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arXiv · 0710.5380

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

Jochen Blath·Alison Etheridge·Mark Meredith
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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 do not, at present, incorporate all of 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 rôle of space in the survival probabilities for the two populations.

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Jochen Blath, Alison Etheridge, Mark Meredith. 2007-10-29. Coexistence in locally regulated competing populations and survival of branching annihilating random walk. https://doi.org/10.1214/105051607000000267

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