## Introduction

Theoreticians whose primary tools are mathematical models operate in such a bewildering variety of ways that someone attempting to summarize the programmatic approach to modeling--a textbook author, for example--is easily frustrated. However, with a bit of effort one begins to see a broad dichotomy among approaches. The two tines of this dichotomy I'll call "tactical" and "strategic." In the first, one identifies a problem or question in biology (or physics or chemistry or ...) and then looks for specific analytical tools to solve or answer it. The second approach addresses the analytical tools themselves and attempts to extend and improve them.

For example, the origin of our first cancer model was the question, are tumors more like ecosystems, which are dominated by competition and segregation, or like tissues, which comprise cooperating cells? Since ordinary differential equations and adaptive dynamics were tools that we had mastered, they are what we applied to the problem. This is an example of the tactical approach. The strategic outlook is well illustrated in a recent paper by our colleagues Kalle Parvinen (who I am visiting in Turku, Finland as I write this), Mikko Heino and Ulf Dieckmann (J. Math. Biol. 67:509). As they note in the introduction, "adaptive dynamics was originally formulated for scalar strategies"--that is, for strategies that can be represented as a real number. However, there are times when the strategy cannot be so represented. For example, the strategy could be vector-valued (characterized by a set of real numbers) or function-valued (represented as a real function of a real variable). In previous work, they develop adaptive dynamics theory for a certain class of models with function-valued strategies using calculus of variations. But models of this class suffer key limitations that restrict the theory's ability to analyze realistic scenarios. So, in the paper quoted, they extend the results to a more general class using optimal control theory. This is what I mean by "strategic."

This dichotomy, of course, is one person's perception. Another person could debate it on the grounds that these two approaches are not mutually exclusive. I would completely agree. Strategic advances must be directed by tactical needs. In other words, one has to have a good reason to extend a modeling theory. On the other hand, when addressing a tactical problem, or even deciding which problem to tackle, one refers to the analytical toolkit at hand, leading to the common perception that modelers tend to shoehorn empirical problems to fit their "favorite" analytical tool. Although such a view (which has been expressed to me many times) is a little unfair, there is some truth in it. But one should not allow such criticisms, among others, to obscure the facts that the dichotomy, if perhaps a little simplistic, is real, and that neither tactical nor strategic programs can survive without the other.