Population-dependent two-sex branching processes with random mating: rates of growth

In Jacob et al. [A General Class of Population-Dependent Two-Sex Processes with Random Mating. Bernoulli 2017, 23, 1737-1758], a new class of two-sex branching processes in discrete time was introduced. These processes present the novelty that, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behavior with only perfect fidelity on the part of female individuals. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. In that article, the authors study conditions leading to the almost sure extinction or to the possible survival with positive probability of such processes. In this work, we continue the research about this class of two-sex processes by investigating the rate of growth of the population in case of survival. In particular, we provide conditions for a simultaneous geometric growth of the number of females and males and conditions for the geometric growth of females with a stable population of males.