Collective Evolution Under Catastrophes

AbstractWe introduce the following discrete time model. Each site of N 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.MSC: 60 ACKNOWLEDGMENTThe author wishes to thank two anonymous referees for their careful reading and thoughtful suggestions.Additional informationNotes on contributorsRinaldo B. SchinaziRINALDO B. SCHINAZI received his Ph.D. in statistics at the University of São Paulo. He has been on the faculty at the University of Colorado at Colorado Springs since 1991. He has been teaching mathematics, writing books in mathematical analysis and probability, and doing research in probability.