Genealogies in bistable waves (Q2082710)

The authors 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. A particular gene appears in two forms (alleles) \(A\) and \(a\), and 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. The authors 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'.