Branching random walk with non-local competition

Abstract

We study the Bolker-Pacala-Dieckmann-Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space, and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non-local competition. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Some ideas in the proof are inspired by works on the non-local Fisher-KPP equation, but the stochasticity of the model creates new difficulties.

major revision

References in corpus (7)

• A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
• Coexistence in locally regulated competing populations and survival of branching annihilating random walk
• Ecological equilibrium for restrained branching random walks
• Voter models with heterozygosity selection
• Systems of branching, annihilating, and coalescing particles
• Survival and complete convergence for a spatial branching system with local regulation