Differential Equations and Mathematical Biology
1st Edition
Published on 26. February 2003
Book
Hardback
408 pages
978-1-58488-296-1 (ISBN)
Description
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.
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.
More details
Series
Language
English
Place of publication
Bosa Roca
United States
Publishing group
Taylor & Francis Inc
Target group
College/higher education
Professional and scholarly
Third or fourth year undergraduates in mathematics, biology, and physical sciences, graduate students and researchers in mathematical biology
Illustrations
3 s/w Tabellen, 84 s/w Abbildungen
1768 equations; 3 Tables, black and white; 84 Illustrations, black and white
Dimensions
Height: 235 mm
Width: 159 mm
Weight
703 gr
ISBN-13
978-1-58488-296-1 (9781584882961)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
New editions
D.S. Jones | Michael Plank | B.D. Sleeman
Differential Equations and Mathematical Biology
Book
11/2009
2nd Edition
Chapman & Hall/CRC
€134.00
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Persons
Content
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 Poincare 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.
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 Poincare 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.