Mathematical biology. I, An introduction

CONTENTS, VOLUME I Preface to the Third Edition vii Preface to the First Edition xi 1. Continuous Population Models for Single Species 1 1.1 Continuous Growth Models . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 InsectOutbreakModel:SpruceBudworm . . . . . . . . . . . . . . . 7 1.3 DelayModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Linear Analysis of Delay Population Models: Periodic Solutions . . . 17 1.5 Delay Models in Physiology: Periodic Dynamic Diseases . . . . . . . 21 1.6 Harvesting a Single Natural Population . . . . . . . . . . . . . . . . 30 1.7 Population Model with Age Distribution . . . . . . . . . . . . . . . . 36 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2. Discrete Population Models for a Single Species 44 2.1 Introduction: Simple Models . . . . . . . . . . . . . . . . . . . . . . 44 2.2 Cobwebbing:AGraphicalProcedureofSolution . . . . . . . . . . . 49 2.3 DiscreteLogistic-TypeModel:Chaos . . . . . . . . . . . . . . . . . 53 2.4 Stability, Periodic Solutions and Bifurcations . . . . . . . . . . . . . 59 2.5 DiscreteDelayModels . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.6 FisheryManagementModel . . . . . . . . . . . . . . . . . . . . . . 67 2.7 Ecological Implications andCaveats . . . . . . . . . . . . . . . . . . 69 2.8 TumourCellGrowth . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3. Models for Interacting Populations 79 3.1 Predator–PreyModels:Lotka–VolterraSystems . . . . . . . . . . . . 79 3.2 Complexity and Stability . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 RealisticPredator–PreyModels . . . . . . . . . . . . . . . . . . . . . 86 3.4 Analysis of a Predator–Prey Model with Limit Cycle Periodic Behaviour: Parameter Domains of Stability . . . . . . . . . . 88 3.5 Competition Models: Competitive Exclusion Principle . . . . . . . . 94 xvi Table of Contents, Volume I 3.6 MutualismorSymbiosis . . . . . . . . . . . . . . . . . . . . . . . . 99 3.7 GeneralModels andCautionaryRemarks . . . . . . . . . . . . . . . 101 3.8 ThresholdPhenomena . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.9 Discrete Growth Models for Interacting Populations . . . . . . . . . . 109 3.10 Predator–PreyModels:DetailedAnalysis . . . . . . . . . . . . . . . 110 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4. Temperature-Dependent Sex Determination (TSD) 119 4.1 Biological Introduction and Historical Asides on the Crocodilia . . . . 119 4.2 Nesting Assumptions and Simple Population Model . . . . . . . . . . 124 4.3 Age-Structured Population Model for Crocodilia . . . . . . . . . . . 130 4.4 Density-DependentAge-StructuredModelEquations . . . . . . . . . 133 4.5 Stability of the Female Population in Wet Marsh Region I . . . . . . . 135 4.6 SexRatio andSurvivorship . . . . . . . . . . . . . . . . . . . . . . . 137 4.7 Temperature-Dependent Sex Determination (TSD) Versus GeneticSexDetermination(GSD) . . . . . . . . . . . . . . . . . . . 139 4.8 RelatedAspects onSexDetermination . . . . . . . . . . . . . . . . . 142 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5. Modelling the Dynamics of Marital Interaction: Divorce Prediction and Marriage Repair 146 5.1 Psychological Background and Data: Gottman and Levenson Methodology . . . . . . . . . . . . . . . . . . 147 5.2 Marital Typology and Modelling Motivation . . . . . . . . . . . . . . 150 5.3 Modelling Strategy and the Model Equations . . . . . . . . . . . . . 153 5.4 Steady States and Stability . . . . . . . . . . . . . . . . . . . . . . . 156 5.5 PracticalResults fromtheModel . . . . . . . . . . . . . . . . . . . . 164 5.6 Benefits, Implications andMarriageRepairScenarios . . . . . . . . . 