A text on the mathematics of diffusion, classical ecology, geology and epidemiology

The back cover of Modeling Differential Equations in Biology explains that, as college level science students only take the rudiments of calculus, this book fills a gap in teaching the biology students how to use differential equations in their research. The text uses actual scientific papers for examples and, therefore, reinforces the relevance of the methodologies. The material is organized from a course taught by the author at Harvard University and follows the contents summarized in Table 1.

Table 1: Contents
1. Introduction
2. Exponential growth with appendix on Taylor’s theorem
3. Introduction to differential equations
4. Stability in a one component system
5. Systems of first order differential equations
6. Phase plane analysis
7. Introduction to vectors
8. Equilibrium in two component, linear systems
9. Stability in non-linear systems
10. Non-linear stability again
11. Matrix notation
12. Remarks about Australian predators
13. Introduction to advection
14. Diffusion equations
15. Two key properties of the advection and diffusion equations
16. The no trawling zone
17. Separation of variables
18. The diffusion equation and pattern formation
19. Stability criteria
20. Summary of advection and diffusion
21. Traveling waves
22. Traveling wave velocities
23. Periodic solutions
24. Fast and slow
25. Estimating elapsed time
26. Switches
27. Testing for periodicity
28. Causes of chaos;
Extra exercises and solutions
Index

The articles accompanying each chapter are current only to the late 1990s when the book was written. For molecular biology, this is ancient history. However, as the book concerns itself with the mathematics of diffusion, classical ecology, geology and epidemiology, this should not be a problem. The author is presumably a mathematician and disavows any previous degree work in life science (but he is obviously a convert by his own enthusiasm for the subject).

Before mentioning some of the pros and cons, I should admit bias here, as the author and I agree on two important points, the first being the place of theory in biology. Unfortunately, theory has never been as acceptable for the professional biologist as it has been in disciplines such as chemistry and physics. Fortunately, this is (very) slowly changing. The author points out that, in some fields, it is the lack of data, rather than theory that holds a particular area of research back. This was obviously written just before the onset of our high throughput screens that now inundate us with data!

The second point is the importance of probability and statistics in biological research. While dismissing the use of this area of study in a single sentence, he quickly qualifies that it is of prime importance in life science and that students should take courses in this area. He merely states that the present text has to do with differential equations and, therefore, focuses on a different type of problem solving. Now on to the core…

What is so good about this text? 
Although the text has many lucid explanations of how nature provides us with excellent modeling opportunities and how simple models may be both quickly constructed and wrong (!), it includes excellent research articles that supplement the text. The author also does a fair job of running through many examples of how a differential equation describes the process that we are studying and allows the researcher to focus on the more important aspects of the complex biological systems. This is illustrated in examples that run the gamut from simple single molecule diffusion across a theoretically simplistic membrane to the mathematics of traveling waves.

What is not quite as good? 
Occasionally, the biologist may be confused or misled by examples where the mathematician makes ‘simplifying assumptions,’ as when a culture dish of bacteria are dividing asynchronously at given time intervals or, certainly, when the math hits without prior explanation. The reader is lulled into a false sense of security in the first few pages, with some very nice examples and discussions and a few equations that appear with good explanations.

Then, all of a sudden, a few new equations appear without the full explanation but the ‘helpful’ hint that all one needs to do is a few simple substitutions and then integrate or differentiate. The biology major may not remember all of the proper techniques right off, so it would be nice to see many worked (and no skipping any steps) examples.

And always remember: (¶/¶x)p = c

Availability:
Modeling Differential Equations in Biology, by Clifford Henry Taubes. Cambridge University Press, New York. 2008. $53.99 (Pb). www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521708432 

John Wass is a statistician based in Chicago, IL. He may be reached at [email protected].