1．Continuous Population Models for Single Species 1
1.1 Continuous Growth Models 1
1.2 Insect Outbreak Model：Spruce Budworm 4
1.3 Delay Models 8
1.4 Linear Analysis of Delay Population Models：Periodic Solutions 12
1.5 Delay Models in Physiology：Dynamic Diseases 15
1.6 Harvesting a Single Natural Population 24
1.7 Population Model with Age Distribution 29
Exercises 33
2．Discrete Population Models for a Single Species 36
2.1 Introduction：Simple Models 36
2.2 Cobwebbing：A Graphical Procedure of Solution 38
2.3 Discrete Logistic Model：Chaos 41
2.4 Stability,Periodic Solutions and Bifurcations 47
2.5 Discrete Delay Models 51
2.6 Fishery Management Model 54
2.7 Ecological Implications and Caveats 57
Exercises 59
3．Continuous Models for Interacting Populations 63
3.1 Predator-Prey Models：Lotka-Volterra Systems 63
3.2 Complexity and Stability 68
3.3 Realistic Predator-Prey Models 70
3.4 Analysis of a Predator-Prey Model with Limit Cycle Periodic Behaviour：Parameter Domains of Stability 72
3.5 Competition Models：Principle of Competitive Exclusion 78
3.6 Mutualism or Symbiosis 83
3.7 General Models and Some General and Cautionary Remarks 85
3.8 Threshold Phenomena 89
Exercises 92
4．Discrete Growth Models for Interacting Populations 95
4.1 Predator-Prey Models：Detailed Analysis 96
4.2 Synchronized Insect Emergence：13 Year Locusts 100
4.3 Biological Pest Control：General Remarks 106
Exercises 107
5．Reaction Kinetics 109
5.1 Enzyme Kinetics：Basic Enzyme Reaction 109
5.2 Michaelis-Menten Theory：Detailed Analysis and the Pseudo-Steady State Hypothesis 111
5.3 Cooperative Phenomena 118
5.4 Autocatalysis,Activation and Inhibition 122
5.5 Multiple Steady States,Mushrooms and Isolas 130
Exercises 137
6．Biological Oscillators and Switches 140
6.1 Motivation,History and Background 140
6.2 Feedback Control Mechanisms 143
6.3 Oscillations and Switches Involving Two or More Species：General Qualitative Results 148
6.4 Simple Two-Species Oscillators：Parameter Domain Determination for Oscillations 156
6.5 Hodgkin-Huxley Theory of Nerve Membranes：FitzHugh-Nagumo Model 161
6.6 Modelling the Control of Testosterone Secretion 166
Exercises 175
7．Belousov-Zhabotinskii Reaction 179
7.1 Belousov Reaction and the Field-Noyes（FN）Model 179
7.2 Linear Stability Analysis of the FN Model and Existence of Limit Cycle Solutions 183
7.3 Non-local Stability of the FN Model 187
7.4 Relaxation Oscillators：Approximation for the Belousov-Zhabotinskii Reaction 190
7.5 Analysis of a Relaxation Model for Limit Cycle Oscillations in the Belousov-Zhabotinskii Reaction 192
Exercises 199
8．Perturbed and Coupled Oscillators and Black Holes 200
8.1 Phase Resetting in Oscillators 200
8.2 Phase Resetting Curves 204
8.3 Black Holes 208
8.4 Black Holes in Real Biological Oscillators 210
8.5 Coupled Oscillators：Motivation and Model System 215
8.6 Singular Perturbation Analysis：Preliminary Transformation 217
8.7 Singular Perturbation Analysis：Transformed System 220
8.8 Singular perturbation Analysis：Two-Time Expansion 223
8.9 Analysis of the Phase Shift Equation and Application to Coupled Belousov-Zhabotinskii Reactions 227
Exercises 231
9．Reaction Diffusion,Chemotaxis and Non-local Mechanisms 232
9.1 Simple Random Walk Derivation of the Diffusion Equation 232
9.2 Reaction Diffusion Equations 236
9.3 Models for Insect Dispersal 238
9.4 Chemotaxis 241
9.5 Non-local Effects and Long Range Diffusion 244
9.6 Cell Potential and Energy Approach to Diffusion 249
Exercises 252
10．Oscillator Generated Wave Phenomena and Central Pattern Generators 254
10.1 Kinematic Waves in the Belousov-Zhabotinskii Reaction 254
10.2 Central Pattern Generator：Experimental Facts in the Swimming of Fish 258
10.3 Mathematical Model for the Central Pattern Generator 261
10.4 Analysis of the Phase-Coupled Model System 268
Exercises 273
11．Biological Waves：Single Species Models 274
11.1 Background and the Travelling Wave Form 274
11.2 Fisher Equation and Propagating Wave Solutions 277
11.3 Asymptotic Solution and Stability of Wavefront Solutions of the Fisher Equation 281
11.4 Density-Dependent Diffusion Reaction Diffusion Models and Some Exact Solutions 286
11.5 Waves in Models with Multi-Steady State Kinetics：The Spread and Control of an Insect Population 297
11.6 Calcium Waves on Amphibian Eggs：Activation Waves on Medaka Eggs 305
Exercises 309
12．Biological Waves：Multi-species Reaction Diffusion Models 311
12.1 Intuitive Expectations 311
12.2 Waves of Pursuit and Evasion in Predator-Prey Systems 315
12.3 Travelling Fronts in the Belousov-Zhabotinskii Reaction 322
