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The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century.

Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics.

This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.

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

Each chapter also contains Exercises.

The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century.

Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics.

This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.

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

Each chapter also contains Exercises.

The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century.

Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics.

This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.

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

Each chapter also contains Exercises.

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

Each chapter also contains Exercises.

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

Each chapter also contains Exercises.

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

Each chapter also contains Exercises.