Differential Equations and Mathematical Biology, Second Edition
 Edition:
 2nd
 Author(s):
 D.S. Jones, Michael Plank, B.D. Sleeman
 ISBN:
 9781420083576
 Format:
 Hardback
 Publication Date:
 November 09, 2009
 Content Details:
 462 pages
 Language:
 English
Also of Interest

About the Book
Book Summary
Deepen students’ understanding of biological phenomena
Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material.
New to the Second Edition
 A section on spiral waves
 Recent developments in tumor biology
 More on the numerical solution of differential equations and numerical bifurcation analysis
 MATLAB^{®} files available for download online
 Many additional examples and exercises
This textbook shows how firstorder ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to firstorder autonomous systems of nonlinear differential equations using the Poincaré phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator–prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor growth and the spread of infectious diseases.
Features
 Uses various differential equations to model biological behavior
 Discusses the modeling of biological phenomena, including the heartbeat cycle, chemical reactions, electrochemical pulses in the nerve, predator–prey models, tumor growth, and epidemics
 Explains how bifurcation and chaotic behavior play key roles in fundamental problems of biological modeling
 Presents a unique treatment of pattern formation in developmental biology based on Turing’s famous idea of diffusiondriven instabilities
 Provides answers to selected exercises at the back of the book
 Offers downloadable MATLAB files on www.crcpress.com
Reviews
"… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions."
—SIAM Review, Sept. 2010, Vol. 52, No. 3"… Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book … The authors succeed because they do a splendid job of integrating their treatment of differential equations with the applications, and they don’t try to do too much. … Each chapter comes with a collection of wellselected exercises, and plenty of references for further reading."
—MAA Reviews, April 2010Praise for the First Edition
"A strength of [this book] is its concise coverage of a broad range of topics. … It is truly remarkable how much material is squeezed into the slim book’s 400 pages."
—SIAM Review, Vol. 46, No. 1
"It is remarkable that without the classical scheme (definition, theorem, and proof) it is possible to explain rather deep results like properties of the Fitz–Hugh–Nagumo model … or the Turing model. … This feature makes the reading of this text pleasant business for mathematicians. … [This book] can be recommended for students of mathematics who like to see applications, because it introduces them to problems on how to model processes in biology, and also for theoretically oriented students of biology, because it presents constructions of mathematical models and the steps needed for their investigations in a clear way and without references to other books."
—EMS Newsletter
"The title precisely reflects the contents of the book, a valuable addition to the growing literature in mathematical biology from a deterministic modeling approach. This book is a suitable textbook for multiple purposes. … Overall, topics are carefully chosen and well balanced. …The book is written by experts in the research fields of dynamical systems and population biology. As such, it presents a clear picture of how applied dynamical systems and theoretical biology interact and stimulate each other—a fascinating positive feedback whose strength is anticipated to be enhanced by outstanding texts like the work under review."
—Mathematical Reviews, Issue 2004g 
Contents
Introduction
Population growth
Administration of drugs
Cell division
Differential equations with separable variables
Equations of homogeneous type
Linear differential equations of the first order
Numerical solution of firstorder equations
Symbolic computation in MATLAB
Linear Ordinary Differential Equations with Constant Coefficients
Introduction
Firstorder linear differential equations
Linear equations of the second order
Finding the complementary function
Determining a particular integral
Forced oscillations
Differential equations of order n
Uniqueness
Systems of Linear Ordinary Differential Equations
Firstorder systems of equations with constant coefficients
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
Numerical solution of linear boundary value problems
Modelling Biological Phenomena
Introduction
Heartbeat
Nerve impulse transmission
Chemical reactions
Predator–prey models
FirstOrder Systems of Ordinary Differential Equations
Existence and uniqueness
Epidemics
The phase plane and the Jacobian matrix
Local stability
Stability
Limit cycles
Forced oscillations
Numerical solution of systems of equations
Symbolic computation on firstorder systems of equations and higherorder equations
Numerical solution of nonlinear boundary value problems
Appendix: existence theory
Mathematics of Heart Physiology
The local model
The threshold effect
The phase plane analysis and the heartbeat model
Physiological considerations of the heartbeat cycle
A model of the cardiac pacemaker
Mathematics of Nerve Impulse Transmission
Excitability and repetitive firing
Travelling waves
Qualitative behavior of travelling waves
Piecewise linear model
Chemical Reactions
Wavefronts for the Belousov–Zhabotinskii reaction
Phase plane analysis of Fisher’s equation
Qualitative behavior in the general case
Spiral waves and λ − ω systems
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
Visualization of solutions
Evolutionary Equations
The heat equation
Separation of variables
Simple evolutionary equations
Comparison theorems
Problems of Diffusion
Diffusion through membranes
Energy and energy estimates
Global behavior of nerve impulse transmissions
Global behavior 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 and perioddoubling
Chaos
Stability of limit cycles
The Poincaré plane
Averaging
Numerical Bifurcation Analysis
Fixed points and stability
Pathfollowing and bifurcation analysis
Following stable limit cycles
Bifurcation in discrete systems
Strange attractors and chaos
Stability analysis of partial differential equations
Growth of Tumors
Introduction
Mathematical model I of tumor growth
Spherical tumor growth based on model I
Stability of tumor growth based on model I
Mathematical model II of tumor growth
Spherical tumor growth based on model II
Stability of tumor growth based on model II
Epidemics
The Kermack–McKendrick model
Vaccination
An incubation model
Spreading in space
Answers to Selected Exercises
Index