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

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

2007-10-29

arxiv.org/abs/0710.5380v1

Annals of Applied Probability 2007, Vol. 17, No. 5,6, 1474-1507

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/105051607000000267 the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051607000000267

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 r\^{o}le of space in the survival probabilities for the two populations.

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