Read This! The MAA Online book review column Mathematical Biology I: An Introduction by J. D. Murray Reviewed by Charles Ashbacher

Change is the nature of things in biology and differential equations are used to model change. The concentrations of some bio-chemicals present in an organism change over time, sometimes dramatically, in response to the changes in the concentrations of other chemicals. Other systems make up a steady-state system, where the concentrations of reactants are damped within fairly rigid bounds. Segments of organisms change due to dynamic feedback with other segments of the same organism, actions in one location can propagate throughout the entire creature. Organisms of one species interact with others of the same species, sometimes in cooperation, and other times in competition. The same thing applies to populations of different species, in some circumstances, all species benefit or suffer and in other circumstances the advantage to one species is based on harm to the other(s). All of these situations are examined in a thorough manner; the models are based on differential equations started from some basic, logical assumptions.

The wide range of subjects covered in J. D. Murray's Mathematical Biology I is very impressive and at times it kept my attention like a novel. My two favorite chapters were four, which covered temperature-dependent sex determination and chapter five, which dealt with modeling the dynamics of marital interaction, divorce prediction and marriage repair. The first paragraph of chapter four really grabs your attention and starts with a simple, but very complex statement. "It is a fascinating subject why some species become extinct and others do not." From this, the families of crocodilia (crocodiles, alligators and gavials) are noted as having survived essentially unchanged for 100 million years, fully 63 million years after the dinosaurs (other reptiles) became extinct. One primary difference between the crocodilia and other reptilian species is that the incubation temperature of the crocodilia egg determines the sex of the hatchling. Wetter areas are cooler, which will produce all females and dry areas are warmer, which will produce all males. Intermediate areas will produce a 50/50 mix of the sexes. This has definite survival advantages, especially after a drastic reduction in the population. With few females to compete for nesting sites, most will build nests in damp areas, meaning that most of the hatchlings will be female. Since the crocodilia males have harems, the large number of females will mean that the population can rapidly recover from any dramatic reduction in numbers.

Using mathematical models to predict marital interaction is certainly a new use of mathematics, and in fact the mathematicians involved in the initial research were originally very skeptical regarding the practicality. It is a model that our intuition tells us could not be feasible. Given the specific environmental conditions, chemicals will always react in a certain way, and certain species hunt other species. However, our emotional prejudice makes us believe that the emotional and social interaction between spouses is too complex to be mathematically modeled. Most people will find it somewhat deflating to learn that even our most complex emotions can be mathematically understood. The testing began with couples being subjected to the Rapid Couples Interaction Scoring (RCISS) examination. A couple is videotaped having an interactive discussion concerning a problem area and the details of their emotions are numerically coded. The apparent validity of the model has enormous implications for all aspects of human interaction. It brought back memories of the psychohistory theme used in the "Foundation" science fiction series by Isaac Asimov.

The remaining chapters deal with basics such as population models for single and multiple species, predator-prey models, chemical reaction dynamics, the spread of infectious diseases, biological oscillators and wave phenomena. Although it was short, and had little mathematical content, I was also impressed with the last chapter on fractals. In the nineteen-eighties, fractals exploded on the mathematical scene and some argued that they could be used to explain most natural phenomena. As Murray sensibly points out, the fact that a fractal looks like a biological or other natural phenomenon does not in any way mean that the phenomenon is generated by a fractal.

This book is one of the most powerful arguments for the power of mathematics that I have ever read. It appears to be a basic human prejudice that biology is somehow a higher order of operation, where some aspects will forever remain beyond our understanding. That belief is severely strained by the models in this book, in that even the most complex of human behaviors can be modeled using what amounts to some very basic equations. I encourage everyone to read it, the mathematics is occasionally lengthy, but never gets much beyond the fundamentals of differential equations.

Publication Data: Mathematical Biology I: An Introduction, Third Edition, by J. D. Murray. Publisher, 2004. Springer-Verlag, 2002. Hardcover, 556 pp., $59.95. ISBN 0-387-95223-3.

An electronic version is also available.

Charles Ashbacher (cashbacher@yahoo.com) teaches at Mount Mercy College in Cedar Rapids, Iowa.

Posted November 9, 2004

Go to... The main Read This! page The index of all reviews The MAA Online front page

Find out... About the Mathematical Association of America Why you should be a member of the MAA How you can help support the activities of the MAA

Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.