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Differential Equations and Mathematical Biology

D. S. Jones, M. J. Plank, and B. D. Sleeman
Chapman & Hall/CRC
Publication Date: 
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Chapman&Hall/CRC Mathematical and Computational Biology Series
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This is the second edition of a book in the Chapman and Hall/CRC Mathematical and Computational Biology series. It is primarily about differential equations — ordinary and partial — with applications to biology. The authors have devised the text to serve three separate, partially overlapping purposes: a basic course in differential equations, a course in biological modeling for students of mathematics and the physical sciences, and a course in modeling with differential equations for students in the life sciences.

There are no surprises in either the treatment of differential equations or the biological applications. The topics and approach to differential equations are standard, and the nearly all the applications have also been treated in comparable books. Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book, not at all flashy. Illustrations, for example, are black and white and mostly line drawings. 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.

The basics of ordinary differential equations are treated in Chapters 1, 2, 3, and 5. No linear algebra background is assumed. Chapter 4 introduces the process of modeling biological phenomena, beginning with a model for the heartbeat. Here the authors note, “…the real test of the model is that it not only agrees qualitatively with the biological process but has the ability to suggest new experiments and bring deeper insight into the biological situation.” Although this is true of all modeling endeavors, it deserves special emphasis here because mathematical biology is still a relatively immature field. Differential equations provide one class of tools; many others remain to be developed.

Specific applications are treated next: heart physiology, nerve impulse transmission, chemical reactions, and predator-prey interactions. A very nice introduction to partial differential equations follows in Chapters 10 and 11. The authors emphasize the importance of diffusion processes in biology and present several examples in Chapter 12; these include Turing’s approach to pattern formation arising from diffusion-driven instability.

The first edition of this book had a chapter on catastrophe theory. In the second edition, this has been replaced by two chapters on bifurcation, chaos and numerical bifurcation analysis. The book concludes with two more application chapters; one describing models of tumor growth and the other epidemics.

Each chapter comes with a collection of well-selected exercises, and plenty of references for further reading.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


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