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.

1. LINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES

1.1 Introduction
1.2 Basic Definitions and Notation
1.3 First-Order 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 Leslie 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

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 First-Order Equations
2.5 Global Stability in First-Order Equations
2.6 The Approximate 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 2.1
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 Model

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 Model
3.6 Predator-Prey Model
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 Owl
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

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-OrderSystems
4.7 First-Order Linear Systems
4.7.1 Constant Coefficients
4.8 Phase-Plane Analysis
4.9 Gershgorin's Theorem
4.10 An 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

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 Poincaré-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 Subcritical and Supercritical Hopf Bifurcations
5.15.2 Strong Delay Kernel

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

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 of the Continuous Age-Structured Model
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