Dynamics of a quantitative character subject only to stabilising selection

The implications of stabilising selection on a quantitative trait, in the absence of other evolutionary forces, are theoretically investigated in a randomly mating population. The dynamics of various statistics that describe the alleles contributing to the trait are determined and used to infer the behaviour of the trait. Dynamical solutions of the distribution of allelic effects and the distribution of the trait are found when all initial distributions of allelic effects are Gaussian and linkage disequilibria are neglected. Some results for the behaviour of the mean and the variance of genotypic effects of the population, when subject to a moving optimum, are derived. When the initial distributions of allelic effects are not Gaussian, but possess a small asymmetry, the mean and the variance of the allelic effects differ only slightly from the Gaussian results. By contrast, the third central moments of allelic effects, are, at all loci, strictly zero in the Gaussian case but are generally non-zero for non-symmetric initial distributions. To leading order in a quantitative measure of the asymmetry of the distribution, we determine the third central moment of allelic effects.