A spatial model for dormancy in random environment

TL;DR

This paper develops a spatial model for dormancy in populations within random environments, analyzing how dormancy influences growth and survival using an extended Parabolic Anderson model with two types of individuals.

Contribution

It introduces a novel two-type branching random walk model incorporating dormancy and random environments, extending the Parabolic Anderson model to analyze population dynamics.

Findings

Dormancy can significantly enhance population survival in hostile environments.

The model quantifies how environmental catalysts and traps affect growth rates.

Large-time asymptotics reveal the impact of dormancy on population size.

Abstract

In this paper, we introduce 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. We consider three specific choices for random environments composed of particles: (1) a Bernoulli field of immobile particles, (2) one moving particle, and (3) a Poisson field of moving particles. In each case, the particles of the random environment can either be interpreted as catalysts, accelerating the branching mechanism, or as traps, aiming to kill the individuals. The different 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. Especially, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk.