Mathematical Biology: I. An Introduction

Table of Contents......Page 16
Preface to the Third Edition......Page 8
Preface to the First Edition......Page 12
1.1 Continuous Growth Models......Page 26
1.2 Insect Outbreak Model: Spruce Budworm......Page 32
1.3 Delay Models......Page 38
1.4 Linear Analysis of Delay Population Models: Periodic Solutions......Page 42
1.5 Delay Models in Physiology: Periodic Dynamic Diseases......Page 46
1.6 Harvesting a Single Natural Population......Page 55
1.7 Population Model with Age Distribution......Page 61
Exercises......Page 65
2.1 Introduction: Simple Models......Page 69
2.2 Cobwebbing: A Graphical Procedure of Solution......Page 74
2.3 Discrete Logistic-Type Model: Chaos......Page 78
2.4 Stability, Periodic Solutions and Bifurcations......Page 84
2.5 Discrete Delay Models......Page 87
2.6 Fishery Management Model......Page 92
2.7 Ecological Implications and Caveats......Page 94
2.8 Tumour Cell Growth......Page 97
Exercises......Page 100
3.1 Predator–Prey Models: Lotka–Volterra Systems......Page 104
3.2 Complexity and Stability......Page 108
3.3 Realistic Predator–Prey Models......Page 111
3.4 Analysis of a Predator–Prey Model with Limit Cycle Periodic Behaviour: Parameter Domains of Stability......Page 113
3.5 Competition Models: Competitive Exclusion Principle......Page 119
3.6 Mutualism or Symbiosis......Page 124
3.7 General Models and Cautionary Remarks......Page 126
3.8 Threshold Phenomena......Page 130
3.9 Discrete Growth Models for Interacting Populations......Page 134
3.10 Predator–Prey Models: Detailed Analysis......Page 135
Exercises......Page 140
4.1 Biological Introduction and Historical Asides on the Crocodilia......Page 144
4.2 Nesting Assumptions and Simple Population Model......Page 149
4.3 Age-Structured Population Model for Crocodilia......Page 155
4.4 Density-Dependent Age-Structured Model Equations......Page 158
4.5 Stability of the Female Population in Wet Marsh Region I......Page 160
4.6 Sex Ratio and Survivorship......Page 162
4.7 Temperature-Dependent Sex Determination (TSD) Versus Genetic Sex Determination (GSD)......Page 164
4.8 Related Aspects on Sex Determination......Page 167
Exercise......Page 169
5. Modelling the Dynamics of Marital Interaction: Divorce Prediction and Marriage Repair......Page 171
5.1 Psychological Background and Data: Gottman and Levenson Methodology......Page 172
5.2 Marital Typology and Modelling Motivation......Page 175
5.3 Modelling Strategy and the Model Equations......Page 178
5.4 Steady States and Stability......Page 181
5.5 Practical Results from the Model......Page 189
5.6 Benefits, Implications and Marriage Repair Scenarios......Page 195
6.1 Enzyme Kinetics: Basic Enzyme Reaction......Page 200
6.2 Transient Time Estimates and Nondimensionalisation......Page 203
6.3 Michaelis–Menten Quasi-Steady State Analysis......Page 206
6.4 Suicide Substrate Kinetics......Page 213
6.5 Cooperative Phenomena......Page 222
6.6 Autocatalysis, Activation and Inhibition......Page 226
6.7 Multiple Steady States, Mushrooms and Isolas......Page 233
Exercises......Page 240
7.1 Motivation, Brief History and Background......Page 243
7.2 Feedback Control Mechanisms......Page 246
7.3 Oscillators and Switches with Two or More Species: General Qualitative Results......Page 251
7.4 Simple Two-Species Oscillators: Parameter Domain Determination for Oscillations......Page 259
7.5 Hodgkin–Huxley Theory of Nerve Membranes: FitzHugh–Nagumo Model......Page 264
7.6 Modelling the Control of Testosterone Secretion and Chemical Castration......Page 269
Exercises......Page 278
8.1 Belousov Reaction and the Field–Körös–Noyes (FKN) Model......Page 282
8.2 Linear Stability Analysis of the FKN Model and Existence of Limit Cycle Solutions......Page 286
8.3 Nonlocal Stability of the FKN Model......Page 290
