Global stability in a multi-species periodic Leslie-Gower model

Abstract

We consider a population model consisting of d species interacting in a p-periodic environment and modelled by a d-dimensional system of Leslie-Gower-type difference equations (coupled Beverton-Holt equations). It is shown that if the interspecific competition (coupling) is sufficiently small and the inherent growth rate of each species is such that in the absence of competition each species will grow to its (positive) individual carrying capacity, then there is a positive asymptotically stable p-periodic state that globally attracts all positive initial states. Three examples are studied numerically in which the competition is large and the principle of competitive exclusion is observed. The rate of decay to extinction is observed to be sensitive to the inherent growth rate of the dying species. The individual carrying capacities are seen to play a determining role in the case of equal and large competition and equal inherent growth rates. © 2011 Taylor & Francis.