Coexistence of Species in a Competition Model on Random Geometric Graphs

TL;DR

This work analyzes the coexistence of two competing species on the infinite component $\mathcal{H}$ of a random geometric graph in $\mathbb{R}^d$, where growth follows Richardson's first-passage percolation and competition follows the voter dynamics. By leveraging the Euclidean-ball shape result and moderate deviations for FPP on RGGs, together with a novel intermediate-condition framework and invasion-time controls via random-walk duality, the authors prove that coexistence occurs with strictly positive annealed probability for any viable finite initial configuration in the supercritical regime $r>r_c(\lambda)$. The approach combines geometric, probabilistic, and percolation techniques to connect growth interfaces with invasion dynamics in a random spatial environment, yielding explicit probabilistic bounds and a robust coexistence mechanism. These results advance understanding of multispecies persistence in spatially random networks and have potential implications for ecological and epidemiological models on random media.

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.

Figures (5)

• Figure 1: Two simulations of the competition model on a RGG.
• Figure 2: The regions $\upchi_{w,n}$, $\upchi_{w',n}$, $\Upphi_n(w)$, $\Upphi_n(w')$ on the event $\Gamma_n$.
• Figure 3: The spanning tree $\mathbb{T}$ of $\mathbb{G}$ and its subgraphs.
• Figure 4: Schematics for worst case of the growth-invasion dynamics in $\mathcal{W}$.
• Figure 5: The regions $\mathcal{R}_n$, $\mathcal{R}_{n+1}$, $\mathcal{B}_n$, and $\mathcal{B}_n'$ before intersecting $\mathop{\mathrm{\mathcal{H}}}\nolimits$.

Theorems & Definitions (18)

• Remark 1.1
• Remark 1.2
• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3: Moderate deviations of first-passage times
• Lemma 3.1
• proof
• Lemma 3.2
• proof