## Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 17, Number 5-6 (2007), 1474-1507.

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

Jochen Blath, Alison Etheridge, and Mark Meredith

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 17, Number 5-6 (2007), 1474-1507.

**Dates**

First available in Project Euclid: 3 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1191419173

**Digital Object Identifier**

doi:10.1214/105051607000000267

**Mathematical Reviews number (MathSciNet)**

MR2358631

**Zentralblatt MATH identifier**

1145.92032

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

**Keywords**

Competing species coexistence branching annihilating random walk interacting diffusions regulated population heteromyopia stepping stone model survival Feller diffusion Wright–Fisher diffusion

#### Citation

Blath, Jochen; Etheridge, Alison; Meredith, Mark. Coexistence in locally regulated competing populations and survival of branching annihilating random walk. Ann. Appl. Probab. 17 (2007), no. 5-6, 1474--1507. doi:10.1214/105051607000000267. https://projecteuclid.org/euclid.aoap/1191419173