Interest in and sophistication with mathematical modeling in environmental science has grown significantly in the last decade, spurred at least in part by issues of global warming. *Mathematics for Ecology and Environmental Sciences* is a collection of papers from a 2004 conference (“Dynamical Systems Theory and Its Applications to Biology and Environmental Sciences”). A companion volume (Mathematics for Life Science and Medicine) focuses more on epidemiology and infectious disease dynamics and is reviewed separately.

The current volume concentrates on population dynamics in a broad sense. There is an introductory essay and seven papers. These are unlike typical conference papers because the authors — at the direction of the conference organizers — made serious efforts to write accessibly for non-experts. Consequently, each paper addresses the biological background and motivation for the mathematical modeling. A modest background in differential equations — including analysis of stability and equilibrium for systems of differential equations — is adequate to understand most of the papers’ mathematical content.

I can perhaps best give the flavor of the book by summarizing a couple of the papers. The first paper develops a mathematical model of physiologically structured populations wherein individual development is affected by the population density. The authors show how to associate a dynamical system to the model and determine the steady states. An example of a physiologically structured population is an environment with juvenile and adult fish. Cycles can occur that begin with juveniles outcompeting their parents for food since they are small-bodied and need less energy for maintenance than their larger-bodied parents. The parents then starve, the juveniles reproduce and then they starve as they are outcompeted in turn by their offspring.

On a cheerier note, a later paper investigates indirect reciprocity — the possibility that altruistic acts within a population are returned not to the recipient but to a third party. The authors describe how this situation is handled by classical game theory, and then discuss how evolutionary game theory shows how strategies for indirect reciprocation can be evolutionarily stable.

This small volume is expensive for its size, but a good introduction to the field for non-specialists who want to sample the area.

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.