170 6. Reaction Kinetics 175 6.1 EnzymeKinetics:BasicEnzymeReaction . . . . . . . . . . . . . . . 175 6.2 Transient Time Estimates and Nondimensionalisation . . . . . . . . . 178 6.3 Michaelis–MentenQuasi-SteadyStateAnalysis . . . . . . . . . . . . 181 6.4 SuicideSubstrateKinetics . . . . . . . . . . . . . . . . . . . . . . . 188 6.5 Cooperative Phenomena . . . . . . . . . . . . . . . . . . . . . . . . 197 6.6 Autocatalysis, Activation and Inhibition . . . . . . . . . . . . . . . . 201 6.7 Multiple Steady States, Mushrooms and Isolas . . . . . . . . . . . . . 208 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7. Biological Oscillators and Switches 218 7.1 Motivation, Brief History and Background . . . . . . . . . . . . . . . 218 7.2 FeedbackControlMechanisms . . . . . . . . . . . . . . . . . . . . . 221 7.3 Oscillators and Switches with Two or More Species: General Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . 226 7.4 Simple Two-Species Oscillators: Parameter Domain Determination for Oscillations . . . . . . . . . . . . . . . . . . . . . 234 Table of Contents, Volume I xvii 7.5 Hodgkin–Huxley Theory of Nerve Membranes: FitzHugh–Nagumo Model . . . . . . . . . . . . . . . . . . . . . . . 239 7.6 Modelling the Control of Testosterone Secretion and ChemicalCastration. . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 8. BZ Oscillating Reactions 257 8.1 Belousov Reaction and the Field–K¨or¨os–Noyes (FKN)Model . . . . 257 8.2 Linear Stability Analysis of the FKN Model and Existence ofLimitCycleSolutions . . . . . . . . . . . . . . . . . . . . . . . . 261 8.3 Nonlocal Stability of the FKN Model . . . . . . . . . . . . . . . . . 265 8.4 Relaxation Oscillators: Approximation for the Belousov–ZhabotinskiiReaction . . . . . . . . . . . . . . . . . . . . 268 8.5 Analysis of a Relaxation Model for Limit Cycle Oscillations in theBelousov–ZhabotinskiiReaction . . . . . . . . . . . . . . . . . 271 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9. Perturbed and Coupled Oscillators and Black Holes 278 9.1 Phase Resetting in Oscillators . . . . . . . . . . . . . . . . . . . . . 278 9.2 Phase Resetting Curves . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.3 BlackHoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 9.4 Black Holes in Real Biological Oscillators . . . . . . . . . . . . . . . 288 9.5 Coupled Oscillators: Motivation and Model System . . . . . . . . . . 293 9.6 Phase Locking of Oscillations: Synchronisation in Fireflies . . . . . . 295 9.7 Singular Perturbation Analysis: Preliminary Transformation . . . . . 299 9.8 Singular Perturbation Analysis: Transformed System . . . . . . . . . 302 9.9 Singular Perturbation Analysis: Two-Time Expansion . . . . . . . . . 305 9.10 Analysis of the Phase Shift Equation and Application to Coupled Belousov–Zhabotinskii Reactions . . . . . . . . . . . . . 310 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10. Dynamics of Infectious Diseases 315 10.1 HistoricalAsideonEpidemics . . . . . . . . . . . . . . . . . . . . . 315 10.2 SimpleEpidemicModels andPracticalApplications . . . . . . . . . 319 10.3 Modelling Venereal Diseases . . . . . . . . . . . . . . . . . . . . . . 327 10.4 Multi-Group Model for Gonorrhea and Its Control . . . . . . . . . . . 331 10.5 AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV) . . . . . . . . . . . . . . . . . . . . . 333 10.6 HIV: Modelling Combination Drug Therapy . . . . . . . . . . . . . . 341 10.7 DelayModel forHIVInfectionwithDrugTherapy . . . . . . . . . . 350 10.8 Modelling the Population Dynamics of Acquired Immunity to Parasite Infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 10.9 Age-DependentEpidemicModel andThresholdCriterion . . . . . . . 