For too long, biology has been thought of as a descriptive discipline, driven by the inductive methods of collating facts to support theories rather than by the more deductive methods appropriate to mathematics. Values, opinions, and science, however, can all rapidly change (as Thomas Kuhn has taught us). Mathematical approaches form a cogent framework, a common language, for developing, testing, and criticizing biological hypotheses. The application of mathematics to biological problems is now a science for the 21st century. While physics and chemistry have a long historical integration with math, biology has often seemed to lag behind—but not anymore.
Mathematical biology is a highly disparate discipline—many available approaches and techniques involving algebra, geometry, number theory, topology, probability, and statistics have applications to biological or medical research problems. Often a student's first (and last) exposure to such applications is during high school, through simple statistical tests on biological data—a χ2test, for instance—yet mathematical biology is infinitely richer than this. We can apply calculus to understand evolution, geometry to understand development, and number theory to understand pattern formation, among other examples. Rapid advances in all areas of biology make the development of cohesive mathematical frameworks for guiding or interpreting progress an essential attribute and skill, which all who are interested in the life sciences need to develop. The release of The Theoretical Biologist's Toolbox, by Marc Mangel, offers a refreshing approach to this goal by providing aspiring (and even established) theoreticians with a user-friendly and thorough introduction to a wide variety of techniques.
Marc Mangel is a professor of applied math and statistics, and fellow of Stevenson College, at the University of California, Santa Cruz. His scholarly output extends to many areas of biology: among the books he has written or coauthored are Decision and Control in Uncertain Resource Systems, Dynamic Modeling in Behavioral Ecology (with Colin Clark), The Ecological Detective: Confronting Models with Data (with Ray Hilborn), and Dynamic State Variable Models in Ecology: Methods and Applications (with Colin Clark). The Theoretical Biologist's Toolbox, which follows in the vein of these titles, is a how-to guide with applications. It is arranged into eight chapters and covers broadly sweeping topics, from the evolutionary ecology of parasitoids and the population biology of disease to topics in ordinary and partial differential equations. Mangel aims to give readers the skills they need to understand problems in theoretical and mathematical biology.
The book is a graduate-level text that covers modeling ideas, differential equations, and probability in the first three chapters, and then, in the five sub-sequent ones, the evolutionary ecology of parasitoids, diseases, fisheries, and stochastic population theory, both the basics and its applications in ecology, evolution, and demography. It is an accessible introduction to different models and analyses in theoretical population biology.
Unsurprisingly, as the book was written around a taught course, it is replete with exercises. These are not confined to the end of each chapter but appear after the introduction of associated material. This approach is a huge asset that underscores the book's pedagogic approach to mathematical biology, allowing easy reference to (and reinforcement of) particular topics. Following an exercise, the material extends into the next section of the chapter and concludes with a section identifying further reading to enable a deeper understanding.
Each chapter is self-contained (apart from some basic calculus skills, which are available from any good reference guide), yet there is considerable benefit to starting at the beginning of this book and finishing at the end—it serves aptly as a learning guide as well as a toolbox. The reader can learn how to use one tool well before moving to the next—although knowing how to use a mallet is invaluable, knowing how to use one with a chisel can lead to works of art.
Developing a skill set and populating your own toolbox will not only provide personal edification but also offer unprecedented opportunity to help resolve a growing number of real-world problems that require sets of mathematical sensibilities. Research in areas such as bioinformatics, epidemiology, population ecology, and immunology recently has seen record growth. The application of mathematics to both large-scale patterns (climate change) and fine-scale patterns (cell regulation and gene expression) is revealing detailed insight into the underlying biological phenomena that generate the patterns. The challenge is open for those who are motivated to learn mathematics and biology. Mangel's book should be an inspiration.
