Competitive Lotka-Volterra systems with random catastrophes: invasion fitnesses and reversal of the competitive hierarchy

Abstract

We study a two-type individual-based population with logistic competition subject to random catastrophes: at the epochs of a Poisson process, each individual survives independently with a random, type-dependent probability. In the large-population limit, the process converges to a competitive Lotka-Volterra system perturbed by multiplicative jumps. We show that the long-time behaviour of this system -exclusion of one type or the other, or coexistence -is governed by two averaged invasion fitnesses, obtained from the classical ones by replacing each growth rate a i with its average {ā}i = a i + rE[ln $Θ$ (i) ], where r is the catastrophe rate and $Θ$ (i) is the survival probability of an individual of type i when a catastrophe occurs. A consequence is that catastrophes can reverse the competitive hierarchy: a type doomed to exclusion in the deterministic system may persist or even exclude its competitor, when catastrophes are frequent enough, even if each catastrophe is individually more harmful to it. In the finite population setting, we also study the invasion of a mutant on the time scale ln K, where K is a scaling parameter, through a coupling with branching processes in the varying random environment generated by the resident population.