Genetic contribution of advantaged ancestors in the biparental Moran model -- finite selection

Abstract: We study a population of N individuals evolving according to a biparental Moran model with two types, one being advantaged compared to the other. The advantage is conferred by a Mendelian mutation, that reduces the death probability of individuals carrying it. We assume that a proportion a of individuals initially carry this mutation, which therefore eventually gets fixed with high probability. After a long time, we sample a gene uniformly from the population, at a new locus, independent of the locus under selection, and calculate the probability that this gene originated from one of the initially advantaged individuals, when population size is large. Our theorem provides quantitative insights, such as the observation that under strong selection, if only 1% of the individuals are initially advantaged, approximately 19% of the population's genome will originate from them after a long time.