A diffusion model in population genetics with dynamic fitness

We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection and drift. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system. We also prove that the genetic variance in the system tends to zero in the long time limit, and relate the dynamics of the trait mean to the variance.