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An Exactly Solved Model for Mutation, Recombination and Selection

Published online by Cambridge University Press:  20 November 2018

Michael Baake  and
Ellen Baake
Michael Baake
Affiliation:
Institut für Mathematik und Informatik Universität Greifswald Jahnstr. 15a 17487 Greifswald Germany, email: [email protected]
Ellen Baake
Affiliation:
Institut für Mathematik und Informatik Universität Greifswald Jahnstr. 15a 17487 Greifswald Germany, email: [email protected]
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Abstract

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It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.

Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Keywords

92D1034L3037N3006A0760J25population geneticsrecombinationnonlinear ODEsmeasure-valued dynamical systemsMöbius inversion
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 1 , 01 February 2003 , pp. 3 - 41
Copyright
Copyright © Canadian Mathematical Society 2003

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