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

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