Duality, ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210-237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33-53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.