Nonspreading solutions and patch formation in an integro-difference model with a strong Allee effect and overcompensation

Abstract

Previous work involving integro-difference equations of a single species in a homogenous environment has emphasized spreading behavior in unbounded habitats. We show that, under suitable conditions, a simple scalar integro-difference equation incorporating a strong Allee effect and overcompensation can produce solutions where the population persists in an essentially bounded domain without spread despite the homogeneity of the environment. These solutions are robust in that they occupy a region of full measure in parameter space. We develop orbit diagrams showing various patterns of nonspreading solutions from stable equilibria, period-two, to higher periodicity. We show that from a relatively uniform initial density with small stochastic perturbations, a population consisting of multiple isolated patches can emerge. In ecological terms, this work suggests a novel endogenous mechanism for the creation of patch boundaries.