Collective behavior under catastrophes

TL;DR

This paper models ecological niches with fitness levels, incorporating normal and catastrophic environmental events, and derives explicit formulas for the distribution and long-term behavior of site fitnesses.

Contribution

It introduces a novel discrete-time model capturing the effects of catastrophes on ecological fitness distributions and computes their explicit stationary distribution.

Findings

Explicit joint fitness distribution for finite sites.

Convergence to a unique stationary distribution.

Stationary distribution explicitly derived.

Abstract

We introduce the following discrete time model. Each natural number represents an ecological niche and is assigned a fitness in $(0,1)$. All the sites are updated simultaneously at every discrete time. At any given time the environment may be normal with probability $p$ or a catastrophe may occur with probability $1-p$. If the environment is normal the fitness of each site is replaced by the maximum of its current fitness and a random number. If there is a catastrophe the fitness of each site is replaced by a random number. We compute the joint fitness distribution of any finite number of sites at any fixed time. We also show convergence of this system to a stationary distribution. This too is computed explicitly.