1．Multi-Species Waves and Practical Applications 1
1.1 Intuitive Expectations 1
1.2 Waves of Pursuit and Evasion in Predator-Prey Systems 5
1.3 Competition Model for the Spatial Spread of the Grey Squirrel in Britain 12
1.4 Spread of Genetically Engineered Organisms 18
1.5 Travelling Fronts in the Belousov-Zhabotinskii Reaction 35
1.6 Waves in Excitable Media 41
1.7 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics 49
1.8 Spiral Waves 54
1.9 Spiral Wave Solutions of λ-ωReaction Diffusion Systems 61
Exercises 67
2．Spatial Pattern Formation with Reaction Diffusion Systems 71
2.1 Role of Pattern in Biology 71
2.2 Reaction Diffusion(Turing)Mechanisms 75
2.3 General Conditions for Diffusion-Driven Instability:Linear Stability Analysis and Evolution of Spatial Pattern 82
2.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion Mechanism 90
2.5 Dispersion Relation,Turing Space,Scale and Geometry Effects in Pattern Formation Models 103
2.6 Mode Selection and the Dispersion Relation 113
2.7 Pattern Generation with Single-Species Models:Spatial Heterogeneity with the Spruce Budworm Model 120
2.8 Spatial Patterns in Scalar Population Interaction Diffusion Equations with Convection:Ecological Control Strategies 125
2.9 Nonexistence of Spatial Patterns in Reaction Difrusion Systems:General and Particular Results 130
Exercises 135
3．Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms 141
3.1 Mammalian Coat Patterns—'How the Leopard Got Its Spots' 142
3.2 Teratologies:Examples of Animal Coat Pattern Abnormalities 156
3.3 A Pattern Formation Mechanism for Butterfly Wing Patterns 161
3.4 Modelling Hair Patternsin a Whorl in Acetabularia 180
4．Pattern Formation on Growing Domains:Alligators and Snakes 192
4.1 Stripe Pattern Formation in the Alligator:Experiments 193
4.2 Modelling Concepts:Determining the Time of Stripe Formation 196
4.3 Stripes and Shadow Stripes on the Alligator 200
4.4 Spatial Patterning of Teeth Primordia in the Alligator:Background and Relevance 205
4.5 Biology of Tooth Initiation 207
4.6 Modelling Tooth Primordium Initiation:Background 213
4.7 Model Mechanism for Alligator Teeth Patterning 215
4.8 Results and Comparison with Experimental Data 224
4.9 Prediction Experiments 228
4.10 Concluding Remarks on Alligator Tooth Spatial Patterning 232
4.11 Pigmentation Pattern Formation on Snakes 234
4.12 Cell-Chemotaxis Model Mechanism 238
4.13 Simple and Complex Snake Pattern Elements 241
4.14 Propagating Pattern Generation with the Cell-Chemotaxis System 248
5．Bacterial Patterns and Chemotaxis 253
5.1 Background and Experimental Results 253
5.2 Model Mechanism for E. coli in the Semi-Solid Experiments 260
5.3 Liquid Phase Model:Intuitive Analysis of Pattern Formation 267
5.4 Interpretation of the Analytical Results and Numerical Solutions 274
5.5 Semi-Solid Phase Model Mechanism for S. typhimurium 279
5.6 Linear Analysis of the Basic Semi-Solid Model 281
5.7 Brief Outline and Results of the Nonlinear Analysis 287
5.8 Simulation Results,Parameter Spaces and Basic Patterns 292
5.9 Numerical Results with Initial Conditions from the Experiments 297
5.10 Swarm Ring Patterns with the Semi-Solid Phase Model Mechanism 299
5.11 Branching Patterns in Bacillus subtilis 306
6．Mechanical Theory for Generating Pattern and Form in Development 311
6.1 Introduction,Motivation and Background Biology 311
6.2 Mechanical Model for Mesenchymal Morphogenesis 319
6.3 Linear Analysis,Dispersion Relation and Pattern Formation Potentiai 330
6.4 Simple Mechanical Models Which Generate Spatial Patterns with Complex Dispersion Relations 334
6.5 Periodic Patterns of Feather Germs 345
6.6 Cartilage Condensations in Limb Morphogenesis and Morphogenetic Rules 350
6.7 Embryonic Fingerprint Formation 358
6.8 Mechanochemical Model for the Epidermis 367
6.9 Formation of Microvilli 374
6.10 Complex Pattern Formation and Tissue Interaction Models 381
Exercises 394
7．Evolution,Morphogenetic Laws,Developmental Constraints and Teratologies 396
7.1 Evolution and Morphogenesis 396
7.2 Evolution and Morphogenetic Rules in Cartilage Formation in the Vertebrate Limb 402
7.3 Teratologies(Monsters) 407
7.4 Developmental Constraints,Morphogenetic Rules and the Consequences for Evolution 411
8．A Mechanical Theory of Vascular Network Formation 416
8.1 Biological Background and Motivation 416
8.2 Cell-Extracellular Matrix Interactions for Vasculogenesis 417
8.3 Parameter Values 425
8.4 Analysis of the Model Equations 427
8.5 Network Patterns:Numerical Simulations and Conclusions 433
9．Epidermal Wound Healing 441
9.1 Brief History of Wound Healing 441
9.2 Biological Background:Epidermal Wounds 444
9.3 Model for Epidermal Wound Healing 447
9.4 Nondimensional Form,Linear Stability and Parameter Values 450
9.5 Numerical Solution for the Epidermal Wound Repair Model 451
9.6 Travelling Wave Solutions for the Epidermal Model 454
9.7 Clinical Implications of the Epidermal Wound Model 461
9.8 Mechanisms of Epidermal Repair in Embryos 468
9.9 Actin Alignment in Embryonic Wounds:A Mechanical Model 471
9.10 Mechanical Model with Stress Alignment of the Actin Filaments in Two Dimensions 482
10．Dermal Wound Healing 491
10.1 Background and Motivation—General and Biological 491
10.2 Logic of Wound Healing and Initial Models 495
10.3 Brief Review of Subsequent Developments 500
10.4 Model for Fibroblast-Driven Wound Healing:Residual Strain and Tissue Remodelling 503
10.5 Solutions of the Model Equations and Comparison with Experiment 507
10.6 Wound Healing Model of Cook(1995) 511
10.7 Matrix Secretion and Degradation 515
10.8 CellMovement in an Oriented Environment 518
10.9 Model System for Dermal Wound Healing with Tissue Structure 521
10.10 One-Dimensional Model for the Structure of Pathological Scars 526
10.11 Open Problems in Wound Healing 530
10.12 Concluding Remarks on Wound Healing 533
11．Growthand Control of Brain Tumours 536
11.1 Medical Background 538
11.2 Basic Mathematical Model of Glioma Growth and Invasion 542
11.3 Tumour Spread In Vitro:Parameter Estimation 550
11.4 Tumour Invasionin the Rat Brain 559
11.5 Tumour Invasion in the Human Brain 563
11.6 Modelling Treatment Scenarios:General Comments 579
11.7 Modelling Tumour Resection in Homogeneous Tissue 580
11.8 Analytical Solution for Tumour Recurrence After Resection 584
11.9 Modelling Surgical Resection with Brain Tissue Heterogeneity 588
11.10 Modelling the Effect of Chemotherapy on Tumour Growth 594
11.11 Modelling Tumour Polyclonality and Cell Mutation 605
12．Neural Models of Pattern Formation 614
12.1 Spatial Patterning in Neural Firng with a Simple Activation-Inhibition Model 614
12.2 AMechanism for Stripe Formation in the Visual Cortex 622
12.3 A Model for the Brain Mechanism Underlying Visual Hallucination Patterns 627
12.4 Neural Activity Model for Shell Patterns 638
12.5 Shamanism and Rock Art 655
Exercises 659
13．Geographic Spread and Control of Epidemics 661
13.1 Simple Model for the Spatial Spread of an Epidemic 661
13.2 Spread ofthe BlackDeath in Europe 1347-1350 664
13.3 Brief History of Rabies:Facts and Myths 669
13.4 The Spatial Spread of Rabies Among Foxes Ⅰ:Background and Simple Model 673
13.5 The Spatial Spread of Rabies Among Foxes Ⅱ:Three-Species(SIR) Model 681
13.6 Control Strategy Based on Wave Propagation into a Nonepidemic Region:Estimate of Width of a Rabies Barrier 696
13.7 Analytic Approximation for the Width of the Rabies Control Break 700
13.8 Two-Dimensional Epizootic Fronts and Effects of Variable Fox Densities:Quantitative Predictions for a Rabies Outbreak in England 704
13.9 Effect of Fox Immunity on the Spatial Spread of Rabies 710
Exercises 720
14．Wolf Territoriality,Wolf-Deer Interaction and Survival 722
14.1 Introduction and Wolf Ecology 722
14.2 Models for Wolf Pack Territory Formation:Single Pack—Home Range Model 729
14.3 Multi-Wolf Pack Territorial Model 734
14.4 Wolf-Deer Predator-Prey Model 745
14.5 Concluding Remarks on Wolf Territoriality and Deer Survival 751
14.6 Coyote Home Range Patterns 753
14.7 Chippewa and Sioux Intertribal Conflict c1750-1850 754
Appendix 757
A．General Results for the Laplacian Operator in Bounded Domains 757
Bibliography 761
Index 791