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 MATLAB® 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 Poincaré-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
