## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 1 (1993), 198-211.

### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 1 (1993), 198-211.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005515

**Digital Object Identifier**

doi:10.1214/aoap/1177005515

**Mathematical Reviews number (MathSciNet)**

MR1202523

**Zentralblatt MATH identifier**

0776.92014

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 60G42: Martingales with discrete parameter 60F99: None of the above, but in this section

**Keywords**

Galton-Watson predator-prey process extinction probability normalizing constants martingales two-district migration model

#### Citation

Alsmeyer, Gerold. On the Galton-Watson Predator-Prey Process. Ann. Appl. Probab. 3 (1993), no. 1, 198--211. doi:10.1214/aoap/1177005515. https://projecteuclid.org/euclid.aoap/1177005515