model 18

1.7 Properties of the Leslie Matrix 20

1.8 Exercises for Chapter 1 28

1.9 References for Chapter 1 33

1.10 Appendix for Chapter 1 34

1.10.1 Maple Program:Turtle Model 34

1.10.2 MATLAB® Program:Turtle Model 34

2 NONLINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES 36

2.1 Introduction 36

2.2 Basic Definitions and Notation 37

2.3 Local Stability in First-Order Equations 40

2.4 Cobwebbing Method for First-Order Equations 45

2.5 Global Stability in First-Order Equations 46

2.6 The Approximate Logistic Equation 52

2.7 Bifurcation Theory 55

2.7.1 Types of Bifurcations 56

2.7.2 Liapunov Exponents 60

2.8 Stability in First-Order Systems 62

2.9 Jury Conditions 67

2.10 An Example: Epidemic Model 69

2.11 Delay Difference Equations 73

2.12 Exercises for Chapter 2 76

2.13 References for Chapter 2 82

2.14 Appendix for Chapter 2 84

2.14.1 Proof of Theorem 2.1 84

2.14.2 A Definition of Chaos 86

2.14.3 Jury Conditions (Schur-Cohn Criteria) 86

2.14.4 Liapunov Exponents for Systems of Difference Equations 87

2.14.5 MATLAB Program: SIR Epidemic Model 88

3 BIOLOGICAL APPLICATIONS OF DIFFERENCE EQUATIONS 89

3.1 Introduction 89

3.2 Population Models 90

3.3 Nicholson-Bailey Model 92

3.4 Other Host-Parasitoid Models 96

3.5 Host-Parasite Model 98

3.6 Predator-Prey Model 99

3.7 Population Genetics Models 103

3.8 Nonlinear Structured Models 110

3.8.1 Density-Dependent Leslie Matrix Models 110

3.8.2 Structured Model for Flour Beetle Populations 116

3.8.3 Structured Model for the Northern Spotted Owl 118

3.8.4 Two-Sex Model 121

3.9 Measles Model with Vaccination 123

3.10 Exercises for Chapter 3 127

3.11 References for Chapter 3 134

3.12 Appendix for Chapter 3 138

3.12.1 Maple Program: Nicholson-Bailey Model 138

3.12.2 Whooping Crane Data 138

3.12.3 Waterfowl Data 139

4 LINEAR DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES 141

4.1 Introduction 141

4.2 Basic Definitions and Notation 142

4.3 First-Order Linear Differential Equations 144

4.4 Higher-Order Linear Differential Equations 145

4.4.1 Constant Coefficients 146

4.5 Routh-Hurwitz Criteria 150

4.6 Converting Higher-Order Equations to First-OrderSystems 152

4.7 First-Order Linear Systems 154

4.7.1 Constant Coefficients 155

4.8 Phase-Plane Analysis 157

4.9 Gershgorin’s Theorem 162

4.10 An Example: Pharmacokinetics Model 163

4.11 Discrete and Continuous Time Delays 165

4.12 Exercises for Chapter 4 169

4.13 References for Chapter 4 172

4.14 Appendix for Chapter 4 173

4.14.1 Exponential of a Matrix 173

4.14.2 Maple Program: Pharmacokinetics Model 175

5 NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES 176

5.1 Introduction 176

5.2 Basic Definitions and Notation 177

5.3 Local Stability in First-Order Equations 180

5.3.1 Application to Population Growth Models 181

5.4 Phase Line Diagrams 184

5.5 Local Stability in First-Order Systems 186

5.6 Phase Plane Analysis 191

5.7 Periodic Solutions 194

5.7.1 Poincaré-Bendixson Theorem 194

5.7.2 Bendixson’s and Dulac’s Criteria 197

5.8 Bifurcations 199

5.8.1 First-Order Equations 200

5.8.2 Hopf Bifurcation Theorem 201

5.9 Delay Logistic Equation 204

5.10 Stability Using Qualitative Matrix Stability 211

5.11 Global Stability and Liapunov Functions 216

5.12 Persistence and Extinction Theory 221

5.13 Exercises for Chapter 5 224

5.14 References for Chapter 5 232

5.15 Appendix for Chapter 5 234

5.15.1 Subcritical and Supercritical Hopf Bifurcations 234

5.15.2 Strong Delay Kernel 235

6 BIOLOGICAL APPLICATIONS OF DIFFERENTIAL EQUATIONS 237

6.1 Introduction 237

6.2 Harvesting a Single Population 238

6.3 Predator-Prey Models 240

6.4 Competition Models 248

6.4.1 Two Species 248

6.4.2 Three Species 250

6.5 Spruce Budworm Model 254

6.6 Metapopulation and Patch Models 260

6.7 Chemostat Model 263

6.7.1 Michaelis-Menten Kinetics 263

6.7.2 Bacterial Growth in a Chemostat 266

6.8 Epidemic Models 271

6.8.1 SI, SIS, and SIR Epidemic Models 271

6.8.2 Cellular Dynamics of HIV 276

6.9 Excitable Systems 279

6.9.1 Van der Pol Equation 279

6.9.2 Hodgkin-Huxley and FitzHugh-Nagumo Models 280

6.10 Exercises for Chapter 6 283

6.11 References for Chapter 6 292

6.12 Appendix for Chapter 6 296

6.12.1 Lynx and Fox Data 296

6.12.2 Extinction in Metapopulation Models 296

7 PARTIAL DIFFERENTIAL EQUATIONS: THEORY, EXAMPLES, AND APPLICATIONS 299

7.1 Introduction 299

7.2 Continuous Age-Structured Model 300

7.2.1 Method of Characteristics 302

7.2.2 Analysis of the Continuous Age-Structured Model 306

7.3 Reaction-Diffusion Equations 309

7.4 Equilibrium and Traveling Wave Solutions 316

7.5 Critical Patch Size 319

7.6 Spread of Genes and Traveling Waves 321

7.7 Pattern Formation 325

7.8 Integrodifference Equations 330

7.9 Exercises for Chapter 7 331

7.10 References for Chapter 7 336

Index 339

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