KEY BENEFIT: This reference 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. Contains applications of mathematical theory to biological examples in each chapter. Focuses on deterministic mathematical models with an emphasis on predicting the qualitative solution behavior over time. Discusses classical mathematical models from population , including 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. KEY MARKET: Readers seeking a solid background in the mathematics behind modeling in biology and exposure to a wide variety of mathematical models in biology.
Author(s): Linda J.S. Allen
Publisher: Pearson
Year: 2006
Language: English
Pages: 368
Tags: Biology;Cell Biology;Developmental Biology;Entomology;Marine Biology;Microbiology;Molecular Biology;Biostatistics;Biological Sciences;Science & Math;Biomathematics;Applied;Mathematics;Science & Math;Biology;Biology & Life Sciences;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique;Mathematics;Algebra & Trigonometry;Calculus;Geometry;Statistics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique
Cover ... 1
AN INTRODUCTION TO MATHEMATICAL BIOLOGY ... 3
Copyright ... 4
© 2007 Pearson Education ... 4
ISBN: D-13-035216-D ... 4
011323.5 A436 2007 570.15118-dc22 ... 4
LCCN 2006042585 ... 4
Dedication ... 5
CONTENTS ... 7
PREFACE ... 13
Chapter 1 LINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES ... 15
1.1 Introduction ... 15
1.2 Basic Definitions and Notation ... 16
1.3 FirstOrder Equations ... 20
1.4 Second-Order and Higher-Order Equations ... 22
1.5 First-Order Linear Systems ... 28
1.6 An Example: Leslie's Age-Structured Model ... 32
1.7 Properties of the Les'ie Matrix ... 34
1.8 Exercises for Chapter 1 ... 42
1.9 References for Chapter 1 ... 47
1.10 Appendix for Chapter 1 ... 48
1.10.1 Maple Program: Turtle Model ... 48
1.10.2 MATLAB® Program: Turtle Model ... 48
Chapter 2 NONLINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES ... 50
2.1 Introduction ... 50
2.2 Basic Definitions and Notation ... 51
2.3 Local Stability in First-order Equations ... 54
2.4 Cobwebbing Method for FirstOrder Equations ... 59
2.5 Global Stability in FirstOrder Equations ... 60
2.6 The A pproximate Logistic Equation ... 66
2.7 Bifurcation Theory ... 69
2.7.1 Types of Bifurcations ... 70
2.7.2 Liapunov Exponents ... 74
2.8 Stability in First-Order Systems ... 76
2.9 Jury Conditions ... 81
2.10 An Example: Epidemic Model ... 83
2.11 Delay Difference Equations ... 87
2.12 Exercises for Chapter 2 ... 90
2.13 References for Chapter 2 ... 96
2.14 Appendix for Chapter 2 ... 98
2.14.1 Proof of Theorem 2.6 ... 98
2.14.2 A Definition of Chaos ... 100
2.14.3 Jury Conditions (Schur-Cohn Criteria) ... 100
2.14.4 Liapunov Exponents for Systems of Difference Equations ... 101
2.14.5 MATLAB Program: SIR Epidemic Mode( ... 102
Chapter 3 BIOLOGICAL APPLICATIONS OF DIFFERENCE EQUATIONS ... 103
3.1 Introduction ... 103
3.2 Population Models ... 104
3.3 Nicholson-Bailey Model ... 106
3.4 Other Host-Parasitoid Models ... 110
3.5 Host-Parasite Models ... 112
3.6 Predator-Prey Models ... 113
3.7 Population Genetics Models ... 117
3.8 Nonlinear Structured Models ... 124
3.8.1 Density-Dependent Leslie Matrix Models ... 124
3.8.2 Structured Model for Flour Beetle Populations ... 130
3.8.3 Structured Model for the Northern Spotted Cowl ... 132
3.8.4 Two-Sex Model ... 135
3.9 Measles Model with Vaccination ... 137
3.10 Exercises for Chapter 3 ... 141
3.11 References for Chapter 3 ... 148