361 10.10 SimpleDrugUseEpidemicModel andThresholdAnalysis . . . . . . 365 10.11 Bovine Tuberculosis Infection in Badgers and Cattle . . . . . . . . . 369 xviii Table of Contents, Volume I 10.12 Modelling Control Strategies for Bovine Tuberculosis in Badgers and Cattle . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11. Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms 395 11.1 Simple Random Walk and Derivation of the Diffusion Equation . . . 395 11.2 ReactionDiffusionEquations . . . . . . . . . . . . . . . . . . . . . . 399 11.3 Models forAnimalDispersal . . . . . . . . . . . . . . . . . . . . . . 402 11.4 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 11.5 NonlocalEffects andLongRangeDiffusion . . . . . . . . . . . . . . 408 11.6 Cell Potential and Energy Approach to Diffusion andLongRangeEffects . . . . . . . . . . . . . . . . . . . . . . . . . 413 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 12. Oscillator-GeneratedWave Phenomena 418 12.1 Belousov–ZhabotinskiiReactionKinematicWaves . . . . . . . . . . 418 12.2 Central Pattern Generator: Experimental Facts in the Swimming ofFish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 12.3 MathematicalModel for theCentralPatternGenerator . . . . . . . . 424 12.4 Analysis of the Phase Coupled Model System . . . . . . . . . . . . . 431 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 13. BiologicalWaves: Single-Species Models 437 13.1 Background and the Travelling Waveform . . . . . . . . . . . . . . . 437 13.2 Fisher–Kolmogoroff Equation and Propagating Wave Solutions . . . . 439 13.3 Asymptotic Solution and Stability of Wavefront Solutions of the Fisher–Kolmogoroff Equation . . . . . . . . . . . . . . . . . . 444 13.4 Density-Dependent Diffusion-Reaction Diffusion Models andSomeExactSolutions . . . . . . . . . . . . . . . . . . . . . . . 449 13.5 Waves in Models with Multi-Steady State Kinetics: Spread and Control of an Insect Population . . . . . . . . . . . . . . 460 13.6 Calcium Waves on Amphibian Eggs: Activation Waves on MedakaEggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 13.7 Invasion Wavespeeds with Dispersive Variability . . . . . . . . . . . 471 13.8 Species InvasionandRangeExpansion . . . . . . . . . . . . . . . . . 478 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 14. Use and Abuse of Fractals 484 14.1 Fractals:BasicConcepts andBiologicalRelevance . . . . . . . . . . 484 14.2 ExamplesofFractals andTheirGeneration . . . . . . . . . . . . . . 487 14.3 Fractal Dimension: Concepts and Methods of Calculation . . . . . . . 490 14.4 Fractals or Space-Filling? . . . . . . . . . . . . . . . . . . . . . . . . 496 Appendices 501 A. Phase Plane Analysis 501 Contents, Volume II xix B. Routh-Hurwitz Conditions, Jury Conditions, Descartes’ Rule of Signs, and Exact Solutions of a Cubic 507 B.1 Polynomials and Conditions . . . . . . . . . . . . . . . . . . . . . . 507 B.2 Descartes’RuleofSigns . . . . . . . . . . . . . . . . . . . . . . . . 509 B.3 Roots of a General Cubic Polynomial . . . . . . . . . . . . . . . . . 510 Bibliography 513 Index 537 CONTENTS, VOLUME II J.D. Murray: Mathematical Biology, II: Spatial Models and Biomedical Applications Preface to the Third Edition Preface to the First Edition 1. Multi-SpeciesWaves and Practical Applications 1.1 Intuitive Expectations 1.2 Waves of Pursuit and Evasion in Predator–Prey Systems 1.3 Competition Model for the Spatial Spread of the Grey Squirrel in Britain 1.4 Spread of Genetically Engineered Organisms 1.5 Travelling Fronts in the Belousov–Zhabotinskii Reaction 1.6 Waves in Excitable Media 1.7 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics 1.8 Spiral Waves 1.9 Spiral Wave Solutions of λ-ω Reaction Diffusion Systems 2. Spatial Pattern Formation with Reaction Diffusion Systems 2.1 Role of Pattern in Biology 2.2 Reaction Diffusion (Turing) Mechanisms 2.3 General Conditions for Diffusion-Driven Instability: Linear Stability Analysis and Evolution of Spatial Pattern 2.