8.4 Relaxation Oscillators: Approximation for the Belousov–Zhabotinskii Reaction......Page 293
8.5 Analysis of a Relaxation Model for Limit Cycle Oscillations in the Belousov–Zhabotinskii Reaction......Page 296
Exercises......Page 302
9.1 Phase Resetting in Oscillators......Page 303
9.2 Phase Resetting Curves......Page 307
9.3 Black Holes......Page 311
9.4 Black Holes in Real Biological Oscillators......Page 313
9.5 Coupled Oscillators: Motivation and Model System......Page 318
9.6 Phase Locking of Oscillations: Synchronisation in Fireflies......Page 320
9.7 Singular Perturbation Analysis: Preliminary Transformation......Page 324
9.8 Singular Perturbation Analysis: Transformed System......Page 327
9.9 Singular Perturbation Analysis: Two-Time Expansion......Page 330
9.10 Analysis of the Phase Shift Equation and Application to Coupled Belousov–Zhabotinskii Reactions......Page 335
Exercises......Page 338
10.1 Historical Aside on Epidemics......Page 340
10.2 Simple Epidemic Models and Practical Applications......Page 344
10.3 Modelling Venereal Diseases......Page 352
10.4 Multi-Group Model for Gonorrhea and Its Control......Page 356
10.5 AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV)......Page 358
10.6 HIV: Modelling Combination Drug Therapy......Page 366
10.7 Delay Model for HIV Infection with Drug Therapy......Page 375
10.8 Modelling the Population Dynamics of Acquired Immunity to Parasite Infection......Page 376
10.9 Age-Dependent Epidemic Model and Threshold Criterion......Page 386
10.10 Simple Drug Use Epidemic Model and Threshold Analysis......Page 390
10.11 Bovine Tuberculosis Infection in Badgers and Cattle......Page 394
10.12 Modelling Control Strategies for Bovine Tuberculosis in Badgers and Cattle......Page 404
Exercises......Page 418
11.1 Simple Random Walk and Derivation of the Diffusion Equation......Page 420
11.2 Reaction Diffusion Equations......Page 424
11.3 Models for Animal Dispersal......Page 427
11.4 Chemotaxis......Page 430
11.5 Nonlocal Effects and Long Range Diffusion......Page 433
11.6 Cell Potential and Energy Approach to Diffusion and Long Range Effects......Page 438
Exercises......Page 441
12.1 Belousov–Zhabotinskii Reaction Kinematic Waves......Page 443
12.2 Central Pattern Generator: Experimental Facts in the Swimming of Fish......Page 447
12.3 Mathematical Model for the Central Pattern Generator......Page 449
12.4 Analysis of the Phase Coupled Model System......Page 456
Exercises......Page 461
13.1 Background and the Travelling Waveform......Page 462
13.2 Fisher–Kolmogoroff Equation and Propagating Wave Solutions......Page 464
13.3 Asymptotic Solution and Stability of Wavefront Solutions of the Fisher–Kolmogoroff Equation......Page 469
13.4 Density-Dependent Diffusion-Reaction Diffusion Models and Some Exact Solutions......Page 474
13.5 Waves in Models with Multi-Steady State Kinetics: Spread and Control of an Insect Population......Page 485
13.6 Calcium Waves on Amphibian Eggs: Activation Waves on Medaka Eggs......Page 492
13.7 Invasion Wavespeeds with Dispersive Variability......Page 496
13.8 Species Invasion and Range Expansion......Page 503
Exercises......Page 507
14.1 Fractals: Basic Concepts and Biological Relevance......Page 509
14.2 Examples of Fractals and Their Generation......Page 512
14.3 Fractal Dimension: Concepts and Methods of Calculation......Page 515
14.4 Fractals or Space-Filling?......Page 521
A. Phase Plane Analysis......Page 526
B.1 Polynomials and Conditions......Page 532
B.2 Descartes\' Rule of Signs......Page 534
B.3 Roots of a General Cubic Polynomial......Page 535
Bibliography......Page 538
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C......Page 564
D......Page 565
F......Page 566
G......Page 567
H......Page 568
L......Page 569
M......Page 570
O......Page 571
P......Page 572
R......Page 573
S......Page 574
W......Page 575
Z......Page 576