Two-Sex Branching Populations

Abstract: We introduce a Two-Sex Galton-Watson Branching Process, to study the dynamics of two-sex populations. It allows individuals to interact with each other in order to form couples (or mating units), and each of these couples produces children for the new generation. We investigate the long-run behavior of this general model for two special cases: (i) when in a given generation each female has equal chance of getting married, where this chance could depend on the total number of marriages in the previous generation as well as on the total numbers of females and males in that generation; (ii) when in a given generation the number of marriages of the daughters of each family could depend on each other as well as on the total numbers of daughters and sons in that family and the total numbers of females and males in that generation. We investigate both the super-critical and the critical cases of these models.

Beside the general model mentioned above, we also study the behavior of a simple, but interesting bisexual model. In it female reproductivity tends to increase with the number of males in the population. With the help of the coupling method, we study the long-run behavior in the super-critical case.

AMS 1991 subject classification:} 60J80, 92A15