Coexistence of Species in a Competition Model on Random Geometric Graphs

TL;DR

This paper studies how two competing species can coexist on random geometric graphs by combining growth and competition models, proving positive probability of coexistence using advanced probabilistic techniques.

Contribution

It introduces a novel analysis of species coexistence on RGGs by integrating Richardson's growth model with voter dynamics and applying probabilistic deviation results.

Findings

Coexistence occurs with positive probability in the model.

The analysis uses moderate deviations in first-passage percolation.

Results focus on specific spatial regions.

Abstract

This paper investigates the coexistence of two competing species on random geometric graphs (RGGs) in continuous time. The species grow by occupying vacant sites according to Richardson's model, while simultaneously competing for occupied sites under the dynamics of the voter model. Coexistence is defined as the event in which both species occupy at least one site simultaneously at any given time. We prove that coexistence occurs with strictly positive annealed probability by applying results from moderate deviations in first-passage percolation and random walk theory, with a focus on specific regions of the space.