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

TL;DR

This paper introduces two lattice-based models of competing populations, demonstrating conditions for their long-term coexistence and linking one model to branching annihilating random walk, with implications for survival probabilities.

Contribution

It presents novel lattice models of competition that allow for analytical results and establishes a connection between population coexistence and branching annihilating random walk.

Findings

Both populations coexist with positive probability under certain parameters.

Branching annihilating random walk survives indefinitely at high branching rates.

Comparison with oriented percolation underpins the analytical results.

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