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Mathematical biology

Author: Murray, James D. Series: Biomathematics texts ; 19 Publisher: Springer, 1993.Edition: 2nd ed.Language: EnglishDescription: 767 p. : Graphs ; 24 cm.ISBN: 038757204XType of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print QA401 .M87 1993
(Browse shelf)
001176951
Available 001176951
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Includes bibliographical references and index

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Mathematical Biology Table of Contents 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 * Denotes sections in which the mathematics is at a higher level. These sections can be omitted without loss of continuity. X Table of Contents 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 .................................. 5.2 Michaelis-Menten Theory: Detailed Analysis and the Pseudo-Steady State Hypothesis 5.3 Cooperative Phenomena .............................................................. 5.4 Autocatalysis, Activation and Inhibition .................................... 5.5 Multiple Steady States, Mushrooms and Isolas ......................... Exercises 109 111 118 122 130 137 6. Biological Oscillators and Switches ................................................ 140 6.1 Motivation, History and Background .......................................... 6.2 Feedback Control Mechanisms ................................................... 6.3 Oscillations and Switches Involving Two or More Species: General Qualitative Results ..................................................... 6.4 Simple Two-Species Oscillators: Parameter Domain Determination for Oscillations ................................................. 6.5 Hodgkin-Huxley Theory of Nerve Membranes: FitzHugh-Nagumo Model .......................................................... 6.6 Modelling the Control of Testosterone Secretion ....................... Exercises 7. Belousov-Zhabotinskii Reaction ..................................................... 7.1 Belousov Reaction and the Field-Noyes (FN) Model . . . . .......... 7.2 Linear Stability Analysis of the FN Model and Existence of Limit Cycle Solutions ............................................................. 7.3 Non-local Stability of the FN Model ............................................ 7.4 Relaxation Oscillators: Approximation for the Belousov-Zhabotinskii Reaction ............................................... 7.5 Analysis of a Relaxation Model for Limit Cycle Oscillations in the Belousov-Zhabotinskii Reaction ..................................... Exercises 140 143 148 156 161 166 175 179 179 183 187 190 192 199 8. Perturbed and Coupled Oscillators and Black Holes ....................... 200 8.1 8.2 8.3 8.4 8.5 Phase Resetting in Oscillators .................................................... Phase Resetting Curves .............................................................. Black Holes ................................................................................. Black Holes in Real Biological Oscillators .................................. Coupled Oscillators: Motivation and Model System ................... 200 204 208 210 215 Table of Contents *8.6 *8.7 *8.8 *8.9 Singular Perturbation Analysis: Preliminary Transformation ....... 217 Singular Perturbation Analysis: Transformed System . . . . .......... 220 Singular perturbation Analysis: Two-Time Expansion . . . ............ 223 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 . .......... 9.2 Reaction Diffusion Equations ........................................................ 9.3 Models for Insect Dispersal ........................................................... 9.4 Chemotaxis ................................................................................... *9.5 Non-local Effects and Long Range Diffusion ................................... *9.6 Cell Potential and Energy Approach to Diffusion ............................ Exercises 232 236 238 241 244 249 252 10. Oscillator Generated Wave Phenomena and Central Pattern Generators 254 10.1 Kinematic Waves in the Belousov-Zhabotinskii Reaction . . .......... 10.2 Central Pattern Generator: Experimental Facts in the Swimming of Fish ....................................................................... *10.3 Mathematical Model for the Central Pattern Generator . . ............ *10.4 Analysis of the Phase-Coupled Model System .............................. Exercises 254 258 261 268 273 11. Biological Waves: Single Species Models ............................................ 274 11.1 Background and the Travelling Wave Form .................................. 11.2 Fisher Equation and Propagating Wave Solutions ........................ 11.3 Asymptotic Solution and Stability of Wavefront Solutions of the Fisher Equation ................................................................ 11.4 Density-Dependent Diffusion Reaction Diffusion Models and Some Exact Solutions .......................................................... 11.5 Waves in Models with Multi-Steady State Kinetics: The Spread and Control of an Insect Population ......................... 11.6 Calcium Waves on Amphibian Eggs: Activation Waves on Medaka Eggs ......................................................................... Exercises 12. Biological Waves: Multi-species Reaction Diffusion Models . . . . 311 12.1 12.2 12.3 12.4 Intuitive Expectations .................................................................. 311 Waves of Pursuit and Evasion in Predator-Prey Systems . ........... 315 Travelling Fronts in the Belousov-Zhabotinskii Reaction . . . ......... 322 Waves in Excitable Media ............................................................ 328 274 277 281 286 297 305 309 XII Table of Contents 12.5 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics *12.6 Linear Stability of Wave Train Solutions of A-w Systems . . ........... 12.7 Spiral Waves ................................................................................. *12.8 Spiral Wave Solutions of A-w Reaction Diffusion Systems . . ......... Exercises 336 340 343 350 356 *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 Table of Contents XIII 16. Neural Models of Pattern Formation ............................................... 16.1 Spatial Patterning in Neural Firing with a Simple Activation-Inhibition Model ........................................................ 16.2 A Mechanism for Stripe Formation in the Visual Cortex . . ...... 16.3 A Model for the Brain Mechanism Underlying Visual Hallucination Patterns ................................................................ 16.4 Neural Activity Model for Shell Patterns ................................... Exercises 481 481 489 494 505 523 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 Multi-group 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 XIV Table of Contents 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 20. Geographic Spread of Epidemics .................................................... 651 Appendices .......................................................................................... 697 1. 2. 3. 4. Phase Plane Analysis ................................................................. Routh-Hurwitz Conditions, Jury Conditions, Descartes' Rule of Signs and Exact Solutions of a Cubic ................................... Hopf Bifurcation Theorem and Limit Cycles ............................. General Results for the Laplacian Operator in Bounded Domains ..................................................................................... 697 702 706 720 723 745 Bibliography Index

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