Article ID: | iaor1997310 |
Country: | United Kingdom |
Volume: | 28 |
Issue: | 1 |
Start Page Number: | 227 |
End Page Number: | 251 |
Publication Date: | Mar 1996 |
Journal: | Advances in Applied Probability |
Authors: | Bomze Immanuel M., Brger Reinhard |
Keywords: | differential equations |
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 a 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 te 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.