An exactly solved model for mutation, recombination and selection

Details An exactly solved model for mutation, recombination and selection An exactly solved model for mutation, recombination and selection

2002-10-28

arxiv.org/abs/math/0210422v1

Can. J. Math. 55 (2003) 3 - 41

Mathematics

Classical Analysis and ODEs

48 pages

Scientific paper

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.

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