On the speed of coming down from infinity for subcritical branching processes with pairwise interactions

TL;DR

This paper analyzes the speed and fluctuations of subcritical cooperative branching processes with pairwise interactions starting from large populations, extending classical models and applying to population genetics.

Contribution

It characterizes the speed of coming down from infinity and second-order fluctuations for BPI processes, including applications to fragmentation-coalescent models.

Findings

Characterized the speed of coming down from infinity for BPI processes.

Analyzed second-order fluctuations of these processes.

Extended results to exchangeable fragmentation-coalescent processes.

Abstract

In this paper, we study the phenomenon of coming down from infinity for subcritical cooperative branching processes with pairwise interactions (BPI processes) under suitable conditions. BPI processes are continuous-time Markov chains that extend classical branching models by incorporating additional mechanisms accounting for both competitive and cooperative interactions between pairs of individuals. Our main focus is on characterising the speed at which BPI processes evolve when starting from a very large initial population in the subcritical regime. In addition, we investigate their second-order fluctuations. Furthermore, our results also apply to a class of exchangeable fragmentation-coalescent processes introduced by Berestycki (2004) and several other models from population genetics.