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion Mechanism 2.5 Dispersion Relation, Turing Space, Scale and Geometry Effects in Pattern Formation Models 2.6 Mode Selection and the Dispersion Relation 2.7 Pattern Generation with Single-Species Models: Spatial Heterogeneity with the Spruce Budworm Model 2.8 Spatial Patterns in Scalar Population Interaction Diffusion Equations with Convection: Ecological Control Strategies xx Contents, Volume II 2.9 Nonexistence of Spatial Patterns in Reaction Diffusion Systems: General and Particular Results 3. Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms 3.1 Mammalian Coat Patterns—‘How the Leopard Got Its Spots’ 3.2 Teratologies: Examples of Animal Coat Pattern Abnormalities 3.3 A Pattern Formation Mechanism for Butterfly Wing Patterns 3.4 Modelling Hair Patterns in a Whorl in Acetabularia 4. Pattern Formation on Growing Domains: Alligators and Snakes 4.1 Stripe Pattern Formation in the Alligator: Experiments 4.2 Modelling Concepts: Determining the Time of Stripe Formation 4.3 Stripes and Shadow Stripes on the Alligator 4.4 Spatial Patterning of Teeth Primordia in the Alligator: Background and Relevance 4.5 Biology of Tooth Initiation 4.6 Modelling Tooth Primordium Initiation: Background 4.7 Model Mechanism for Alligator Teeth Patterning 4.8 Results and Comparison with Experimental Data 4.9 Prediction Experiments 4.10 Concluding Remarks on Alligator Tooth Spatial Patterning 4.11 Pigmentation Pattern Formation on Snakes 4.12 Cell-Chemotaxis Model Mechanism 4.13 Simple and Complex Snake Pattern Elements 4.14 Propagating Pattern Generation with the Cell-Chemotaxis System 5. Bacterial Patterns and Chemotaxis 5.1 Background and Experimental Results 5.2 Model Mechanism for E. coli in the Semi-Solid Experiments 5.3 Liquid Phase Model: Intuitive Analysis of Pattern Formation 5.4 Interpretation of the Analytical Results and Numerical Solutions 5.5 Semi-Solid Phase Model Mechanism for S. typhimurium 5.6 Linear Analysis of the Basic Semi-Solid Model 5.7 Brief Outline and Results of the Nonlinear Analysis 5.8 Simulation Results, Parameter Spaces, Basic Patterns 5.9 Numerical Results with Initial Conditions from the Experiments 5.10 Swarm Ring Patterns with the Semi-Solid Phase Model Mechanism 5.11 Branching Patterns in Bacillus subtilis 6. Mechanical Theory for Generating Pattern and Form in Development 6.1 Introduction, Motivation and Background Biology 6.2 Mechanical Model for Mesenchymal Morphogenesis 6.3 Linear Analysis, Dispersion Relation and Pattern Formation Potential Contents, Volume II xxi 6.4 Simple Mechanical Models Which Generate Spatial Patterns with Complex Dispersion Relations 6.5 Periodic Patterns of Feather Germs 6.6 Cartilage Condensation in Limb Morphogenesis and Morphogenetic Rules 6.7 Embryonic Fingerprint Formation 6.8 Mechanochemical Model for the Epidermis 6.9 Formation of Microvilli 6.10 Complex Pattern Formation and Tissue Interaction Models 7. Evolution, Morphogenetic Laws, Developmental Constraints and Teratologies 7.1 Evolution and Morphogenesis 7.2 Evolution and Morphogenetic Rules in Cartilage Formation in the Vertebrate Limb 7.3 Teratologies (Monsters) 7.4 Developmental Constraints, Morphogenetic Rules and the Consequences for Evolution 8. A Mechanical Theory of Vascular