The asymmetric multitype contact process

TL;DR

This paper analyzes an asymmetric multitype contact process on a lattice, showing that the dominant type persists and spreads linearly over time, and characterizes all stationary distributions.

Contribution

It provides a rigorous proof that the stronger type survives and dominates, and fully characterizes the stationary distributions of the process.

Findings

Type 1 has a positive probability of never going extinct.

Type 1 takes over a linearly growing ball if initially present.

The process converges to a convex combination of stationary distributions.

Abstract

In the multitype contact process, vertices of a graph can be empty or occupied by a type 1 or a type 2 individual; an individual of type $i$ dies with rate 1 and sends a descendant to a neighboring empty site with rate $\lambda_i$. We study this process on $\Z^d$ with $\lambda_1 > \lambda_2$ and $\lambda_1$ larger than the critical value of the (one-type) contact process. We prove that, if there is at least one type 1 individual in the initial configuration, then type 1 has a positive probability of never going extinct. Conditionally on this event, type 1 takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove that the process started from any initial configuration converges to a convex combination of distributions in this set.