Genealogies in bistable waves

Abstract

We study a model of selection acting on a diploid population (one in which each individual carries two copies of each gene) living in one spatial dimension. We suppose a particular gene appears in two forms (alleles) A and a, and that individuals carrying $AA$ have a higher fitness than $aa$ individuals, while $Aa$ individuals have a lower fitness than both $AA$ and $aa$ individuals. The proportion of advantageous A alleles expands through the population approximately according to a travelling wave. We prove that on a suitable timescale, the genealogy of a sample of A alleles taken from near the wavefront converges to a Kingman coalescent as the population density goes to infinity. This contrasts with the case of directional selection in which the corresponding limit is thought to be the Bolthausen-Sznitman coalescent. The proof uses ‘tracer dynamics’.