  In this paper, we consider a specialist predator-prey patchy model over the
closed stream network. We study the dynamics and the asymptotic profiles of
positive steady states according to the mortality rate of the specialist
predators, advection and diffusion rates. We verify that the specialist
predators can successfully invade as long as the mortality rate is sufficiently
small. On the other hand, the impacts of diffusion and advection on the
asymptotic profiles of positive steady states and on the concentration of the
species are given.
