J F Crow in Journal of Biology (free registration required to read the article) discusses the relevance of mathematical models in evolutionary studies — in a piece that has some interesting academic gossip too:
In 1959 Ernst Mayr (…) flung down the gauntlet […] at the feet of the three great population geneticists RA Fisher, Sewall Wright and JBS Haldane (…): “But what, precisely,” he said, “has been the contribution of this mathematical school to the evolutionary theory, if I may be permitted to ask such a provocative question?” His skepticism arose in part from the fact that the mathematical theory at the time had little to say about speciation, Mayr’s major interest. But his criticism was more broadly addressed to the utility of the entire approach. A particular focus was the simplification that he called “beanbag genetics”, in which “Evolutionary genetics was essentially presented as an input or output of genes, as the adding of certain beans to a beanbag and the withdrawing of others.” […].
Mayr was, however, criticizing textbook simplifications, rather than the actual work of the three pioneers. Far from treating gene frequency changes as analogous to the consequence of beans jostling at random in a bag, both Fisher and Wright considered gene interactions in detail. Fisher (…) showed that, despite interactions between genes, natural selection acts on the additive component of the genetic variance. It is as if nature were familiar with least squares. The beanbag criticism was particularly inappropriate for Wright (…), who specifically devised his ‘shifting balance’ theory as a way for a population to go from one harmonious gene combination (Mayr would say “integrated genotype”) to another when intermediates were disadvantageous.
Who was to answer Mayr’s criticism? Fisher was already dead, and in any case preferred attack to defense, and Wright was too gentle – though admittedly not always when Mayr was involved: returning from Italy where he had received the prestigious Balzan Prize in 1984, Wright told me that the value of the prize was considerably diminished when he discovered that Mayr had won it the year before. In the event, however, it was Haldane (…) who took up the challenge. And he did it with flair and gusto. The result was “A defense of beanbag genetics” […]. This was Haldane at his best – witty, spirited, informed, interesting and convincing.
But the larger question remains: what indeed has been the contribution of mathematical theory to evolution? Mathematics is not central to evolution in the way it has been in theoretical physics. Solid advances have been made without using mathematics, much being due to Mayr himself […]. And these continue. Yet, I shall argue that mathematical ideas have made important, and often essential, contributions, and still do. Many concepts that are now established were arrived at mathematically, although their origins have since been forgotten.