Collective behavior under catastrophes

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.