Table of Contents
INTRODUCTION
Population Growth
Administration of Drugs
Cell Division
Differential Equations with Separable Variables
General Properties
Equations of Homogeneous Type
Linear Differential Equations of the First Order
LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Introduction
First-Order Linear Differential Equations
Linear Equations of the Second Order
Finding the Complementary Function
Determining a Particular Integral
Forced Oscillations
Differential Equations of the Order n
Uniqueness
Appendix: Existence Theory
SIMULTANEOUS EQUATIONS WITH CONSTANT COEFFICIENTS
Simultaneous Equations of the First Order
Replacement of One Differential Equation by a System
The General System
The Fundamental System
Matrix Notation
Initial and Boundary Value Problems
Solving the Inhomogeneous Differential Equation
Appendix: Symbolic Computation
MODELLING BIOLOGICAL PHENOMENA
Introduction
Heart Beat
Blood Flow
Nerve Impulse Transmission
Chemical Reactions
Predator-Prey Models
FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
Existence and Uniqueness
Epidemics
The Phase Plane
Local Stability
Stability
Limit Cycles
Forced Oscillations
Appendix: Existence Theory
Appendix: Computing Trajectories
MATHEMATICS OF HEART PHYSIOLOGY
The Local Model
The Threshold Effect
Phase Plane Analysis and the Heart Beat Model
Physiological Considerations of the Heart Beat Cycle
A Model of the Cardiac Pacemaker
MATHEMATICS OF NERVE IMPULSE TRANSMISSION
Excitability and Repetitive Firing
Travelling Waves
Qualitative Behaviour of Travelling Waves
CHEMICAL REACTIONS
Wavefronts for the Belousov-Zhabotinskii Reaction
Phase Plane Analysis of Fisher's Equation
Qualitative Behaviour in the General Case
PREDATOR AND PREY
Catching Fish
The Effect of Fishing
The Volterra-Lotka Model
PARTIAL DIFFERENTIAL EQUATIONS
Characteristics for Equations of the First Order
Another View of Characteristics
Linear Partial Differential Equations of the Second Order
Elliptic Partial Differential Equations
Parabolic Partial Differential Equations
Hyperbolic Partial Differential Equations
The Wave Equation
Typical Problems for the Hyperbolic Equation
The Euler-Darboux Equation
EVOLUTIONARY EQUATIONS
The Heat Equation
Separation of Variables
Simple Evolutionary Equations
Comparison Theorems
PROBLEMS OF DIFFUSION
Diffusion through Membranes
Energy and Energy Estimates
Global Behaviour of Nerve Impulse Transmissions
Global Behaviour in Chemical Reactions
Turing Diffusion Driven Instability and Pattern Formation
Finite Pattern Forming Domains
BIFURCATION AND CHAOS
Bifurcation
Bifurcation of a Limit Cycle
Discrete Bifurcation
Chaos
Stability
The Poincaré Plane
Averaging
Appendix: Programs
GROWTH OF TUMOURS
Introduction
A Mathematical Model of Tumour Growth
A Spherical Tumour
Stability
EPIDEMICS
The Kermack-McKendrick Model
Vaccination
An Incubation Model
Spreading in Space
ANSWERS TO EXERCISES
INDEX