Genealogies in bistable waves

Consider a diploid population (one in which each individual carries two copies of each gene) living in one spatial dimension. 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 can prove that on a suitable timescale, the genealogy (family tree) of a sample of A alleles taken from near the wavefront converges to a Kingman coalescent as the population density goes to infinity. We conjecture that if the relative fitness of Aa individuals is increased then the behaviour of the genealogy changes as the travelling wave changes from a pushed wave to a pulled wave.

Joint work with Alison Etheridge.