Exact solutions for the selection–mutation equilibrium in the Crow–Kimura evolutionary model

• We find stationary distributions of the Crow–Kimura model for specific fitness landscapes.
• We propose a generating function method for the equilibrium distributions.
• We give simple analytical expressions for a number of examples.
• Our method generalizes the random variable technique.

We reformulate the eigenvalue problem for the selection–mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations.