Mathematical models are a critical part of evolutionary biology. Sometimes they are used to analyze data, sometimes they are used to make theoretical predictions, but in either case, they represent the purest expression of what biologists suppose about the relationship between the patterns they observe in the natural world and the processes that produce them. As a result, they are often at the core of the most important presentations of research in the scientific literature and at professional conferences. Unfortunately they also tend to be very abstract, and can be a stumbling point in those presentations*.
The conference Evolution 2012 has now been raging for three days, and the air is thick with fascinating models, and also incomprehensible ones, so I wanted to take a moment to highlight a couple of speakers who I felt presented a mathematical model in an exceptionally clear and excellent way.
First, Carl Boettiger, in his half hour talk “Detecting evolutionary regime shifts with comparative phylogenetics.” He spoke about developing a model to analyze trait data in a phylogenetic context to ask whether the evolution of the pharyngeal jaw in Labroid fishes freed them from the suction feeding habits of their ancestors and allowed a burst of morphological and ecological diversification in the group. This question, whether a “key innovation” can release a constraint and result in diversification is a long-standing one in evolution, and no explicit mathematical model capable of being applied to real data had been developed to address it. After explaining this, Boettiger launched into a comprehensive, clear explanation of the derivation of his model, what each of the terms meant and then proceeded to analyze it. Perhaps not surprisingly, he found that the pharyngeal jaw did indeed seem to serve as a key innovation, releasing a constraint and resulting in diversification. (Boettiger’s talk is, amazingly, already on-line if you want to check it out)
The second was Gideon Bradburd. Bradburd gave a 15 minute talk “A Bayesian method for estimating genetic differentiation due to isolation by geographic and ecological distance.” Identifying the factors that cause population divergence is a key task in evolution, ecology and conservation biology. One of the ways to go about doing this is to look for correlations between population genetic divergence and geographical or environmental factors. One of the typical ways that people have gone about doing that is by using a statistical method called a partial mantel test. Unfortunately, the partial Mantel test has some undesirable statistical properties that cause it to be misleading under some circumstances. Enter the new model. In just a few slides, Bradburd clearly explained the data required, the math behind how he models genetic divergence, and how the explanatory variables fit in. He then used the model to ask whether elevation influenced genetic divergence in the wild ancestor of corn, teosinte. He found that the model appeared to perform very well and give strong evidence that elevation was important.
Talks like these make the meetings fun to attend, and represent the kind of clarity I strive for in my own presentations.
*See the much discussed recent paper in PNAS (Fawcett and Higginson 2012), in which it is shown that the density of equations in a scientific publication has a negative impact on its citation rate.
Nothing in Biology Makes Sense! 
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