- Series
- Addison-Wesley
- Author
- Linda J.S. Allen
- Publisher
- Pearson
- Cover
- Softcover
- Edition
- 1
- Language
- English
- Total pages
- 360
- Pub.-date
- July 2006
- ISBN13
- 9780130352163
- ISBN
- 0130352160
- Related Titles

ISBN | Product | Product | Price CHF | Available | |
---|---|---|---|---|---|

Introduction to Mathematical Biology, An |
9780130352163 Introduction to Mathematical Biology, An |
155.40 | approx. 7-9 days |

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

** **

• **Applications of mathematical theory to biological examples in each chapter.**

- 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.

- In this way, the advantages and disadvantages of the various model formulations can be compared.

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

• **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.

• **Discussion of classical mathematical models from population biology** - Includes 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.

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

- Can be covered very briefly or in more detail, depending on the students' background.

- Difference equation models are presented in Chapters 1, 2, and 3.

• **Coverage of ordinary differential equation models** - Includes an introduction to partial differential equation models in biology.

• **Exercises at the end of each chapter** to reinforce concepts discussed.

• **MATLAB and Maple programs in the appendices** - Encourages students to use these programs to visualize the dynamics of various models.

- Can be modified for other types of models or adapted to other programming languages.

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

• **Lists of useful references for additional biological applications **at the end of each chapter.

Preface **xi**

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

**1.1 **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 MATLABÂ® 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 PoincarĂ©-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**