Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory

The allele frequency spectrum has attracted considerable interest for the simultaneous inference of the demographic and adaptive history of populations. In a recent study, Evans et al. (2007) developed a forward diffusion equation describing the allele frequency spectrum, when the population is subject to size changes, selection and mutation. From the diffusion equation, the authors derived a system of ordinary differential equations (ODEs) for the moments in a Wright–Fisher diffusion with varying population size and constant selection. Here, we present an explicit solution for this system of ODEs with variable population size, but without selection, and apply this result to derive the expected spectrum of a sample for time-varying population size. We use this forward-in-time-solution of the allele frequency spectrum to obtain the backward-in-time-solution previously derived via coalescent theory by Griffiths and Tavaré (1998). Finally, we discuss the applicability of the theoretical results to the analysis of nucleotide polymorphism data.