3.12 Appendix for Chapter 3 ... 152
3.12.1 Maple Program; Nicholson-Bailey Model ... 152
3.12.2 Whooping Crane Data ... 152
3.12.3 Waterfowl Data ... 153
Chapter 4 LINEAR DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES ... 155
4.1 Introduction ... 155
4.2 basic Definitions and Notation ... 156
4.3 First-Order Linear Differential Equations ... 158
4.4 Higher-Order Linear Differential Equations ... 159
4.4.1 Constant Coefficients ... 160
4.5 Routh-Hurwitz Criteria ... 164
4.6 Converting Higher-Order Equations to First-Order Systems ... 166
4.7 First-Order Linear Systems ... 168
4.7.1 Constant Coefficients ... 169
4.8 Phase Plane Analysis ... 171
4.9 Gershgorin's Theorem ... 176
4.10 Tn Example: Pharmacokinetics Model ... 177
4.11 Discrete and Continuous Time Delays ... 179
4.12 Exercises for Chapter 4 ... 183
4.13 References for Chapter 4 ... 186
4.14 Appendix for Chapter 4 ... 187
4.14.1 Exponential of a Matrix ... 187
4.14.2 Maple Program; Pharmacokinetics Model ... 189
Chapter 5 NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES ... 190
5.1 introduction ... 190
5.2 Basic Definitions and Notation ... 191
5.3 Local Stability in First-Order Equations ... 194
5.3.1 Application to Population Growth Models ... 195
5.4 Phase Line Diagrams ... 198
5.5 Local Stability in First-Order Systems ... 200
5.6 Phase Plane Analysis ... 205
5.7 Periodic Solutions ... 208
5.7.1 Poincare-Bendixson Theorem ... 208
5.7.2 Bendixson's and Dulac's Criteria ... 211
5.8 Bifurcations ... 213
5.8.1 First-Order Equations ... 214
5.8.2 Hopf Bifurcation Theorem ... 215
5.9 Delay Logistic Equation ... 218
5.10 Stability using Qualitative Matrix Stability ... 225
5.11 Global Stability and Liapunov Functions ... 230
5.12 Persistence and Extinction Theory ... 235
5.13 Exercises for Chapter 5 ... 238
5.14 References for Chapter 5 ... 246
5.15 Appendix for Chapter 5 ... 248
5.15.1 Suberitical and Supercritical Hopf Bifurcations ... 248
5.15.2 Strong Delay Kernel ... 249
Chapter 6 BIOLOGICAL APPLICATIONS OF DIFFERENTIAL EQUATIONS ... 251
6.1 Introduction ... 251
6.2 Harvesting a Single Population ... 252
6.3 Predator-Prey Models ... 254
6.4 Competition Models ... 262
6.4.1 Two Species ... 262
6.4.2 Three Species ... 264
6.5 Spruce Budworm Model ... 268
6.6 Metapopulation and Patch Models ... 274
6.7 Chemostat Model ... 277
6.7.1 Michaelis-Menten Kinetics ... 277
6.7.2 Bacterial Growth in a Chemostat ... 280
6.8 Epidemic Models ... 285
6.8.1 SI, SIS, and SIR Epidemic Models ... 285
6.8.2 Cellular Dynamics of HIV ... 290
6.9 Excitable Systems ... 293
6.9.1 Van der Pol Equation ... 293
6.9.2 Hodgkin-Huxley and FitzHugh-Nagumo Models ... 294
6.10 Exercises for Chapter 6 ... 297
6.11 References for Chapter 6 ... 306
6.12 Appendix for Chapter 6 ... 310
6.12.1 Lynx and Fox Data ... 310
6.12.2 Extinction in Metapopulation Models ... 310
Chapter 7 PARTIAL DIFFERENTIAL EQUATIONS: THEORY, EXAMPLES, AND APPLICATIONS ... 313
7.1 Introduction ... 313
7.2 Continuous Age-Structured Model ... 314
7.2.1 Method of Characteristics ... 316
7.2.2 Analysis o the Continuous Age-Structured Mode! ... 320
7.3 Reaction-Diffusion Equations ... 323
7.4 Equilibrium and Traveling Wave Solutions ... 330
7.5 Critical Patch Size ... 333
7.6 Spread of Genes and Traveling Waves ... 335
7.7 Pattern Formation ... 339
7.8 Integrodifference Equations ... 344
7.9 Exercises for Chapter 7 ... 345
7.10 References for Chapter 7 ... 350
INDEX ... 353
Back Cover ... 365
