Stationary gene frequency distribution in the environment fluctuating between two distinct states

A general method is given to obtain a stationary distribution in a “stochastic” one-dimensional dynamical system in which an environmental parameter specifying the dynamical system is a stationary Markov process with only two states. By applying this method, the exact stationary gene frequency distribution is obtained for a genic selection model in the environment fluctuating between two distinct states. Several limiting stationary distributions are obtained therefrom, and one of them is shown to coincide with a stationary solution of the diffusion equation heuristically derived by us for more general cases. Discussion is given on the relationship between the diffusion equations obtained by various authors starting from discrete, non-overlapping generation models.