Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology

Conditions are derived that we conjecture are necessary and sufficient for the existence of stationary densities for a class of two-dimensional diffusion processes. The derivation of the conditions rests on the assumption that a two-dimensional stationary density (which can be viewed as a stable “internal equilibrium”) exists if and only if all “boundary equilibria” are unstable in the sense that small perturbations lead to moving away from the boundaries with high probability. For the models considered, the boundary equilibria are one-dimensional stationary densities and equilibrium points. To demonstrate the usefulness of the conditions, three random environment models are analyzed: a three-allele selection model, a two-species competition model, and a two-locus selection model. Several of the results obtained have been verified by alternate methods.