Influence of protection zone on a diffusive host-generalist parasitoid model

In this paper, we consider a host-generalist parasitoid diffusion model with mutual interference among the generalist parasitoids and non-flux boundary conditions, where the generalist parasitoids are introduced to control the invasion of the hosts. We first prove the uniform persistence, and then the existence and global asymptotic stability of positive constant equilibrium solution. We find that the model exhibits bistable structure when parameter [Formula: see text], which shows that our model has no spatiotemporal patterns. When a protection zone [Formula: see text] is introduced in the model, we demonstrate that the stationary patterns emerge under some conditions. More precisely, when [Formula: see text], there exists a threshold value [Formula: see text], such that the positive non-constant stationary solutions of our model do exist for [Formula: see text]. When [Formula: see text] and [Formula: see text], the model admits positive non-constant stationary solutions. In addition, letting [Formula: see text], we obtain another threshold value [Formula: see text], such that the model admits a unique asymptotical stable positive non-constant stationary solutions for [Formula: see text]. Our results show that protection zone plays a crucial role in the formation of stationary patterns for our model, which is a strong contrast to the case without protection zone.