Bisexual branching processes in a genetic context: the extinction problem for Y-linked genes

In this paper the inheritance of a Y-linked gene with alleles R and r in a population with both females and males is modelled using a two-type bisexual branching process. It is assumed that the reproductive distribution associated with the R allele can differ from the associated with the r allele and that females prefer to mate with a male having the R allele rather than with a male with the r allele. Under these assumptions, we provide some conditions for the extinction and/or survival of both alleles in the population. These conditions depend on the magnitudes of the average number of females and males per mating unit. Moreover, the almost sure extinction of the r allele is independent of the behaviour of the R allele. On the other hand, the survival of the R allele with positive probability may depend strongly on the reproductive behaviour of the other allele. Theoretical results are illustrated by means of simulated examples and some open problems are proposed to the reader as conjectures.