Limiting fitness distributions in evolutionary dynamics
This resolves the dynamic insufficiency of Fisher's fundamental theorem for evolutionary biology, providing a more general statistical framework for modeling fitness distributions.
The authors tackled the modeling of Darwinian evolution by showing that evolving fitness distributions converge to a one-parameter family with a fixed parabolic skewness-kurtosis relationship, encompassing both positive and negative selection, and found that mean fitness follows a power-law over time, matching experimental data but differing from fitness wave theory.
Darwinian evolution can be modeled in general terms as a flow in the space of fitness (i.e. reproductive rate) distributions. In 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.