  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.
