The goal of A Course in Mathematical Biology is to introduce students problem solving in the context of biology, and to do so by integrating analytical and computational tools for modeling biological processes. According to the introduction, this book is directed principally toward undergraduates in mathematics, physics, biology and other quantitative sciences.
The contents fall neatly into three parts. The first part addresses basic techniques of analytical modeling: differential and difference equations, probability and cellular automata, parameter estimation and model comparison.
The second part concentrates on computational tools for modeling biological processes. The authors use Maple as their standard software, but examples and exercises in this part could be easily adapted to use MATLAB or Mathematica instead. The last part of the book provides open-ended problems from physiology, ecology and epidemiology. A concluding section presents “solved projects”, a detailed review of modeling efforts in cell competition and chemotaxis developed by students during a workshop.
The mathematical threshold is set pretty high in this text. It is accessible to upper level mathematics and physics majors, but I would expect most biology majors, for example, to find this pretty tough sledding. It is not so much that the prerequisites are extensive; no more than basic calculus and some familiarity with linear algebra and differential equations are assumed. It is more that a relatively high level of mathematical sophistication and comfort with mathematical thinking are required. There is simply a lot of new mathematics introduced as the book progresses. By the time we have reached page 100, we see, for example, linear stability analysis for difference equations, Jury conditions for the eigenvalues of the Jacobian, and bifurcation analysis. Shortly thereafter, the authors plunge into partial differential equations and the reaction-diffusion equation. Too much, too fast for many potential readers!
A small thing, but one which shows up more and more frequently in textbooks at a similar level, is the treatment of existence and uniqueness for solutions of differential equations. Here the authors define Lipschitz continuity, immediately state the Picard-Lindelöf theorem, and then stop. This, I think, has vanishingly small value to the student. Why not either spend some time discussing the issues of existence and uniqueness with examples and pictures, or state that you’re going to assume that all equations under discussion have unique solutions and leave it at that?
Complaints aside, this is a fine text for those students with an appropriate background. There is a nice mix of biological applications, ranging from the more common examples in epidemiology and population genetics to forest ecology, the polymerase chain reaction, movement of flagellated bacteria, and ocular dominance patterns. The chapter on parameter estimation and model comparison is well-done, but at a rather sophisticated level — it begins with a discussion of the likelihood function and accelerates from there. The authors do not give much attention to the verification and validation of models. This is an important topic — particularly for biological models — and deserves a more extensive treatment.
The authors have provided an annotated section of “Further Reading” as well as an extensive bibliography. Both of these are valuable for a field that has grown enormously over the last ten years.
Bill Satzer (firstname.lastname@example.org) 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.
Part I: Theoretical Modeling Tools
Chapter 1: Introduction
Chapter 2: Discrete-Time Models
Chapter 3: Ordinary Differential Equations
Chapter 4: Partial Differential Equations
Chapter 5: Stochastic Models
Chapter 6: Cellular Automata and Related Models
Chapter 7: Estimating Parameters
Part II: Self-Guided Computer Tutorial
Chapter 8: Maple Course
Part III: Projects
Chapter 9: Project Descriptions
Chapter 10: Solved Projects
Appendix: Further Reading