Predator-prey dynamics on infinite trees: a branching random walk approach

TL;DR

This paper analyzes predator-prey dynamics on infinite trees using branching random walks, providing new insights and results on the chase-escape and birth-and-assassination processes, including asymptotic behaviors and criticality outcomes.

Contribution

It introduces a coupling between predator-prey processes and branching random walks, enabling new probabilistic analyses and results for these models.

Findings

Asymptotic tail behavior of infected individuals in subcritical and critical regimes

Almost sure extinction of birth-and-assassination process at criticality

Recovery of known results and derivation of new insights using probabilistic coupling

Abstract

We are interested in predator-prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase-escape process, introduced by Kordzakhia with a variant by Bordenave, and the birth-and-assassination process, introduced by Aldous & Krebs. We exhibit a coupling between these processes and branching random walks starting from a random point and killed at the barrier 0. This sheds new light on the chase-escape and birth-and-assassination processes, which allows us to recover by probabilistic means previously known results and also to obtain new results. For instance, we find the asymptotic behavior of tail of the number of infected individuals in both the subcritical and critical regimes for the chase-escape process, and show that the birth-and-assassination process ends almost surely at criticality.