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

Blath, Jochen; Etheridge, Alison; Meredith, Mark

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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.