A stochastic model for competing growth on $\mathbb{R}^d$

TL;DR

This paper introduces a stochastic model for two competing infections spreading in continuous space, analyzing conditions for unbounded growth and extending existing shape theorems to this competitive setting.

Contribution

It presents a novel continuum model for competing infections, establishing conditions for mutual unbounded growth and extending shape theorems to this context.

Findings

Mutual unbounded growth probability is zero for almost all infection rate pairs.

The model generalizes previous shape theorems to competitive infection scenarios.

Identifies a countable set of infection rates where unbounded growth can occur.

Abstract

A stochastic model, describing the growth of two competing infections on $\mathbb{R}^d$, is introduced. The growth is driven by outbursts in the infected region, an outburst in the type 1 (2) infected region transmitting the type 1 (2) infection to the previously uninfected parts of a ball with stochastic radius around the outburst point. The main result is that with the growth rate for one of the infection types fixed, mutual unbounded growth has probability zero for all but at most countably many values of the other infection rate. This is a continuum analog of a result of H\"{a}ggstr\"{o}m and Pemantle. We also extend a shape theorem of Deijfen for the corresponding model with just one type of infection.