On the Galton-Watson Predator-Prey Process

Abstract

We consider a probabilistic, discrete-time predator-prey model of the following kind: There is a population of predators and a second one of prey. The predator population evolves according to an ordinary supercritical Galton-Watson process. Each prey is either killed by a predator in which case it cannot reproduce, or it survives and reproduces independently of all other population members and according to the same offspring distribution with mean greater than 1. The resulting process $(X_n, Y_n)_{n \geq 0}$, where $X_n$ and $Y_n$, respectively, denote the number of predators and prey of the $n$th generation, is called a Galton-Watson predator-prey process. The two questions of almost certain extinction of the prey process $(Y_n)_{n \geq 0}$ given $X_n \rightarrow \infty$, and of the right normalizing constants $d_n, n \geq 1$ such that $Y_n/d_n$ has a positive limit on the set of nonextinction, are completely answered. Proofs are based on a reformulation of the model as a certain two-district migration model.