# Introduction to Mathematical Biology, An, 1st edition

Published by Pearson (July 19, 2006) Â© 2007

**Linda J.S. Allen**

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For advanced undergraduate and beginning graduate courses on Modeling offered in departments of Mathematics.

This text 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. Undergraduate courses in calculus, linear algebra, and differential equations are assumed.

â€¢ **Applications of mathematical theory to biological examples in each chapter.**

â€“Â Â Â Â Â Â Similar biological applications may appear in more than one chapter. For example, epidemic models and predator-prey models are formulated in terms of difference equations in Chapters 2 and 3 and as differential equations in Chapter 6.

â€“Â Â Â Â Â Â In this way, the advantages and disadvantages of the various model formulations can be compared.

â€“Â Â Â Â Â Â Chapters 3 and 6 are devoted primarily to biological applications. The instructor may be selective about the applications covered in these two chapters.

â€¢ **Focus on deterministic mathematical models, **models formulated as difference equations or ordinary differential equations, with an emphasis on predicting the qualitative solution behavior over time.

â€¢ **Discussion of classical mathematical models from population biology** â€“ Includes the Leslie matrix model, the Nicholson-Bailey model, and the Lotka-Volterra predator-prey model.

â€“ Also discusses more recent models, such as a model for the Human Immunodeficiency Virus - HIV and a model for flour beetles.

â€¢ **A review of the basic theory of linear difference equations and linear differential equations** (Chapters 1 and 4, respectively).

â€“Â Â Â Â Â Â Can be covered very briefly or in more detail, depending on the studentsâ€™ background.

â€“Â Â Â Â Â Â Difference equation models are presented in Chapters 1, 2, and 3.

â€¢ **Coverage of ordinary differential equation models** â€“ Includes an introduction to partial differential equation models in biology.

â€¢ **Exercises at the end of each chapter** to reinforce concepts discussed.

â€¢ **MATLAB and Maple programs in the appendices** â€“ Encourages students to use these programs to visualize the dynamics of various models.

â€“Â Â Â Â Â Â Can be modified for other types of models or adapted to other programming languages.

â€“Â Â Â Â Â Â Research topics assigned on current biological models that have appeared in the literature can be part of an individual or a group research project.

â€¢ **Lists of useful references for additional biological applications **at the end of each chapter.

Preface **xi**

1 LINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES **1**

1.1Introduction1

1.2Basic Definitions and Notation2

1.3First-Order Equations6

1.4Second-Order and Higher-Order Equations8

1.5First-Order Linear Systems14

1.6An Example: Leslieâ€™s Age-Structured Model18

1.7Properties of the Leslie Matrix20

1.8Exercises for Chapter 128

1.9References for Chapter 133

1.10Appendix for Chapter 134Â Â Â Â 1.10.1 Maple Program:Turtle Model

34Â Â Â Â 1.10.2 MATLABÂ® Program:Turtle Model

34

2 NONLINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES **36**

2.1Introduction36

2.2Basic Definitions and Notation37

2.3Local Stability in First-Order Equations40

2.4Cobwebbing Method for First-Order Equations45

2.5Global Stability in First-Order Equations46

2.6The Approximate Logistic Equation52

2.7Bifurcation Theory55Â Â Â Â 2.7.1 Types of Bifurcations

56Â Â Â Â 2.7.2 Liapunov Exponents

60

2.8Stability in First-Order Systems62

2.9Jury Conditions67

2.10An Example: Epidemic Model69

2.11Delay Difference Equations73

2.12Exercises for Chapter 276

2.13References for Chapter 282

2.14Appendix for Chapter 284Â Â Â Â 2.14.1 Proof of Theorem 2.1

84Â Â Â Â 2.14.2 A Definition of Chaos

86Â Â Â Â 2.14.3 Jury Conditions (Schur-Cohn Criteria)

86Â Â Â Â 2.14.4 Liapunov Exponents for Systems of Difference Equations

87Â Â Â Â 2.14.5 MATLAB Program: SIR Epidemic Model

88

3 BIOLOGICAL APPLICATIONS OF DIFFERENCE EQUATIONS **89**

3.1Introduction89

3.2Population Models90

3.3Nicholson-Bailey Model92

3.4Other Host-Parasitoid Models96

3.5Host-Parasite Model98

3.6Predator-Prey Model99

3.7Population Genetics Models103

3.8Nonlinear Structured Models110Â Â Â Â 3.8.1 Density-Dependent Leslie Matrix Models

