UK Nonlinear News, February 2004
2003. 92 figures. 335pp. Springer-Verlag.
Softcover. ISBN 1-85233-536-X
Because the mathematical biosciences have come into such high fashion (for which fact, trust me, I make no complaint) there has arisen a demand for courses in mathematical biology, courses that will introduce mathematically inclined students to the delights of the other side. Such courses come in many different flavours, as many as there are teachers one imagines. Some are aimed at biology students also, some at bioengineers or other engineering students, while others are aimed more strictly at mathematics majors, to give them a flavour of biological applications while still teaching them mathematical techniques. It is this latter kind of course at which Britton's book is aimed.
When I first picked up the book and skimmed through it, I was left with the superficial impression that it is just Jim Murray without the steroids. After all, Murray's book  is one that you could kill with. Literally. (I'm thinking here of tying the volumes together and dropping them from a height. If you did this with Britton's book, it would hardly raise a bump on a student's head. I hasten to add that I haven't yet tried either of these things.) Britton's book goes through the usual topics of population dynamics, a bit of genetics, infectious diseases, biological motion (diffusion, dispersion, chemotaxis, etc), basic enzyme kinetics, reaction-diffusion equations and pattern formation, mechanochemical models, and tumour models. It covers these topics briefly, with minimal digression, the end result being a relatively small book, containing material that could reasonably be covered in a couple of one-semester courses. The focus is entirely on deterministic modeling, and there is no discussion of computational methods.
However, to condemn "Essential Mathematical Biology" just because you couldn't kill someone with it would be to do it a grave disservice, for in brevity and simplicity lies the great strength of this book. It explains its chosen topics clearly and simply, not including extraneous material, and is written at a level that can be understood and appreciated by undergraduate students. Indeed, the level of writing is superb in its clarity and elegance. For this reason it has become a popular teaching text in the UK, as I have been reliably informed. Rather than having to use Murray's wonderful tome (tomes now, I should say), teachers can use a smaller book, containing the essential material, and at the right level and quantity. Just as useful as the writing style are the appendices and the hints. Not only does Britton give elementary presentations of some basic mathematical techniques (difference equations, ODEs and PDEs) he also gives extensive hints for the exercises, bordering on complete solutions in some cases. This is a resource that will surely prove extremely useful for all teachers of such a course. There is even an accompanying web page ( http://www.maths.bath.ac.uk/~nfb/book/) in which additional material may be found, although limited in extent as yet.
So, given the manifold strengths of this book, why would I not use it myself? Well, I can hardly help feeling that, if you want to teach students how to apply math to biology, you really need an awful lot more biology than is given here. As a reviewer, I get to air my pet gripes, and this is one of them. (I have others, so don't ask.) Why should we teach students 82 pages of population dynamics, 30 pages of Turing instabilities, and 6 pages of neuroscience? Isn't this a little lop-sided? Isn't neurophysiology one of the few areas in which the data is quantitative and reproducible, the models well-developed, and the connection between the two explicit? Has anyone yet measured activator and inhibitor concentrations in embryological systems, or is it still in the realm of mathematical fairy tales? Should we really be training math students to believe that an analysis of the Lotka-Volterra equations is mathematical biology? Personally, I believe not.
Where does this leave mathematical biology courses, then? Well, I don't think there really is a good book for undergraduate mathematical biology as yet. The books by Murray  and Keener and Sneyd  are just too thick and heavy, being designed more as graduate texts. Edelstein-Keshet  is more of a mathematics book than anything else, although it does an excellent job in many ways. Hoppensteadt and Peskin  contains a lot of interesting material (and I have taught from this book myself a number of times) but is more like two books in one, as is the edited book by Fall et al. . Taube  has considerable merit, particularly in his unique method of presentation, but is not always suitable for teaching undergraduate mathematics students. Each of these books has strengths and weaknesses, but not one of them covers a broad range of mathematical biology, including computation, data and modeling, at a level and length appropriate for a typical upper-level undergraduate course. I suppose that, in some ways, the search for the perfect book is a futile one; the field of mathematical biology is now so vast and varied, that it is impossible to do it justice, and please every reviewer, in a single undergraduate text.
Nevertheless, my personal prejudices aside, there is no denying that "Essential Mathematical Biology" is superbly designed for the purpose it serves, and will, I am sure, become a popular text book across the world.
 J.D. Murray, Mathematical Biology, Springer, 3rd edition,
 J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, 1998
 L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, 1988
 F. Hoppensteadt and C. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer-Verlag, 2nd edition, 2001
 C. Fall, E. Marland, J. Wagner, J. Tyson (Eds), Computational Cell Biology, Springer-Verlag, 2002
 C.H. Taubes, Modeling Differential Equations in Biology, Prentice Hall, 2000.
UK Nonlinear News thanks Springer-Verlag for providing a review copy of this book.