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Preprint 26-2006

Coexistence in locally regulated competing populations and survival of branching annihilating random walk

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Author(s) : Jochen Blath , Alison Etheridge , Mark Meredith

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 26-2006

MSC 2000

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes
60J85 Applications of branching processes
60J70 Applications of diffusion theory
92D25 Population dynamics

Abstract :
We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability. As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the r\^ole of space in the survival probabilities for the two populations.

Keywords : competing species, coexistence, branching annihilating random walk, interacting diffusions, regulated population, heteromyopia, stepping stone model, survival, Feller diffusion, Wright-Fisher diffusion

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