Competition on $\mathbb{Z}^d$ driven by branching random walk

TL;DR

This paper studies a competitive coloring process on integer lattices driven by branching random walks, exploring conditions for infinite growth and coexistence of two competing species with probabilistic interactions.

Contribution

It introduces a new model of competing branching random walks on $ olinebreak bz^d$ and provides partial results on their long-term behavior and coexistence.

Findings

Partial answers on infinite site coloring by each species.

Conditions under which both species can coexist.

Formulation of many open problems in the model.

Abstract

A competition process on $\mathbb{Z}^d$ is considered, where two species compete to color the sites. The entities are driven by branching random walks. Specifically red (blue) particles reproduce in discrete time and place offspring according to a given reproduction law, which may be different for the two types. When a red (blue) particle is placed at a site that has not been occupied by any particle before, the site is colored red (blue) and keeps this color forever. The types interact in that, when a particle is placed at a site of opposite color, the particle adopts the color of the site with probability $p\in[0,1]$. Can a given type color infinitely many sites? Can both types color infinitely many sites simultaneously? Partial answers are given to these questions and many open problems are formulated.