Threshold dynamics of a Wolbachia-driven mosquito suppression model on two patches

The release of Wolbachia-infected mosquitoes is a promising and biologically safe measure for controlling wild mosquitoes. Numerous studies have been devoted to finding optimal control strategies using mathematical tools. However, the effects of dispersal of uninfected and infected mosquitoes remain poorly understood. To characterize the spatial discretization of release sites, we investigate a two-patch mosquito suppression model with time delay and impulsive release. Specifically, we assume that the waiting period between two consecutive releases is equal to the sexual lifespan of infected males. We confirm the well-posedness and monotonicity of the solution and explore the existence and stability of equilibria. By some technical skills, sufficient conditions for the bistable dynamics are provided. Then, the existence of the unstable separatrix is established by some sharp estimates when choosing constant functions as initial values. More interestingly, the monotonicity of this separatrix in the release number is proved, implying the existence of an optimal release strategy. We further find that uniform release on two patches is more effective than single-patch release. Additionally, the higher the cytoplasmic incompatibility intensity, the more likely wild mosquitoes are to be suppressed.