The Interplay Between Dispersal and Allee Effects in a Two-Patch Discrete-Time Model

We consider a discrete-time two-patch model with dispersal and a nonlinear non-monotone growth rate suggesting that the population in a patch may undergo an Allee effect. We assume that newborns do not disperse while other individuals in the population do. We first analyze the model without dispersal and demonstrate that, under certain conditions, the model exhibits two positive equilibria with both the extinction and larger positive equilibrium being locally asymptotically stable. We also provide conditions where a unique positive equilibrium exists and is globally asymptotically stable. We then study the model with dispersal and show that, if at least one patch undergoes an Allee effect in the absence of dispersal, then introducing dispersal may produce multiple stable positive equilibria. As a result, when dispersal parameters are varied, a discontinuous switching phenomenon between two attractors may be introduced into a patch that otherwise has continuous behavior. While the existence and stability of these equilibria is established theoretically for small dispersal proportions, it is shown through numerical simulation that this phenomena may also occur for large dispersal proportions.