  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.
