Mathematics for the Life Sciences

From the beginning the goal is clear. The author says: “There are several outstanding mathematical biology books at the advanced undergraduate and beginning graduate level … largely inaccessible to biologists, simply because they require more mathematical background than most biologists have.” The book then proceeds to try to close the gap between the mathematics biologists typically learn and what they need to know to read more advanced books and papers.

This is a good deal more than a calculus book adapted toward biological applications. The calculus is there — as a kind of primer for some, review for others — but the majority of the book is devoted to the use of mathematical models, probability and inferential methods, discrete and continuous dynamical systems in the life sciences. This is a mathematics book, but it is more concerned with how its mathematical ideas are put to use rather than with their precise expression. Students of the life sciences, the author notes, primarily need to learn to draw conclusions based on reliable methods and good evidence. But the rigor and precision appropriate for future mathematicians would be counterproductive here. At several points through the book the author presents and uses ideas (such as the Akaike information criterion- a modeling tool for balancing quantitative accuracy against complexity) that are relatively easy to explain in general terms but whose rigorous treatment would be quite difficult.

The focus of the book is mathematical modeling. The author’s aim here is to give students enough experience with mathematical modeling that they can read and appreciate scientific work with mathematical content. Compared with other texts on mathematical modeling there are relatively few examples here. However, the ones that are presented are treated in detail. Two types of model are considered: mechanistic models (a priori models based on assumptions about principles underlying the phenomena being modeled) and empirical models (based on data).

Throughout the book the author includes case studies in his problem sets that continue and develop as new material from the text is introduced. These include applications in areas like measles infections, demographics, the growth of organisms, dopamine and psychosis, and nerve pulse times. This practice imparts a sense of continuity and progression to the exercises.

After Part 1 treats calculus and the general elements of modeling, Parts 2 and 3 present some important tools for modeling. Part 2 introduces the basic ideas and applications of probability together with a fair bit of statistics. Part 3 takes up dynamical systems; it first concentrates on the dynamics of a single population in both continuous and discrete forms, and then goes on to more or less conventional separate treatments of discrete and continuous dynamics in the last two chapters. Part 2 is probably the strongest part of the book. Its discussion of probability begins with probability distributions, and this seems exactly the right way into the subject for life science students.

This is a clearly written text that is sensitive to the needs and capabilities of life science students. My main reservation about the book is that the text often seems rather verbose. 

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.