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Stationary distributions under mutation-selection balance: structure and properties

Part of: Markov processes Groups and semigroups of linear operators, their generalizations and applications Special classes of linear operators

Published online by Cambridge University Press:  01 July 2016

Reinhard Bürger  and
Immanuel M. Bomze
Reinhard Bürger*
Affiliation:
Universität Wien
Immanuel M. Bomze*
Affiliation:
Universität Wien
*
* Postal address: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria.
** Postal address: I.S.O.C., Universität Wien, Universitätsstraße 5, A-1010 Wien, Austria.
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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 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.

Keywords

PROBABILITY MEASURESPERRÖN-FROBENIUS THEORYINTEGRO-DIFFERENTIAL EQUATIONSPOSITIVE OPERATORSCONTINUUM-OF-ALLELES

MSC classification

Primary: 92D10: Genetics
Secondary: 60J35: Transition functions, generators and resolvents 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 47B65: Positive operators and order-bounded operators 47D03: Groups and semigroups of linear operators
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 227 - 251
Copyright
Copyright © Applied Probability Trust 1996 

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