: 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. 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

Linear Difference Equations, Theory, and Examples

1

(35)

Introduction

1

(1)

Basic Definitions and Notation

2

(4)

First-Order Equations

6

(2)

Second-Order and Higher-Order Equations

8

(6)

First-Order Linear Systems

14

(4)

An Example: Leslie's Age-Structured Model

18

(2)

Properties of the Leslie Matrix

20

(8)

Exercises for Chapter 1

28

(5)

References for Chapter 1

33

(1)

Appendix for Chapter 1

34

(2)

Maple Program: Turtle Model

34

(1)

MATLAB® Program: Turtle Model

34

(2)

Nonlinear Difference Equations, Theory, and Examples

36

(53)

Introduction

36

(1)

Basic Definitions and Notation

37

(3)

Local Stability in First-Order Equations

40

(5)

Cobwebbing Method for First-Order Equations

45

(1)

Global Stability in First-Order Equations

46

(6)

The Approximate Logistic Equation

52

(3)

Bifurcation Theory

55

(7)

Types of Bifurcations

56

(4)

Liapunov Exponents

60

(2)

Stability in First-Order Systems

62

(5)

Jury Conditions

67

(2)

An Example: Epidemic Model

69

(4)

Delay Difference Equations

73

(3)

Exercises for Chapter 2

76

(6)

References for Chapter 2

82

(2)

Appendix for Chapter 2

84

(5)

Proof of Theorem 2.6

84

(2)

A Definition of Chaos

86

(1)

Jury Conditions (Schur-Cohn Criteria)

86

(1)

Liapunov Exponents for Systems of Difference Equations

87

(1)

MATLAB Program: SIR Epidemic Model

88

(1)

Biological Applications of Difference Equations

89

(52)

Introduction

89

(1)

Population Models

90

(2)

Nicholson-Bailey Model

92

(4)

Other Host-Parasitoid Models

96

(2)

Host-Parasite Model

98

(1)

Predator-Prey Model

99

(4)

Population Genetics Models

103

(7)

Nonlinear Structured Models

110

(13)

Density-Dependent Leslie Matrix Models

110

(6)

Structured Model for Flour Beetle Populations

116

(2)

Structured Model for the Northern Spotted Owl

118

(3)

Two-Sex Model

121

(2)

Measles Model with Vaccination

123

(4)

Exercises for Chapter 3

127

(7)

References for Chapter 3

134

(4)

Appendix for Chapter 3

138

(3)

Maple Program: Nicholson-Bailey Model

138

(1)

Whooping Crane Data

138

(1)

Waterfowl Data

139

(2)

Linear Differential Equations: Theory and Examples

141

(35)

Introduction

141

(1)

Basic Definitions and Notation

142

(2)

First-Order Linear Differential Equations

144

(1)

Higher-Order Linear Differential Equations

145

(5)

Constant Coefficients

146

(4)

Routh-Hurwitz Criteria

150

(2)

Converting Higher-Order Equations to First-Order Systems

152

(2)

First-Order Linear Systems

154

(3)

Constant Coefficients

155

(2)

Phase-Plane Analysis

157

(5)

Gershgorin's Theorem

162

(1)

An Example: Pharmacokinetics Model

163

(2)

Discrete and Continuous Time Delays

165

(4)

Exercises for Chapter 4

169

(3)

References for Chapter 4

172

(1)

Appendix for Chapter 4

173

(3)

Exponential of a Matrix

173

(2)

Maple Program: Pharmacokinetics Model

175

(1)

Nonlinear Ordinary Differential Equations: Theory and Examples

176

(61)

Introduction

176

(1)

Basic Definitions and Notation

177

(3)

Local Stability in First-Order Equations

180

(4)

Application to Population Growth Models

181

(3)

Phase Line Diagrams

184

(2)

Local Stability in First-Order Systems

186

(5)

Phase Plane Analysis

191

(3)

Periodic Solutions

194

(5)

Poincare-Bendixson Theorem

194

(3)

Bendixson's and Dulac's Criteria

197

(2)

Bifurcations

199

(5)

First-Order Equations

200

(1)

Hopf Bifurcation Theorem

201

(3)

Delay Logistic Equation

204

(7)

Stability Using Qualitative Matrix Stability

211

(5)

Global Stability and Liapunov Functions

216

(5)

Persistence and Extinction Theory

221

(3)

Exercises for Chapter 5

224

(8)

References for Chapter 5

232

(2)

Appendix for Chapter 5

234

(3)

Subcritical and Supercritical Hopf Bifurcations

234

(1)

Strong Delay Kernel

235

(2)

Biological Applications of Differential Equations

237

(62)

Introduction

237

(1)

Harvesting a Single Population

238

(2)

Predator-Prey Models

240

(8)

Competition Models

248

(6)

Two Species

248

(2)

Three Species

250

(4)

Spruce Budworm Model

254

(6)

Metapopulation and Patch Models

260

(3)

Chemostat Model

263

(8)

Michaelis-Menten Kinetics

263

(3)

Bacterial Growth in a Chemostat

266

(5)

Epidemic Models

271

(8)

SI, SIS, and SIR Epidemic Models

271

(5)

Cellular Dynamics of HIV

276

(3)

Excitable Systems

279

(4)

Van der Pol Equation

279

(1)

Hodgkin-Huxley and FitzHugh-Nagumo Models

280

(3)

Exercises for Chapter 6

283

(9)

References for Chapter 6

292

(4)

Appendix for Chapter 6

296

(3)

Lynx and Fox Data

296

(1)

Extinction in Metapopulation Models

296

(3)

Partial Differential Equations: Theory, Examples, and Applications

299

(40)

Introduction

299

(1)

Continuous Age-Structured Model

300

(9)

Method of Characteristics

302

(4)

Analysis of the Continuous Age-Structured Model

306

(3)

Reaction-Diffusion Equations

309

(7)

Equilibrium and Traveling Wave Solutions

316

(3)

Critical Patch Size

319

(2)

Spread of Genes and Traveling Waves

321

(4)

Pattern Formation

325

(5)

Integrodifference Equations

330

(1)

Exercises for Chapter 7

331

(5)

References for Chapter 7

336

(3)

Index

339

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