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An Exactly Solved Model for Mutation, Recombination and Selection

  Published:2003-02-01
 Printed: Feb 2003
  • Michael Baake
  • Ellen Baake
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Abstract

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 M\"obius 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: population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, Möbius inversion population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, Möbius inversion
MSC Classifications: 92D10, 34L30, 37N30, 06A07, 60J25 show english descriptions Genetics {For genetic algebras, see 17D92}
Nonlinear ordinary differential operators
Dynamical systems in numerical analysis
Combinatorics of partially ordered sets
Continuous-time Markov processes on general state spaces
92D10 - Genetics {For genetic algebras, see 17D92}
34L30 - Nonlinear ordinary differential operators
37N30 - Dynamical systems in numerical analysis
06A07 - Combinatorics of partially ordered sets
60J25 - Continuous-time Markov processes on general state spaces
 

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