12.4 Waves in Excitable Media 328
12.5 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics 336
12.6 Linear Stability of Wave Train Solutions of λ-ω Systems 340
12.7 Spiral Waves 343
12.8 Spiral Wave Solutions of λ-ω Reaction Diffusion Systems 350
Exercises 356
13．Travelling Waves in Reaction Diffusion Systems with Weak Diffusion：Analytical Techniques and Results 360
13.1 Reaction Diffusion System with Limit Cycle Kinetics and Weak Diffusion：Model and Transformed System 360
13.2 Singular Perturbation Analysis：The Phase Satisfies Burgers'Equation 363
13.3 Travelling Wavetrain Solutions for Reaction Diffusion Systems with Limit Cycle Kinetics and Weak Diffusion：Comparison with Experiment 367
14．Spatial Pattern Formation with Reaction/Population Interaction Diffusion Mechanisms 372
14.1 Role of Pattern in Developmental Biology 372
14.2 Reaction Diffusion（Turing）Mechanisms 375
14.3 Linear Stability Analysis and Evolution of Spatial Pattern：General Conditions for Diffusion-Driven Instability 380
14.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion Mechanism 387
14.5 Dispersion Relation,Turing Space,Scale and Geometry Effects in Pattern Formation in Morphogenetic Models 397
14.6 Mode Selection and the Dispersion Relation 408
14.7 Pattern Generation with Single Species Models：Spatial Heterogeneity with the Spruce Budworm Model 414
14.8 Spatial Patterns in Scalar Population Interaction-Reaction Diffusion Equations with Convection：Ecological Control Strategies 419
14.9 Nonexistence of Spatial Patterns in Reaction Diffusion Systems：General and Particular Results 424
Exercises 430
15．Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms 435
15.1 Mammalian Coat Patterns-‘How the Leopard Got Its Spots’ 436
15.2 A Pattern Formation Mechanism for Butterfly Wing Patterns 448
15.3 Modelling Hair Patterns in a Whorl in Acetabularia 468
16．Neural Models of Pattern Formation 481
16.1 Spatial Patterning in Neural Firing with a Simple Activation Inhibition Model 481
16.2 A Mechanism for Stripe Formation in the Visual Cortex 489
16.3 A Model for the Brain Mechanism Underlying Visual Hallucination Patterns 494
16.4 Neural Activity Model for Shell Patterns 505
Exercises 523
17．Mechanical Models for Generating Pattern and Form in Development 525
17.1 Introduction and Background Biology 525
17.2 Mechanical Model for Mesenchymal Morphogenesis 528
17.3 Linear Analysis,Dispersion Relation and Pattern Formation Potential 538
17.4 Simple Mechanical Models Which Generate Spatial Patterns with Complex Dispersion Relations 542
17.5 Periodic Patterns of Feather Germs 554
17.6 Cartilage Condensations in Limb Morphogenesis 558
17.7 Mechanochemical Model for the Epidermis 566
17.8 Travelling Wave Solutions of the Cytogel Model 572
17.9 Formation of Microvilli 579
17.10 Other Applications of Mechanochemical Models 586
Exercises 590
18．Evolution and Developmental Programmes 593
18.1 Evolution and Morphogenesis 593
18.2 Evolution and Morphogenetic Rules in Cartilage Formation in the Vertebrate Limb 599
18.3 Developmental Constraints,Morphogenetic Rules and the Consequences for Evolution 606
19．Epidemic Models and the Dynamics of Infectious Diseases 610
19.1 Simple Epidemic Models and Practical Applications 611
19.2 Modelling Venereal Diseases 619
19.3 Multigroup Model for Gonorrhea and Its Control 623
19.4 AIDS：Modelling the Transmission Dynamics of the Human Immunodeficiency Virus（HIV） 624
19.5 Modelling the Population Dynamics of Acquired Immunity to Parasite Infection 630
19.6 Age Dependent Epidemic Model and Threshold Criterion 640
19.7 Simple Drug Use Epidemic Model and Threshold Analysis 645
Exercises 649
20．Geographic Spread of Epidemics 651
20.1 Simple Model for the Spatial Spread of an Epidemic 651
20.2 Spread of the Black Death in Europe 1347-1350 655
20.3 The Spatial Spread of Rabies Among Foxes Ⅰ：Background and Simple Model 659
20.4 The Spatial Spread of Rabies Among Foxes Ⅱ：Three Species（SIR）Model 666
20.5 Control Strategy Based on Wave Propagation into a Non-epidemic Region：Estimate of Width of a Rabies Barrier 681
20.6 Two-Dimensional Epizootic Fronts and Effects of Variable Fox Densities：Quantitative Predictions for a Rabies Outbreak in England 689
Exercises 696
Appendices 697
1．Phase Plane Analysis 697
2．Routh-Hurwitz Conditions,Jury Conditions,Descartes'Rule of Signs and Exact Solutions of a Cubic 702
3．Hopf Bifurcation Theorem and Limit Cycles 706
4．General Results for the Laplacian Operator in Bounded Domains 720
Bibliography 723
Index 745