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Matteo Smerlak emailed me notifying me of this paper, which sounds really interesting:
Abstract. In the most general terms, Darwinian evolution is a flow in the space of fitness distributions. In the limit where mutations are infinitely frequent and have infinitely small fitness effects (the "diffusion approximation", Tsimring et al. have showed that this flow admits "fitness wave" solutions: Gaussian-shape fitness distributions moving towards higher fitness values at constant speed. Here we show more generally that evolving fitness distributions are attracted to a one-parameter family of distributions with a fixed parabolic relationship between skewness and kurtosis. Unlike fitness waves, this statistical pattern encompasses both positive and negative (a.k.a. purifying) selection and is not restricted to rapidly adapting populations. Moreover we find that the mean fitness of a population under the selection of pre-existing variation is a power-law function of time, as observed in microbiological evolution experiments but at variance with fitness wave theory. At the conceptual level, our results can be viewed as the resolution of the "dynamic insufficiency" of Fisher's fundamental theorem of natural selection. Our predictions are in good agreement with numerical simulations.
I asked him if he could write a blog article summarizing this paper, and he gladly agreed! I hope he will write it here on the wiki.
If you see this, Matteo, you can announce any progress you make here! Or, if you have questions, you can ask them here.