110Â Â Â Â 3.8.2 Structured Model for Flour Beetle Populations

116Â Â Â Â 3.8.3 Structured Model for the Northern Spotted Owl

118Â Â Â Â 3.8.4 Two-Sex Model

121

3.9Measles Model with Vaccination123

3.10Exercises for Chapter 3127

3.11References for Chapter 3134

3.12Appendix for Chapter 3138Â Â Â Â 3.12.1 Maple Program: Nicholson-Bailey Model

138Â Â Â Â 3.12.2 Whooping Crane Data

138Â Â Â Â 3.12.3 Waterfowl Data

139

4 LINEAR DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES **141**

4.1Introduction141

4.2Basic Definitions and Notation142

4.3First-Order Linear Differential Equations144

4.4Higher-Order Linear Differential Equations145Â Â Â Â 4.4.1 Constant Coefficients

146

4.5Routh-Hurwitz Criteria150

4.6Converting Higher-Order Equations to First-OrderSystems152

4.7First-Order Linear Systems154Â Â Â Â 4.7.1 Constant Coefficients

155

4.8Phase-Plane Analysis157

4.9Gershgorinâ€™s Theorem162

4.10An Example: Pharmacokinetics Model163

4.11Discrete and Continuous Time Delays165

4.12Exercises for Chapter 4169

4.13References for Chapter 4172

4.14Appendix for Chapter 4173Â Â Â Â 4.14.1 Exponential of a Matrix

173Â Â Â Â 4.14.2 Maple Program: Pharmacokinetics Model

175

5 NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES **176**

5.1Introduction176

5.2Basic Definitions and Notation177

5.3Local Stability in First-Order Equations180Â Â Â Â 5.3.1 Application to Population Growth Models

181

5.4Phase Line Diagrams184

5.5Local Stability in First-Order Systems186

5.6Phase Plane Analysis191

5.7Periodic Solutions194Â Â Â Â 5.7.1 PoincarÃ©-Bendixson Theorem

194Â Â Â Â 5.7.2 Bendixsonâ€™s and Dulacâ€™s Criteria

197

5.8Bifurcations199Â Â Â Â 5.8.1 First-Order Equations

200Â Â Â Â 5.8.2 Hopf Bifurcation Theorem

201

5.9Delay Logistic Equation204

5.10Stability Using Qualitative Matrix Stability211

5.11Global Stability and Liapunov Functions216

5.12Persistence and Extinction Theory221

5.13Exercises for Chapter 5224

5.14References for Chapter 5232

5.15Appendix for Chapter 5234Â Â Â Â 5.15.1 Subcritical and Supercritical Hopf Bifurcations

234Â Â Â Â 5.15.2 Strong Delay Kernel

235

6 BIOLOGICAL APPLICATIONS OF DIFFERENTIAL EQUATIONS **237**

6.1Introduction237

6.2Harvesting a Single Population238

6.3Predator-Prey Models240

6.4Competition Models248Â Â Â Â 6.4.1 Two Species

248Â Â Â Â 6.4.2 Three Species

250

6.5Spruce Budworm Model254

6.6Metapopulation and Patch Models260

6.7Chemostat Model263Â Â Â Â 6.7.1 Michaelis-Menten Kinetics

263Â Â Â Â 6.7.2 Bacterial Growth in a Chemostat

266

6.8Epidemic Models271Â Â Â Â 6.8.1 SI, SIS, and SIR Epidemic Models

271Â Â Â Â 6.8.2 Cellular Dynamics of HIV

276

6.9Excitable Systems279Â Â Â Â 6.9.1 Van der Pol Equation

279Â Â Â Â 6.9.2 Hodgkin-Huxley and FitzHugh-Nagumo Models

280

6.10Exercises for Chapter 6283

6.11References for Chapter 6292

6.12Appendix for Chapter 6296Â Â Â Â 6.12.1 Lynx and Fox Data

296Â Â Â Â 6.12.2 Extinction in Metapopulation Models

296

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

7.1Introduction299

7.2Continuous Age-Structured Model300Â Â Â Â 7.2.1 Method of Characteristics

302Â Â Â Â 7.2.2 Analysis of the Continuous Age-Structured Model

306

7.3Reaction-Diffusion Equations309

7.4Equilibrium and Traveling Wave Solutions316

7.5Critical Patch Size319

7.6Spread of Genes and Traveling Waves321

7.7Pattern Formation325

7.8Integrodifference Equations330

7.9Exercises for Chapter 7331

7.10References for Chapter 7336

Index **339**

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