Review of Article by Hoppensteadt:
This is a fairly short article, and so it cannot be very detailed obviously.
I have a few minor suggestions that may help the context a bit:
(1) Historically, there were at least two large fields where the mixture of mathematics and biology became ingrained and very important: (a) Neurophysiology: ever since Hodgkin and Huxley's pioneering work, this field has traditionally used the tools of differential equation models to represent neuronal action potentials, excitable systems, etc.
(b) Mathematical Ecology: Here the tradition started with Volterra. The role of mathematical reasoning and modeling, has been relatively well-accepted in population dynamics and ecology as a result.
Because these areas started quite early (before the term "Mathematical Biology" became popular, practitioners in these fields no longer question the relevance of such methods.
I also think that it would be important to highlight the cross-fertilization that has taken place. There are techniques from mathematics (nonlinear dynamics, combinatorics, probability, etc) that have revolutionized certain areas in biology (genomics being one example). There are also biological problems (developmental biology and pattern formation) that have motivated some very novel mathematics. (Reaction-diffusion equations, and their behaviour has arisen as an outcome of interest in developmental biology and Turing's 1950's work on morphogenesis.)
There is a huge variety of excellent texts in Mathematical Biology. One that could be added to the list is Sneyd & Keener, who are mainly concerned with physiological applications of mathematics.
I think it is simply excellent. It gives a very clear idea of the connections between mathematical work and biological problems, and emphasises that these connections sweep across the entire range of mathematics. Although brief, the article gives some feel for the details, by picking some particular areas and indicating the connections between the mathematical theory and the biological application. As well as indicating how such biological problems can occasionally play a role either in the development of the mathematics or in helping bring some seemingly uninteresting part of mathematics to centre stage as is its wider implications are realised (chaos in 1-dimensional difference equations is a particularly good example of this).
Such comments as I have are very minor.
First, I would be inclined to include one of the Bernoulli's in the pantheon in the first paragraph (arguably the first biological application of calculus was in the mid-1700s, in relation to cowpox vaccination, by Bernoulli's).
Second, in the second paragraph it would probably be too big a stretch to include number theory among the areas of a mathematics with biological application, but I would nevertheless be inclined to do so (eg: 17 and 13 year periodical cicadas).
My third and fourth comments go a bit wider, each suggesting an extra sentence or two.
Third, I think it might be nice to add a short paragraph indicating that earlier applications of mathematics in biology were essentially analytic rather than computational, but the exponential rise in computational power means a great deal of mathematical biology now consists of extremely complex numerical simulations or statistical analyses. This is a change over time, and I think the widening of the kinds of applications is itself worthy of note.
Fourth, and here I am riding one of my own hobby-horses, if the third suggestion is accepted, then I would add to it a sentence indicating that sometimes there are dangers in people with little mathematical training of knowledge producing large “mathematical biology” simulations whose outcome is not at all well understood; I have elaborated this in a short essay on “Uses and abuses of mathematics in biology” (Science, 303, 790-793, 2004).
In summary, the first two points are very minor, the fourth merely a tentative suggestion, but I think the third is an important point.
I think the list of references is excellent.