A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

Let k ≥ 0 be an integer, T = (T(t))t ∈R a C0-group of bounded operators and A the infinitesimal generator of T. We prove that if, and , and if the spectrum of A is equal to {λ}, then A is bounded and (A – λ)k+1 = 0. Examples are given to show that these conditions are, essentially, the best possible.