Stationary distributions under mutation-Selection balance: Structure and properties

Stationary distributions under mutation-Selection balance: Structure and properties

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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: ,
Keywords: differential equations
Abstract:

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.

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