Numerical and exact solutions for continuum of alleles models

Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards'' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method.