After the last glaciation, oak trees re-colonized Europe beginning in havens in Spain , Italy and the Balkans and spreading into northern Europe . The oak trees spread at a rate of 50 to 500 meters per year, very fast considering that acorns fall very near the trunks of the trees. How do we explain this rapid rate of re-colonization?

This is one of several biological mysteries explored in *Mathematical Modeling for the Life Sciences*. The author first considers a deterministic diffusion model and uses standard reaction diffusion equations to model the expansion of the oak trees. However, the rate predicted by this model is too low. Using a spatial branching process model, however, the author can account for the observed spreading rate and provide a convincing biological explanation. Animals, particularly jays, carry a minority of acorns away, and the carrying distance varies from a hundred meters to several kilometers. Furthermore, the jays tend to bury the acorns separately in favorable vegetation transition zones. The stochastic branching model mixes two distributions for the spreading of acorns: the majority fall near the parent tree, but a minority fall according to a long-range distribution with a higher probability for large deviations.

The whole discussion of this example occupies about three pages of text, and this is typical of the book. The outlook is “panoramic”, the focus is on real issues in the life sciences, and the discussion of each application is usually quite brief. The author does a good job in balancing mathematical rigor and biological interest. Theorems are presented, but often without proof or elaboration. Detailed calculations are presented when the author judges that necessary to explain the underlying biological issue.

The author considers several other fascinating applications in the life sciences. These include pest control of the spruce budworm, the apparently chaotic population dynamics of the coleopter *Tribolium*, game theory for the interaction of hawks and doves, domestication of pearl millet, the Polymerase Chain Reaction (PCR) for DNA replication, and mapping of the Qualitative Trait Locus in genetics. The mathematics associated with these applications runs from discrete and continuous dynamical systems to Markov chains and diffusion, branching processes, and maximum likelihood estimation.

This is a book best suited to advanced undergraduates or beginning graduate students. The prerequisites include some familiarity with ordinary and partial differential equations, probability and statistics. The Dominated Convergence Theorem is invoked at one point, but a good background in advanced calculus is generally sufficient in most places. There are appendices on ordinary differential equations, evolution equations, probability and statistics, but these are very brief summaries.

This book was originally published in French in 2000. The English edition is quite readable, although there are occasional odd choices of words as well as a sentence here and there that doesn’t quite make sense. This would be a good choice for the main text or for supplemental reading in a course on mathematical applications to biology, particularly with appropriate instructor support to expand and amplify the abbreviated discussions.

Bill Satzer (wjsatzer@mmm.com) 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.