The two-type continuum Richardson model: Non-dependence of the survival of both types on the initial configuration

TL;DR

This paper proves that in a two-type competing growth model in continuous space, the survival of both infection types does not depend on the initial configuration, extending previous results without restrictive assumptions.

Contribution

It provides a proof that the coexistence of both infection types is independent of initial conditions, removing previous limitations on radius distribution assumptions.

Findings

Survival of both types is independent of initial configuration.

The model's robustness extends to deterministic radius cases.

Generalization includes immune regions and delayed initial infections.

Abstract

We consider the model of Deijfen et al. for competing growth of two infection types in R^d, based on the Richardson model on Z^d. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen et al. conjectured that the initial configuration basically is irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which e.g. do not include the case of a deterministic radius. Here we give a proof that doesn't rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.