Mathematical biology. (English) Zbl 0682.92001

Murray, James D.

Biomathematics, 19. Berlin etc.: Springer-Verlag. xiv, 767 p. DM 98.00 (1989).

This is neither a textbook nor a treatise on its subject matter, but it must be one of the best books ever written on applying mathematics to modelling biological phenomena. It should be an ideal complement for courses on ordinary differential equations or nonlinear partial differential equations due to the wealth of examples it contains. And, if you must or like to read about applying mathematics to biology, besides the large amount of material you find a list of several hundreds of references. The models considered are motivated phenomenologically (trying to derive the phenomenological equations from basic principles would make the book hopelessly long or unreadable). Even though some situations appear to be modeled more “truthfully” then others, once the models are set up simplifying assumptions have to be introduced in order to make further analysis possible. The important thing is that the analysis is always carried out with on eye on the biological interpretation of the results. This is the key to the appeal of the book.
A “look along the diagonal” into the contents of the book will take you through the basics of single-type population models in continuous and discrete time. When considering nonlinear flows in discrete time, the concepts of chaos (sensitivity to initial conditions) and bifurcation (parametric dependence of the periodic orbits and equilibrium states) appear in a natural way. The next degree of complexity corresponds to the study of populations made up of several types of interacting individuals, again in continuous and discrete time. Individual chapters are devoted to reaction kinetics, biological oscillators and the famous Belousov- Zhabotinskij reaction (a chemical system in which periodic solutions can be seen).
In chapter 8 the behaviour of periodic orbits under perturbations and interactions is studied. Singular perturbation techniques make their appearance. Chapter 9 is devoted to the basics of reaction diffusion systems and in chapter 10 oscillator waves or quasiwaves are described. This phenomenon corresponds to aggregates of non-interacting oscillators whose phases have spatial structure. It is in chapter 11 where real waves make their appearance. A solution \(u(t,x)\) to a scalar wave equation is a wave if it is of the type \(u(t,x)=\phi (x\pm vt);\) where the velocity of propagation v may be a function of the amplitude u. Diffusion-reaction with interaction between populations is studied in chapter 12.
The effect of small diffusions on periodic orbits of interacting populations is taken up in chapter 13 and the analysis of formation and stability of spatial patterns is carried out in chapter 14. These results are applied to a variety of systems in which “spatial” patterns exist. In particular, the analysis of models exhibiting spatial pattern formation, similar to coat or wing patterns or hair patterns, is carried out in chapter 15. Chapters 19 and 20 are devoted to complementary aspects of the dynamics and spatial spread of contagious diseases (epidemics). Probably the relevance of the models prompted the author to set these topics into two separate chapters, instead of treating the material as application of techniques developed in earlier chapters.
Although no stochastic modeling is included, many stochastic models will lead to equations like those described in the book. (Imagine adding several hundred pages more!).

Reviewer: H.Gzyl

MSC:

92-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology
92-02	Research exposition (monographs, survey articles) pertaining to biology
92D25	Population dynamics (general)
92-XX	Biology and other natural sciences

Keywords:

deterministic models; epidemiology; population models; nonlinear flows; chaos; bifurcation; periodic orbits; equilibrium states; reaction kinetics; biological oscillators; Belousov-Zhabotinskij; perturbations; Singular perturbation techniques; reaction diffusion systems; oscillator waves; quasiwaves; non-interacting oscillators; small diffusions; interacting populations; spatial patterns; pattern formation; spatial spread of contagious diseases