Mathematics – Probability
Scientific paper
2011-06-20
Mathematics
Probability
35 pages, 2 figures. Accepted for publication in the Annals of Applied Probability
Scientific paper
We extend the spatial Lambda-Fleming-Viot process introduced in [BEV10] to incorporate recombination. The process models allele frequencies in a population which is distributed over the two-dimensional torus T(L) of sidelength L and is subject to two kinds of reproduction events :small events of radius O(1) and much rarer large events of radius O(L^{alpha}) for some alpha in (0,1]. We investigate the correlation between the times to the most recent common ancestor of alleles at two linked loci for a sample of size two from the population. These individuals are initially sampled from `far apart' on the torus. As L tends to infinity, depending on the frequency of the large events, the recombination rate and the initial distance between the two individuals sampled, we obtain either a complete decorrelation of the coalescence times at the two loci, or a sharp transition between a first period of complete correlation and a subsequent period during which the remaining times needed to reach the most recent common ancestor at each locus are independent. We use our computations to derive approximate probabilities of identity by descent as a function of the separation at which the two individuals are sampled.
Etheridge Alison M.
Veber Amandine
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