On Asymptotic Behavior of Extinction Moment of Critical Bisexual Branching Process in Random Environment

TL;DR

This paper analyzes the extinction time of a critical bisexual branching process in a random environment, showing it scales as the square of the logarithm of the initial population, with broad applicability across various mating functions.

Contribution

It establishes the asymptotic order of extinction time for a general class of bisexual branching processes in random environments, including monogamous and polygamous models.

Findings

Extinction time scales as ^2 for large initial populations.

Results hold for a broad class of mating functions.

Applicable to models reducing bisexual processes to simple ones.

Abstract

We consider a critical bisexual branching process in a random environment generated by independent and identically distributed random variables. Assuming that the process starts with a large number of pairs $N$, we prove that its extinction time is of the order $\ln^2 N$. Interestingly, this result is valid for a general class of mating functions. Among them are the functions describing the monogamous and polygamous behavior of couples, as well as the function reducing the bisexual branching process to the simple one.