Fisher's lost model of runaway sexual selection.

The bizarre elaboration of sexually selected traits such as the peacock's tail was a puzzle to Charles Darwin and his 19th century followers. Ronald A. Fisher crafted an ingenious solution in the 1930s, positing that female preferences would become genetically correlated with preferred traits due to nonrandom mating. These genetic correlations would translate selection for preferred traits into selection for stronger preferences, leading to a self-reinforcing process of ever-elaborating traits and preferences. It is widely believed that Fisher provided only a verbal model of this "runaway" process. However, in correspondence with Charles Galton Darwin, Fisher also laid out a simple mathematical model that purportedly confirms his verbal prediction of runaway sexual selection. Unfortunately, Fisher's model contains inconsistencies that render his quantitative conclusions inaccurate. Here, we correct Fisher's model and show that it contains all the ingredients of a working runaway process. We derive quantitative predictions of his model using numerical techniques that were unavailable in Fisher's time. Depending on parameter values, mean traits and preferences may increase until genetic variance is depleted by selection, exaggerate exponentially while their variances remain stable, or both means and variances may increase super-exponentially. We thus present the earliest mathematical model of runaway sexual selection.