Quasi-limiting behaviour of the sub-critical Multitype Bisexual Galton-Watson Branching Process

TL;DR

This paper studies the long-term behavior of bisexual subcritical Galton-Watson processes, focusing on their quasi-stationary distributions and extinction properties, which are more complex due to the lack of the branching property.

Contribution

It introduces new methods to analyze the quasi-limiting behavior of bisexual Galton-Watson processes by connecting extinction to eigenvalues of a concave operator.

Findings

Existence of quasi-stationary distributions established.

Convergence to these distributions demonstrated.

Link between extinction and eigenvalues of a concave operator identified.

Abstract

We investigate the quasi-limiting behaviour of bisexual subcritical Galton-Watson branching processes. While classical subcritical Galton-Watson processes have been extensively analyzed, bisexual Galton-Watson branching processes present unique difficulties because of the lack of the branching property. To prove the existence of and convergence to one or several quasi-stationary distributions, we leverage on recent developments linking bisexual Galton-Watson branching processes extinction to the eigenvalue of a concave operator.