The bisexual Galton-Watson process with promiscuous mating: extinction probabilities in the supercritical case

Abstract

We consider the bisexual Galton-Watson process (BGWP) with promiscuous mating, that is, a branching process which behaves like an ordinary Galton-Watson process as long as at least one male is produced in each generation. For the case of Poissonian reproduction, it was pointed out by Daley, Hull and Taylor that the extinction probability of such a BGWP apparently behaves like a constant times the respective probability of its asexual counterpart (where males do not matter) providing the number of ancestors grows to infinity. They further mentioned that they had no theoretical justification for this phenomenon. In the present article we will prove upper and lower bounds for the ratio between the two extinction probabilities and introduce a recursive algorithm that can easily be implemented on a computer to produce very accurate approximations for that ratio. The final section contains a number of numerical results that have been obtained by use of this algorithm.