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Coalescence in a random background. (English) Zbl 1060.60100

A single genetic locus which carries two alleles labelled \(P\) and \(Q\) is considered. For the Wright-Fisher model a system of algebraic equations is given for the probabilities of identity in state for a sample of size two from such a population. In order to obtain the diffusion approximation it is convenient to work with the continuous time counterpart of the Wright-Fisher model, the Moran model. For such a model, assuming that the frequencies of the selected alleles, \(P\) and \(Q\), have reached stationarity, it is written the generator of the process that encodes the backward in time evolution of the selected allele frequencies and the numbers of ancestors of the sample of neutral alleles alive at time \(t\) before the present, labeled according to their background (\(P\) or \(Q\)). The parameters of the model are rescaled and the form of the generator of the corresponding diffusion appraximation is established. Also the existence of a stochastic process with this generator and convergence of the rescaled processes to this limit are established. A system of differential equations for the distribution of coalescence times and hence, for the probability of identity in state in a sample of size two, is written. An iterative solution to the system is established. The extension to larger samples and more complex genetic backgrounds is indicated. Also, numerical examples are presented that illustrate the accuracy and predictions of the diffusion approximation.

MSC:

60K37 Processes in random environments
60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 Applications of branching processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:

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