The role of strong Allee effect and protection zone on a diffusive prey-predator model

The influence of strong Allee effect and protection zone on the dynamical behavior and patterns of the reaction-diffusive model is investigated. Firstly, the existence and stability of the positive constant steady-states are determined. Secondly, we analyze the existence and nonexistence of nonconstant positive steady-states by the implicit function theorem, energy method and Leray-Schauder degree theory. Following, the dynamical properties are also obtained and in particular it is shown that the overexploitation phenomenon may occur. Finally, we show that the overexploitation phenomenon can be avoided by setting up a suitable protection zone for the prey. Hence, the strong Allee effect and protection zone enrich the complexity of the system.