Essential Mathematical Biology Nicholas F. Britton Springer-Verlag, New York, 2003. $34.95 (335 pp.). ISBN 1-85233-536-X
Those of us in mathematical biology like to imagine our field on the verge of achieving critical opalescence, in which the distinct characteristics of mathematicians, biologists, and even physicists are dissolved into a glittering new unity. More precise measurement techniques in molecular biology, evolutionary biology, and ecology are pushing strategic modelers like me (who ask, “Could this mechanism produce that phenomenon in principle?”) to become predictive modelers (who ask, “Are these measured rate constants and interactions consistent with those experimental results?”). Biologists who collaborate closely with quantitative modelers may overcome the sting of Ernest Rutherford’s remark that all of science is either physics or stamp collecting. Physicists, conversely, are realizing that it is biologists who now collect the most interesting stamps, as evidenced by the increasing number of physics journals that focus on biological questions.
As the field develops, it is a pleasure and challenge to share the wide spectrum of problems and approaches with eager undergraduates from various backgrounds. Several textbooks are available, now including Essential Mathematical Biology by Nicholas Britton. The author, director of the University of Bath’s Centre for Mathematical Biology, exemplifies interdisciplinary approaches by bringing powerful applied mathematics to bear on such problems as the implications of habitat loss, coordination of social-insect colonies, and evolution of sex in plants, the fascination of which should not be underestimated.
The biological topics covered in Britton’s new book are reasonably broad; they include models of competition and predation, diseases, evolution, motion, reaction kinetics, pattern formation, and tumor growth. The mathematical topics include the classics of mathematical biology: difference and differential equations, first-order partial differential equations for age-structured populations, and reaction–diffusion systems. The discussion of mathematical methods is supplemented by appendixes outlining the fundamental techniques.
I was disappointed that the author made this textbook so traditional and did not attempt to bring the new spirit of mathematical biology to life. The topics and the approach rarely capitalize on what we have learned in recent decades. Mathematical Models in Biology, by Leah Edelstein-Keshet (Random House, 1988) covers much of the same ground and often does a better job of motivating questions and explaining the mathematics. James D. Murray’s Mathematical Biology (Springer-Verlag, 2003), a two-volume update of a 1989 text, also covers much of the same ground, although in a mathematically—and physically—heftier form.
I appreciate Britton’s including basic models of evolutionary and game theory, strangely neglected in other books, but wish he had discussed the stochastic models that play a fundamental role in helping scientists interpret the recent flood of genetic-sequence data. The sections on metapopulations—populations viewed as unions of smaller units—introduce an important and neglected topic, but the author makes no connection with work on the genetic structure of populations. The chapter on tumor modeling is a welcome addition (although more in-depth material on this topic is included in Murray’s book), but the generic models Britton considers give little sense of what makes different types of tumors behave differently.
Of course, difficult choices had to be made to keep the book in a compact and reasonably priced package. Undergraduate students unfamiliar with the complexity of modern biology could easily be overwhelmed by detailed models; they benefit most from understanding general principles. Nonetheless, any text must carefully motivate questions, and I often found that Britton’s book fell short. For example, the bistable equation, which describes signal propagation in an excitable medium, is introduced with the mathematician’s beloved “let f satisfy f′(0) < 0, f′(1) < 0, f(u) < 0 for u in (0, a), f(u) > 0 for u in (a, 1),” but the equation’s half-page coverage gives no idea under what circumstances such a function might arise. The section on delay-differential equations, which play a particularly important role in biological modeling, describes a generic system with a developmental delay and concludes, “More generally, the effect of delay is often to destabilize steady states.” It does not, however, mention physiological examples or look at the period of the resulting oscillations.
More frustrating than the sketchy motivations was the publisher’s failure to include a full bibliography of papers mentioned in the text and its decision to provide only a very sketchy index that omits authors of cited papers and major topics such as the bistable equation.
In short, Essential Mathematical Biology would serve well as a template for an advanced undergraduate or beginning graduate course in mathematical biology, but would require supplementation to bring the biology up to date, add exercises, clarify the motivations, and tie together different threads. A more advanced research physicist with a growing interest in biological philately might better taste the excitement of the field by having lunch with some biologists.
