A finite locus effect diffusion model for the evolution of a quantitative trait

A diffusion model is constructed for the joint distribution of absolute locus effect sizes and allele frequencies for loci contributing to an additive quantitative trait under selection in a haploid, panmictic population. The model is designed to approximate a discrete model exactly in the limit as both population size and the number of loci affecting the trait tend to infinity. For the case when all loci have the same absolute effect size, formal multiple-timescale asymptotics are used to predict the long-time response of the population trait mean to selection. For the case where loci can take on either of two distinct effect sizes, not necessarily with equal probability, numerical solutions of the system indicate that response to selection of a quantitative trait is insensitive to the variability of the distribution of effect sizes when mutation is negligible.