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We propose and study a two patch Rosenzweig-MacArthur prey-predator model with immobile prey and predator using two dispersal strategies. The first dispersal strategy is driven by the prey-predator interaction strength, and the second dispersal is prompted by the local population density of predators which is referred as the passive dispersal. The dispersal strategies using by predator are measured by the proportion of the predator population using the passive dispersal strategy which is a parameter ranging from 0 to 1. We focus on how the dispersal strategies and the related dispersal strengths affect population dynamics of prey and predator, hence generate different spatial dynamical patterns in heterogeneous environment. We provide local and global dynamics of the proposed model. Based on our analytical and numerical analysis, interesting findings could be summarized as follow: (1) If there is no prey in one patch, then the large value of dispersal strength and the large predator population using the passive dispersal in the other patch could drive predator extinct at least locally. However, the intermediate predator population using the passive dispersal could lead to multiple interior equilibria and potentially stabilize the dynamics; (2) For symmetric patches (i.e., all the life history parameters are the same except the dispersal strengths), the large predator population using the passive dispersal can generate multiple interior attractors; (3) The dispersal strategies can stabilize the system, or destabilize the system through generating multiple interior equilibria that lead to multiple attractors; and (4) The large predator population using the passive dispersal could lead to no interior equilibrium but both prey and predator can coexist through fluctuating dynamics for almost all initial conditions.

A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects