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On the Galton-Watson predator-prey process Gerold Alsmeyer

Summary: 1
On the Galton-Watson predator-prey process
Gerold Alsmeyer
Mathematisches Seminar
Universit¨at Kiel
Ludewig-Meyn-Straße 4
D-24098 Kiel
We consider a probabilistic, discrete-time predator-prey model of the fol-
lowing 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 dis-
tribution with mean > 1. The resulting process (Xn, Yn)n0, where
Xn and Yn, resp., 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 (Yn)n0 given
Xn , and of the right normalizing constants dn, n 1 such that
Yn/dn has a positive limit on the set of non-extinction are completely
answered. Proofs are based on an reformulation of the model as a certain


Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster


Collections: Mathematics