2nd Edition

Differential Equations and Mathematical Biology

By D.S. Jones, Michael Plank, B.D. Sleeman Copyright 2009
    462 Pages 100 B/W Illustrations
    by Chapman & Hall

    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 first-order 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 first-order 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.


    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 first-order equations

    Symbolic computation in MATLAB

    Linear Ordinary Differential Equations with Constant Coefficients


    First-order linear differential equations

    Linear equations of the second order

    Finding the complementary function

    Determining a particular integral

    Forced oscillations

    Differential equations of order n


    Systems of Linear Ordinary Differential Equations

    First-order 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



    Nerve impulse transmission

    Chemical reactions

    Predator–prey models

    First-Order Systems of Ordinary Differential Equations

    Existence and uniqueness


    The phase plane and the Jacobian matrix

    Local stability


    Limit cycles

    Forced oscillations

    Numerical solution of systems of equations

    Symbolic computation on first-order systems of equations and higher-order 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 λω 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 of a limit cycle

    Discrete bifurcation and period-doubling


    Stability of limit cycles

    The Poincaré plane


    Numerical Bifurcation Analysis

    Fixed points and stability

    Path-following and bifurcation analysis

    Following stable limit cycles

    Bifurcation in discrete systems

    Strange attractors and chaos

    Stability analysis of partial differential equations

    Growth of Tumors


    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


    The Kermack–McKendrick model


    An incubation model

    Spreading in space

    Answers to Selected Exercises



    D.S. Jones, FRS, FRSE is Professor Emeritus in the Department of Mathematics at the University of Dundee in Scotland.

    M.J. Plank is a senior lecturer in the Department of Mathematics and Statistics at the University of Canterbury in Christchurch, New Zealand.

    B.D. Sleeman, FRSE is Professor Emeritus in the Department of Applied Mathematics at the University of Leeds in the UK.

    "… 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 well-selected exercises, and plenty of references for further reading."
    MAA Reviews, April 2010

    Praise 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