The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect
We propose a reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone. Ecological examples include marine protected areas where it is illegal to fish, or buffer zones left between the territories of rival predators. The prey are subject to a strong Allee effect, so excessive predation may lead to the extinction of both species. The exclusion zone mitigates this problem by providing the prey with a refuge in which to proliferate without predation. Thus, paradoxically, a smaller predator territory may be able to support a more substantial population than a larger one. Using a topological degree argument, we show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. We also show that as the predator domain becomes asymptotically small, the total predator population does not vanish, and in some cases may actually be maximized in this limit of shrinking predation area. Conversely, we show that as the predator domain becomes large, it may exhibit thresholding behavior, passing suddenly from a regime with coexistence solutions to one in which extinction becomes unavoidable, highlighting the need for careful analysis in the management of predator--prey systems.