Network Formation 8.1 Biological Background and Motivation 8.2 Cell–Extracellular Matrix Interactions for Vasculogenesis 8.3 Parameter Values 8.4 Analysis of the Model Equations 8.5 Network Patterns: Numerical Simulations and Conclusions 9. Epidermal Wound Healing 9.1 Brief History of Wound Healing 9.2 Biological Background: Epidermal Wounds 9.3 Model for Epidermal Wound Healing 9.4 Nondimensional Form, Linear Stability and Parameter Values 9.5 Numerical Solution for the Epidermal Wound Repair Model 9.6 Travelling Wave Solutions for the Epidermal Model 9.7 Clinical Implications of the Epidermal Wound Model 9.8 Mechanisms of Epidermal Repair in Embryos 9.9 Actin Alignment in Embryonic Wounds: A Mechanical Model 9.10 Mechanical Model with Stress Alignment of the Actin Filaments in Two Dimensions 10. Dermal Wound Healing 10.1 Background and Motivation—General and Biological 10.2 Logic of Wound Healing and Initial Models 10.3 Brief Review of Subsequent Developments 10.4 Model for Fibroblast-DrivenWound Healing: Residual Strain and Tissue Remodelling xxii Contents, Volume II 10.5 Solutions of the Model Equation Solutions and Comparison with Experiment 10.6 Wound Healing Model of Cook (1995) 10.7 Matrix Secretion and Degradation 10.8 Cell Movement in an Oriented Environment 10.9 Model System for Dermal Wound Healing with Tissue Structure 10.10 One-Dimensional Model for the Structure of Pathological Scars 10.11 Open Problems in Wound Healing 10.12 Concluding Remarks on Wound Healing 11. Growth and Control of Brain Tumours 11.1 Medical Background 11.2 Basic Mathematical Model of Glioma Growth and Invasion 11.3 Tumour Spread In Vitro: Parameter Estimation 11.4 Tumour Invasion in the Rat Brain 11.5 Tumour Invasion in the Human Brain 11.6 Modelling Treatment Scenarios: General Comments 11.7 Modelling Tumour Resection (Removal) in Homogeneous Tissue 11.8 Analytical Solution for Tumour Recurrence After Resection 11.9 Modelling Surgical Resection with Brain Tissue Heterogeneity 11.10 Modelling the Effect of Chemotherapy on Tumour Growth 11.11 Modeling Tumour Polyclonality and Cell Mutation 12. Neural Models of Pattern Formation 12.1 Spatial Patterning in Neural Firing with a Simple Activation–Inhibition Model 12.2 A Mechanism for Stripe Formation in the Visual Cortex 12.3 A Model for the Brain Mechanism Underlying Visual Hallucination Patterns 12.4 Neural Activity Model for Shell Patterns 12.5 Shamanism and Rock Art 13. Geographic Spread and Control of Epidemics 13.1 Simple Model for the Spatial Spread of an Epidemic 13.2 Spread of the Black Death in Europe 1347–1350 13.3 Brief History of Rabies: Facts and Myths 13.4 The Spatial Spread of Rabies Among Foxes I: Background and Simple Model 13.5 Spatial Spread of Rabies Among Foxes II: Three-Species (SIR) Model 13.6 Control Strategy Based on Wave Propagation into a Non-epidemic Region: Estimate of Width of a Rabies Barrier 13.7 Analytic Approximation for the Width of the Rabies Control Break Contents, Volume II xxiii 13.8 Two-Dimensional Epizootic Fronts and Effects of Variable Fox Densitics: Quantitative Predictions for a Rabies Outbreak in England 13.9 Effect of Fox Immunity on Spatial Spread of Rabies 14. Wolf Territoriality,Wolf–Deer Interaction and Survival 14.1 Introduction and Wolf Ecology 14.2 Models forWolf Pack Territory Formation: Single Pack—Home Range Model 14.3 Multi-Wolf Pack Territorial Model 14.4 Wolf–Deer Predator–Prey Model 14.5 Concluding Remarks on Wolf Territoriality and Deer Survival 14.6 Coyote Home Range Patterns 14.7 Chippewa and Sioux Intertribal Conflict c1750–1850 Appendix A. General Results for the Laplacian Operator in Bounded Domains Bibliography Index