The Multi-type Bisexual Galton-Watson Branching Process

TL;DR

This paper analyzes a multi-type bisexual Galton-Watson process with a superadditive mating function, establishing extinction criteria, long-term behavior, and convergence properties using concave Perron-Frobenius theory.

Contribution

It introduces a novel concave reproduction operator for the bisexual Galton-Watson process and derives conditions for extinction and convergence.

Findings

Established a necessary and sufficient condition for almost sure extinction.

Proved a law of large numbers for the process.

Identified conditions for convergence in $L^1$ to a non-degenerate limit.

Abstract

In this work we study the bisexual Galton-Watson process with a finite number of types, where females and males mate according to a ''mating function'' and form couples of different types. We assume that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. In the absence of a linear reproduction operator which is the key to understand the behaviour of the model in the asexual case, we build a concave reproduction operator and use a concave Perron-Frobenius theory to ensure the existence of eigenelements. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the almost sure long-time convergence of the rescaled process through the identification of a supermartingale, and we give sufficient conditions to ensure a convergence in $L^1$ to a non-degenerate limit.