1．Continuous Population Models for Single Species

1.1 Continuous Growth Models

1.2 Insect Outbreak Model：Spruce Budworm

1.3 Delay Models

1.4 Linear Analysis of Delay Population Models：Periodic Solutions

1.5 Delay Models in Physiology：Dynamic Diseases

1.6 Harvesting a Single Natural Population

1.7 Population Model with Age Distribution

Exercises

2．Discrete Population Models for a Single Species

2.1 Introduction：Simple Models

2.2 Cobwebbing：A Graphical Procedure of Solution

2.3 Discrete Logistic Model：Chaos

2.4 Stability,Periodic Solutions and Bifurcations

2.5 Discrete Delay Models

2.6 Fishery Management Model

2.7 Ecological Implications and Caveats

Exercises

3．Continuous Models for Interacting Populations

3.1 Predator-Prey Models：Lotka-Volterra Systems

3.2 Complexity and Stability

3.3 Realistic Predator-Prey Models

3.4 Analysis of a Predator-Prey Model with Limit Cycle Periodic Behaviour：Parameter Domains of Stability

3.5 Competition Models：Principle of Competitive Exclusion

3.6 Mutualism or Symbiosis

3.7 General Models and Some General and Cautionary Remarks

3.8 Threshold Phenomena

Exercises

4．Discrete Growth Models for Interacting Populations

4.1 Predator-Prey Models：Detailed Analysis

4.2 Synchronized Insect Emergence：13 Year Locusts

4.3 Biological Pest Control：General Remarks

Exercises

5．Reaction Kinetics

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

6．Biological Oscillators and Switches

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

8．Perturbed and Coupled Oscillators and Black Holes

8.1 Phase Resetting in Oscillators

8.2 Phase Resetting Curves

8.3 Black Holes

8.4 Black Holes in Real Biological Oscillators

8.5 Coupled Oscillators：Motivation and Model System

8.6 Singular Perturbation Analysis：Preliminary Transformation

8.7 Singular Perturbation Analysis：Transformed System

8.8 Singular perturbation Analysis：Two-Time Expansion

8.9 Analysis of the Phase Shift Equation and Application to Coupled Belousov-Zhabotinskii Reactions

Exercises

9．Reaction Diffusion,Chemotaxis and Non-local Mechanisms

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

10．Oscillator Generated Wave Phenomena and Central Pattern Generators

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

11．Biological Waves：Single Species Models

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

12.1 Intuitive Expectations

12.2 Waves of Pursuit and Evasion in Predator-Prey Systems

12.3 Travelling Fronts in the Belousov-Zhabotinskii Reaction

12.4 Waves in Excitable Media

12.5 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics

12.6 Linear Stability of Wave Train Solutions of λ-ω Systems

12.7 Spiral Waves

12.8 Spiral Wave Solutions of λ-ω Reaction Diffusion Systems

Exercises

13．Travelling Waves in Reaction Diffusion Systems with Weak Diffusion：Analytical Techniques and Results

13.1 Reaction Diffusion System with Limit Cycle Kinetics and Weak Diffusion：Model and Transformed System

13.2 Singular Perturbation Analysis：The Phase Satisfies Burgers'Equation

13.3 Travelling Wavetrain Solutions for Reaction Diffusion Systems with Limit Cycle Kinetics and Weak Diffusion：Comparison with Experiment

14．Spatial Pattern Formation with Reaction/Population Interaction Diffusion Mechanisms

14.1 Role of Pattern in Developmental Biology

14.2 Reaction Diffusion（Turing）Mechanisms

14.3 Linear Stability Analysis and Evolution of Spatial Pattern：General Conditions for Diffusion-Driven Instability

14.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion Mechanism

14.5 Dispersion Relation,Turing Space,Scale and Geometry Effects in Pattern Formation in Morphogenetic Models

14.6 Mode Selection and the Dispersion Relation

14.7 Pattern Generation with Single Species Models：Spatial Heterogeneity with the Spruce Budworm Model

14.8 Spatial Patterns in Scalar Population Interaction-Reaction Diffusion Equations with Convection：Ecological Control Strategies

14.9 Nonexistence of Spatial Patterns in Reaction Diffusion Systems：General and Particular Results

Exercises

15．Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms

15.1 Mammalian Coat Patterns-‘How the Leopard Got Its Spots’

15.2 A Pattern Formation Mechanism for Butterfly Wing Patterns

15.3 Modelling Hair Patterns in a Whorl in Acetabularia

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

17．Mechanical Models for Generating Pattern and Form in Development

17.1 Introduction and Background Biology

17.2 Mechanical Model for Mesenchymal Morphogenesis

17.3 Linear Analysis,Dispersion Relation and Pattern Formation Potential

17.4 Simple Mechanical Models Which Generate Spatial Patterns with Complex Dispersion Relations

17.5 Periodic Patterns of Feather Germs

17.6 Cartilage Condensations in Limb Morphogenesis

17.7 Mechanochemical Model for the Epidermis

17.8 Travelling Wave Solutions of the Cytogel Model

17.9 Formation of Microvilli

17.10 Other Applications of Mechanochemical Models

Exercises

18．Evolution and Developmental Programmes

18.1 Evolution and Morphogenesis

18.2 Evolution and Morphogenetic Rules in Cartilage Formation in the Vertebrate Limb

18.3 Developmental Constraints,Morphogenetic Rules and the Consequences for Evolution

19．Epidemic Models and the Dynamics of Infectious Diseases

19.1 Simple Epidemic Models and Practical Applications

19.2 Modelling Venereal Diseases

19.3 Multigroup Model for Gonorrhea and Its Control

19.4 AIDS：Modelling the Transmission Dynamics of the Human Immunodeficiency Virus（HIV）

19.5 Modelling the Population Dynamics of Acquired Immunity to Parasite Infection

19.6 Age Dependent Epidemic Model and Threshold Criterion

19.7 Simple Drug Use Epidemic Model and Threshold Analysis

Exercises

20．Geographic Spread of Epidemics

20.1 Simple Model for the Spatial Spread of an Epidemic

20.2 Spread of the Black Death in Europe 1347-1350

20.3 The Spatial Spread of Rabies Among Foxes Ⅰ：Background and Simple Model

20.4 The Spatial Spread of Rabies Among Foxes Ⅱ：Three Species（SIR）Model

20.5 Control Strategy Based on Wave Propagation into a Non-epidemic Region：Estimate of Width of a Rabies Barrier

20.6 Two-Dimensional Epizootic Fronts and Effects of Variable Fox Densities：Quantitative Predictions for a Rabies Outbreak in England

Exercises

Appendices

1．Phase Plane Analysis

2．Routh-Hurwitz Conditions,Jury Conditions,Descartes'Rule of Signs and Exact Solutions of a Cubic

3．Hopf Bifurcation Theorem and Limit Cycles

4．General Results for the Laplacian Operator in Bounded Domains

Bibliography

Index