Dormancy in random environment: Symmetric exclusion

TL;DR

This paper models dormancy in a random environment using a two-type branching random walk influenced by a symmetric exclusion process, analyzing how dormancy affects population growth and survival.

Contribution

It introduces a novel two-type branching random walk model incorporating dormancy and a symmetric exclusion environment, extending the Parabolic Anderson model.

Findings

Quantifies the impact of dormancy on population growth.

Identifies large-time asymptotics of expected population size.

Extends the Parabolic Anderson model to include dormancy effects.

Abstract

In this paper, we study a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of the random environment. However, the branching mechanism is governed by a random environment which dictates the branching rates, namely the simple symmetric exclusion process. We will interpret the presence of the exclusion particles either as catalysts, accelerating the branching mechanism, or as traps, aiming to kill the individuals. The difference between active and dormant individuals is defined in such a way that dormant individuals are protected from being trapped, but do not participate in migration or branching. We quantify the influence of dormancy on the growth resp. survival of the population by identifying the large-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman-Kac formula. In particular, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk.