Branching random walk with non-local competition

TL;DR

This paper analyzes a population model with particles that reproduce, move randomly, and compete over large areas, proving survival and spread properties even with non-local competition, which introduces complex dependencies.

Contribution

It provides a rigorous analysis of the BPDL model with non-local competition, establishing survival and shape theorems in a challenging non-monotone setting.

Findings

Proved global survival of the population.

Established a shape theorem for population spread.

Handled non-local competition with infinite range.

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.