Coalescent patterns in diploid exchangeable population models. (English) Zbl 1054.92039

Möhle, Martin; Sagitov, Serik

J. Math. Biol. 47, No. 4, 337-352 (2003).

Summary: A class of two-sex population models is considered with \(N\) females and equal number \(N\) of males constituting each generation. Reproduction is assumed to undergo three stages: 1) random mating, 2) exchangeable reproduction, 3) random sex assignment. Treating individuals as pairs of genes at a certain locus we introduce the diploid ancestral process (the past genealogical tree) for \(n\) such genes sampled in the current generation. Neither mutation nor selection are assumed.
A convergence criterium for the diploid ancestral process is proved as \(N\) goes to infinity while \(n\) remains unchanged. Conditions are specified when the limiting process (coalescent) is the Kingman coalescent and situations are discussed when the coalescent allows for multiple mergers of ancestral lines.

MSC:

92D15	Problems related to evolution
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60F17	Functional limit theorems; invariance principles
60F05	Central limit and other weak theorems

Keywords:

ancestral process; coalescent; diploid model; exchangeability; generator; neutrality; population genetics; two-sex model; weak convergence