Alternative to the diffusion equation in population genetics.

Since its inception by Kimura in 1955 [M. Kimura, Proc. Natl. Acad. Sci. U.S.A. 41, 144 (1955)], the diffusion equation has become a standard technique of population genetics. The diffusion equation is however only an approximation, valid in the limit of large populations and small selection. Moreover, useful quantities such as the fixation probabilities are not easily extracted from it and need the concomitant use of a forward and backward equation. We show here that the partial differential equation governing the probability generating function can be used as an alternative to the diffusion equation with none of its drawbacks: it does not involve any approximation, it has well-defined initial and boundary conditions, and its solutions are finite polynomials. We apply this technique to derive analytical results for the Moran process with selection, which encompasses the Kimura diffusion equation.