Limiting fitness distributions in evolutionary dynamics

TL;DR

This paper broadens the understanding of fitness distribution evolution by identifying a new family of distributions that include both positive and negative selection, and shows that mean fitness follows a power-law over time, aligning with experimental data.

Contribution

It introduces a more general statistical pattern for evolving fitness distributions, extending beyond fitness wave solutions, and explains the power-law growth of mean fitness.

Findings

Fitness distributions tend toward a family with fixed skewness-kurtosis relationship.

Mean fitness follows a power-law function of time under selection from pre-existing variation.

Results align with microbiological evolution experiments and numerical simulations.

Abstract

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.