A two-species competition model on Z^d

TL;DR

This paper analyzes a stochastic two-species competition model on the integer lattice Z^d, demonstrating conditions under which both species can coexist indefinitely based on geometric properties of the model.

Contribution

It introduces a new two-species competition model on Z^d and establishes conditions for long-term coexistence based on the shape of the limit set.

Findings

Positive probability of coexistence under uniform curvature assumption

Coexistence depends on initial finite populations of each species

Model extends understanding of spatial competition dynamics

Abstract

We consider a two-type stochastic competition model on the integer lattice Z^d. The model describes the space evolution of two ``species'' competing for territory along their boundaries. Each site of the space may contain only one representative (also referred to as a particle) of either type. The spread mechanism for both species is the same: each particle produces offspring independently of other particles and can place them only at the neighboring sites that are either unoccupied, or occupied by particles of the opposite type. In the second case, the old particle is killed by the newborn. The rate of birth for each particle is equal to the number of neighboring sites available for expansion. The main problem we address concerns the possibility of the long-term coexistence of the two species. We have shown that if we start the process with finitely many representatives of each type, then, under the assumption that the limit set in the corresponding first passage percolation model is uniformly curved, there is positive probability of coexistence.