Two-Sex Branching Processes with Offspring and Mating in a Random Environment

Abstract

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

The Multi-type Bisexual Galton-Watson Branching Process

“…Since Daley's work, extinction conditions have been studied for models with a more general family of superadditive mating functions (see for instance [Hul82], [Bru84], [DHT86]) and, in the last decades, results on the limit behaviour of this kind of processes were obtained (see for example [AR96], [AR02], [GM96], [GM97]). From these works, new models of two-sex populations have been developed, such as processes in random or varying environment ( [Ma06], [MM09], [MMR03]), processes with immigration ([GMM00], [GM96], [MX06]), processes with mating function depending on the number of couples ( [MMR02], [MMR06], [XW05]) and more recently, processes with random mating ([JMM17], [MM19]). The interested reader can also consult the surveys of Hull [Hul03], Alsmeyer [Als02] and Molina [Mol10] for a wide description of the work accomplished on this family of processes.…”