UK Nonlinear News, May 2004

Mathematical Biology (3rd edition): I: An Introduction II: Spatial Models and Biomedical Applications

J.D. Murray

By Jaroslav Stark

Springer Verlag
Vol. I: 2002 0-387-95223-3, 551 pages, Vol. II: 2003; 0-387-95228-4, 811 pages.
Corrected 2nd Printings 2004.

Mathematical biology is currently a field undergoing explosive growth. On one hand this is driven by demand from life scientists, who are beginning to appreciate that modelling is going to be an essential tool in making sense of the increasingly complex biological systems that they are attempting to understand. On the other, more and more mathematicians are becoming convinced that many of the most exciting developments in 21st century mathematics will be motivated by biological problems, just as much of the mathematical progress of the last two centuries was driven by the physical sciences.

Given this surge of interest, it is perhaps surprising that there is still only a handful of general text-books in this area. One of the first, which by now has become a classic, is the first edition of Jim Murray's Mathematical Biology, published in 1989. For many years, this was essential reading for any PhD student entering the field, and it is probably fair to say that it shaped a whole generation of mathematical biologists in the UK, if not elsewhere. Following a minor update for the second edition in 1993, Jim Murray has now added so much material that the third edition has appeared in two volumes, which at some 1360 pages are nearly double the page count of the first edition. Clearly, such a vast undertaking represents a huge challenge in terms of organization and presentation, particularly in a field as diverse as mathematical biology. The author has solved this problem in an interesting fashion.

The first volume, as its subtitle suggests, provides a broad introduction to mathematical biology. It is essentially a somewhat cut-down version of the original first edition, with the addition of three new chapters. Two of these are case studies based on the author's own research and describe the development of models for two rather unusual systems, namely temperature sensitive sex determination in alligators and marital conflict. The third new chapter is the final chapter of this volume and consists of an overview of the use of fractals in mathematical biology. This is somewhat out-of-place and perhaps too brief to be useful to someone who is not already familiar with the topic. In addition, most chapters have been expanded with typically one or two new sections per chapter, as well as more detailed updating where appropriate. There has also been some re-arrangement of material, so that for instance the chapter on Continuous Models for Interacting Populations has been merged with that for Discrete Models (losing the section on periodic locust outbreaks in the process), whilst the Dynamics of Infectious Diseases has been substantially expanded and brought forward. Finally, most (but not all) of the chapters dealing with spatially extended systems have been moved to the second volume.

Overall, I think that these changes have been very successful. As a result, this first volume is now an even better introduction to mathematical biology than the first edition and should prove suitable in a wide variety of contexts at the final year undergraduate and beginning graduate level. From a personal point of view, I was a little bit disappointed by the omission of topics such as molecular biology, cell biology and immunology. Modelling in these areas has seen dramatic growth during the past decade (and is likely to continue), and yet as far as I am aware there is a complete lack of text books covering these areas. However, I appreciate that since biology is such a vast subject, it is impossible to cover the whole of mathematical biology in a single text. The material included in a particular volume thus has to reflect the personal interests and tastes of the author. In that context, I think that Jim Murray has made an excellent choice, giving a very coherent and manageable account. Including other topics would probably have quickly led to something that was completely indigestible.

By contrast, the second volume has a different style. It takes the majority of the chapters dealing with spatially extended systems from the earlier editions, updates and expands them, and then adds a considerable number of new topics, all based on the author's own research. As such, it comes close to representing the collected works of Jim Murray, his students and collaborators. This is clearly material dear to the author's own heart and he has put an immense amount of effort into collecting historical background, interesting anecdotes and beautiful photographs which all serve to lighten the vast amount of detailed mathematical development. Being such a personal account, it is well structured and the reader never loses their way in over 750 pages of text, supported by some 750 references. Much of this is held together by a number of key themes such as reaction-diffusion systems, travelling waves and the author's own mechano-chemical theory of pattern formation.

This is a monumental work, longer than the whole of the first edition, and will serve as an important resource for anyone working in this field. It also provides a huge range of challenging problems which could serve as motivation for applied mathematicians looking for new mathematical questions, even if they do not have a particular interest in the underlying biological systems.

Together, both volumes are a tremendous achievement, and I think will turn out to be classics in their own right. As I have indicated above, they serve different purposes and will appeal to different audiences. The first is an ideal introduction to mathematical biology for graduate students, the second a more specialized reference book for active researchers in spatially extended systems. As such, it is unlikely that both volumes will appeal equally to many readers, but I would strongly recommend anyone with an interest in modern nonlinear mathematics to have a close look at least at one or the other.

UK Nonlinear News would like to thank Springer-Verlag for providing a copy of these volumes for review.

Both volumes area currently on special offer from until 31.7.2004.

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