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Preprint 16-2008

Coexistence in locally regulated competing populations and survival of BARW: Full technical proofs and additional remarks

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

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 16-2008

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 :
Note: This paper is the full version of Blath, Etheridge & Meredith (2007). It has also successfully undergone the peer-reviewing process of Annals of Applied Probability, but proved too long to be published in its entirety. It contains full technical details and additional remarks. 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 role 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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