Fixation in bisexual models with variable population sizes

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.