On the Galton-Watson Predator-Prey Process

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 (Xn,Yn)n.0, where Xn and Yn, respectively, denote the number of predators and prey of the nth generation, is called a Galton-Watson predator-prey process. The two questions of almost certain extinction of the prey process (Yn)n.0 given Xn.., and of the right normalizing constants dn,n.1 such that Yn/dn 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.