Duality and asymptotics for a class of nonneutral discrete Moran models. (English) Zbl 1181.92067

Huillet, Thierry; Möhle, Martin

J. Appl. Probab. 46, No. 3, 866-893 (2009).

A multitype (\(K\)-sex) Moran population model is introduced which coincides with the standard haploid Moran model when \(K=1\) and with Kämmerle’s two-sex Moran model if \(K=2\). A duality result is established for the “forward” Markov chain \(X_i\) which counts the descendants in the \(i\)-th generation and the “backward” ancestral chain \(Y_i\) which counts the ancestral mating units. Weak convergence of normalized \(Y\) to an Ornstein-Uhlenbeck process is established as the size of population tends to infinity. Asymptotics of the mean and variance of \(Y\) are given. The extinction probabilities of the forward chain \(X\) and the stationary distribution of \(Y\) are investigated.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

92D15	Problems related to evolution
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
92D10	Genetics and epigenetics

Keywords:

multitype Moran model; ancestral process; descendants process; duality; Ornstein-Uhlenbeck process

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