The Galton-Watson predator-prey process

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive pro- bability of survival of both populations is given. In this paper, we introduce a new probabilistic model for the relationship between the sizes of a population of predators and a population of prey. This process is formulated in terms of a modified two-type discrete-time branching process. The population of predators is described by an ordinary Galton-Watson process. In the population of prey, each surviving particle of the nth generation produces offspring independently of all other particles. However, not all of these prey offspring are allowed to reproduce. Instead, each member of the (n + 1)th generation of predators consumes a random number of these offspring prey particles. (If there are not enough prey offspring to go around, then they are all eaten and the prey population is extinct.) The main result of this paper is a necessary and sufficient condition for there to be positive probability of indefinitely long survival for both populations.