Fixation in bisexual models with variable population sizes. (English) Zbl 0880.92029

Möhle, M.

J. Appl. Probab. 34, No. 2, 436-448 (1997).

Summary: A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely.
The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.

MSC:

92D10	Genetics and epigenetics
60G42	Martingales with discrete parameter
60G09	Exchangeability for stochastic processes

Keywords:

fixation; homozygosity; population dynamics; Wright-Fisher model; Moran model