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. - See more at: http://www.pearsonhighered.com/educator/product/Introduction-to-Mathematical-Biology-An/9780130352163.page#sthash.7adEVTrk.dpuf
Features
• 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.
Author(s): Linda J.S. Allen
Publisher: Pearson
Year: 2006
Language: English
Pages: xii+368
PREFACE
Chapter 1 LINEAR DIFFERENCE EQUATIONS; THEORY; AND EXAMPLES
1.1 Introduction
1.2 Basic Definitions and Notation
1.3 FirstOrder Equations
1.4 Second-Order and Higher-Order Equations
1.5 First-Order Linear Systems
1.6 An Example: Leslie's Age-Structured Model
1.7 Properties of the Les'ie Matrix
1.8 Exercises for Chapter 1
1.9 References for Chapter 1
1.10 Appendix for Chapter 1
1.10.1 Maple Program: Turtle Model
1.10.2 MATLAB® Program: Turtle Model
Chapter 2 NONLINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES
2.1 Introduction
2.2 Basic Definitions and Notation
2.3 Local Stability in First-order Equations
2.4 Cobwebbing Method for FirstOrder Equations
2.5 Global Stability in FirstOrder Equations
2.6 The A pproximate Logistic Equation
2.7 Bifurcation Theory
2.7.1 Types of Bifurcations
2.7.2 Liapunov Exponents
2.8 Stability in First-Order Systems
2.9 Jury Conditions
2.10 An Example: Epidemic Model
2.11 Delay Difference Equations
2.12 Exercises for Chapter 2
2.13 References for Chapter 2
2.14 Appendix for Chapter 2
2.14.1 Proof of Theorem 2.6
2.14.2 A Definition of Chaos
2.14.3 Jury Conditions (Schur-Cohn Criteria)
2.14.4 Liapunov Exponents for Systems of Difference Equations
2.14.5 MATLAB Program: SIR Epidemic Mode(
Chapter 3 BIOLOGICAL APPLICATIONS OF DIFFERENCE EQUATIONS
3.1 Introduction
3.2 Population Models
3.3 Nicholson-Bailey Model
3.4 Other Host-Parasitoid Models
3.5 Host-Parasite Models
3.6 Predator-Prey Models
3.7 Population Genetics Models
3.8 Nonlinear Structured Models
3.8.1 Density-Dependent Leslie Matrix Models
3.8.2 Structured Model for Flour Beetle Populations
3.8.3 Structured Model for the Northern Spotted Cowl
3.8.4 Two-Sex Model
3.9 Measles Model with Vaccination
3.10 Exercises for Chapter 3
3.11 References for Chapter 3
3.12 Appendix for Chapter 3
3.12.1 Maple Program; Nicholson-Bailey Model
3.12.2 Whooping Crane Data
3.12.3 Waterfowl Data
Chapter 4 LINEAR DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES
4.1 Introduction
4.2 basic Definitions and Notation
4.3 First-Order Linear Differential Equations
4.4 Higher-Order Linear Differential Equations
4.4.1 Constant Coefficients
4.5 Routh-Hurwitz Criteria
4.6 Converting Higher-Order Equations to First-Order Systems
4.7 First-Order Linear Systems
4.7.1 Constant Coefficients
4.8 Phase Plane Analysis
4.9 Gershgorin's Theorem
4.10 Tn Example: Pharmacokinetics Model
4.11 Discrete and Continuous Time Delays
4.12 Exercises for Chapter 4
4.13 References for Chapter 4
4.14 Appendix for Chapter 4
4.14.1 Exponential of a Matrix
4.14.2 Maple Program; Pharmacokinetics Model
Chapter 5 NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES
5.1 introduction
5.2 Basic Definitions and Notation
5.3 Local Stability in First-Order Equations
5.3.1 Application to Population Growth Models
5.4 Phase Line Diagrams
5.5 Local Stability in First-Order Systems
5.6 Phase Plane Analysis
5.7 Periodic Solutions
5.7.1 Poincare-Bendixson Theorem
5.7.2 Bendixson's and Dulac's Criteria
5.8 Bifurcations
5.8.1 First-Order Equations
5.8.2 Hopf Bifurcation Theorem
5.9 Delay Logistic Equation
5.10 Stability using Qualitative Matrix Stability
5.11 Global Stability and Liapunov Functions
5.12 Persistence and Extinction Theory
5.13 Exercises for Chapter 5
5.14 References for Chapter 5
5.15 Appendix for Chapter 5
5.15.1 Suberitical and Supercritical Hopf Bifurcations
5.15.2 Strong Delay Kernel
Chapter 6 BIOLOGICAL APPLICATIONS OF DIFFERENTIAL EQUATIONS
6.1 Introduction
6.2 Harvesting a Single Population
6.3 Predator-Prey Models
6.4 Competition Models
6.4.1 Two Species
6.4.2 Three Species
6.5 Spruce Budworm Model
6.6 Metapopulation and Patch Models
6.7 Chemostat Model
6.7.1 Michaelis-Menten Kinetics
6.7.2 Bacterial Growth in a Chemostat
6.8 Epidemic Models
6.8.1 SI, SIS, and SIR Epidemic Models
6.8.2 Cellular Dynamics of HIV
6.9 Excitable Systems
6.9.1 Van der Pol Equation
6.9.2 Hodgkin-Huxley and FitzHugh-Nagumo Models
6.10 Exercises for Chapter 6
6.11 References for Chapter 6
6.12 Appendix for Chapter 6
6.12.1 Lynx and Fox Data
6.12.2 Extinction in Metapopulation Models
Chapter 7 PARTIAL DIFFERENTIAL EQUATIONS: THEORY, EXAMPLES, AND APPLICATIONS
7.1 Introduction
7.2 Continuous Age-Structured Model
7.2.1 Method of Characteristics
7.2.2 Analysis o the Continuous Age-Structured Mode!
7.3 Reaction-Diffusion Equations
7.4 Equilibrium and Traveling Wave Solutions
7.5 Critical Patch Size
7.6 Spread of Genes and Traveling Waves
7.7 Pattern Formation
7.8 Integrodifference Equations
7.9 Exercises for Chapter 7
7.10 References for Chapter 7
INDEX
