Optimality for a diffusive predator-prey system in a spatially heterogeneous environment incorporating a prey refuge

•A diffusive predator-prey system in a spatially heterogeneous environment incorporating a prey refuge is proposed.•The locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous Neumann boundary condition.•The existence and nonexistence of nonconstant steady states are obtained by Leray-Schauder degree theory.•Optimal harvesting strategy for maximum sustainable yield (MSY) and maximum economic yield (MEY) are obtained. In this paper, we consider the optimal harvesting controls for a diffusive predator-prey system, which is formulated by prey refuge and modified Leslie-Gower functional response. The locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous Neumann boundary condition. The nonexistence and existence of nonconstant steady states are obtained by Leray-Schauder degree theory. Also, the optimal harvesting strategies to get maximum sustainable yield (MSY) and maximum economic yield (MEY) are correspondingly given.