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

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.

Spatial Point Processes and Moment Dynamics in the Life Sciences: A Parsimonious Derivation and Some Extensions

“…For instance, Blath et al (2007) investigated a stochastic, lattice model of two competing species by means of stochastic differential equations, to see if coexistence could be achieved purely by spatial properties of competition, as had been previously been observed in a numerical study of a lattice-free moment model Murrell and Law (2003). Among other things, this work illustrates the importance of parsimony in constructing agent-based models.…”

“…Among other things, this work illustrates the importance of parsimony in constructing agent-based models. Blath et al (2007) made a conjecture on coexistence (Conjecture 2.5), but the full stochastic competition model of Murrell and Law (2003) was too complicated to admit a rigorous proof.…”

“…There are other mathematical paths that do not rely on a closure assumption, for instance working directly with the stochastic process, using a perturbation approximation, or working with the full hierarchy of moments, as discussed in Sect. 7 (Blath et al 2007;Bruna and Chapman 2012b;Ovaskainen et al 2014).…”

Coexistence and Duality in Competing Species Models

“…This article is an introduction to stochastic spatial generalizations of the Lotka-Volterra model. For definiteness, our discussions are centered around two well-known generalizations, one due to Neuhauser and Pacala [22] and the other due to Blath, Etheridge and Meredith [1]. We explain in Section 2 how these models are natural extensions of the Lotka-Volterra model.…”

“…The models in [22,1] can be briefly described as follows in terms of the spatial structures and the dynamics. First, the model by Neuhauser and Pacala is a dynamical model (started at time 0) for individuals in two populations of different types.…”