Singular solutions of the di¤usion equation of population genetics

The forward di¤usion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene …xation, with the “weight”associated with the spikes corresponding to the probability of loss or …xation. The forward di¤usion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0; 1) that excludes the frequencies 0 and 1. Previously, the probabilities of the absorbing frequencies 0 and 1 were found by appeal to the backward di¤usion equation, while those in the continuous range (0; 1) were found from the forward di¤usion equation. Our uni…ed approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a di¤usion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when …xation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the …xation probability.