You are here

Biology in Time and Space: A Partial Differential Equation Modeling Approach

James P. Keener
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Andrew Krause
, on
Biology in Time and Space is an impressive introduction to spatial models of biological phenomena and their analysis. The book is a tour de force exploration of a huge variety of physical phenomena, mathematical models, as well as analytical and numerical tools for their analysis. As Michael Reed is quoted as saying on the back cover, this book is undoubtedly now one of the few foundational texts in Mathematical Biology. The book takes an extremely modern approach to the subject and its pedagogical presentation, with frequent links to websites and videos, as well as copious numerical codes for producing and exploring the Figures in the text. This is in addition to an excellent set of exercises, and a brief primer on MATLAB as an appendix.
The book is targeted at advanced undergraduate or beginning graduate mathematics students who have a background in multivariable calculus and ordinary differential equations including basic dynamical systems theory (i.e phase plane analysis), and some familiarity with stochastic processes such as random walks. Chapter 1 contains a very quick overview of these three topics. The rest of the book loosely falls into two halves according to the preface, with Chapters 1-10 developing successfully more complicated kinds of models and their analysis, and Chapters 11-14 primarily focusing on applications. This distinction is not precise, however, as Chapter 13 briefly introduces some new kinds of models (integrodifference equations), and there are an enormous number of biological examples and applications discussed in the first part of the book. While the focus is on PDE-type models, there is a frequent interplay of stochastic processes, numerical simulations, and a variety of analytical techniques for understanding these models in terms of their biological significance.
While the book is intended to serve somewhat as an introduction to PDEs and their analytical/numerical analysis, it is sometimes too brief to really get across many of the nuances and generalities of any particular method. For instance, Chapter 9 studies advection-reaction equations, introducing the method of characteristics (numerically and analytically), age-structured models (which include integral boundary terms), upwinding and the CFL condition, and the use of moment-generating functions to derive and analyze Burgers' equation. Chapter 9 also discusses a host of biological applications from age-structured epidemics, red blood cell production, polymerization of gels and blood clotting, oxygen exchange between capillaries, protein-mediated friction, and muscle contraction. This is all done in about 33 pages, and despite knowing something about all of these things before reading, I had to take several breaks to properly digest what was written. This terse and rapid style allows the author to cover a huge breadth of material with relatively little background, though it is somewhat demanding on the reader, and it is not always clear where to look for more information on any particular idea presented.
There are many important contrasts between this book and the classical two-volume set Mathematical Biology by James D. Murray. Murray's books are much longer, with many more references and more comprehensive pacing. Turing instabilities, for example, take up at least twice as much space in Murray as opposed to the present text. The preface also mentions the two classical areas of application in mathematical biology of population dynamics and physiology, with the easier often more easily grasped by students, but the latter more quantitatively accessible. The present book quickly dives into examples of both, with perhaps much more emphasis on quantitative physiological phenomena compared to Murray (especially the earlier parts of both books). This is useful from the perspective of quickly getting to modern research topics in physiology and medicine, but would require some care from a teaching perspective in terms of properly motivating these more unfamiliar physiological scenarios.
There are a few typographical errors (e.g. Equation (8.12)), and a few typesetting choices that, at least to me, look bad (e.g. all parentheses seem to be the same size, independent of the size of expressions). Nevertheless, the book is extremely well-designed in terms of the use of mathematics, textual descriptions, and Figures, with a number of really well-put-together discussions synthesizing numerical and analytical insights in terms of biologically-meaningful understanding. 
Overall, I would highly recommend this book to anyone interested in a modern crash course in Mathematical Biology, or who wants to see an exciting application-oriented introduction to PDEs and their analysis. I plan to incorporate several topics and examples discussed in my own lectures this next year.


Dr. Andrew Krause is an Assistant Professor in Applied and Computational Mathematics at Durham University. His research is primarily in mathematical biology and nonlinear dynamical systems. More information can be found at