The Multi-type Bisexual Galton-Watson Branching Process

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.

Assessing the probability and time of weed control failure due to herbicide resistance evolution

“…The weed survival probability is not notably affected by cross-pollination between resistant types ( Figure S3 a). For the sexual multitype Galton-Watson process, all theoretical results were derived exclusively for superadditive mating functions (Hull, 1998; Fritsch, Villemonais, and Zalduendo, 2022). However, random mating in a population of several types results in a non-superadditive mating function, not allowing any analysis of the process but solely simulations.…”