Mathematical biology

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