LAUSR

We investigate a reaction-diffusion-advection system which characterizes the interactions between the predator and prey in advective environments, such as streams or rivers. In contrast with non-advective environments, the dynamics of this system is more complicated. It turns out that there exists a critical mortality rate of the predator and two critical advection rates, which classify the dynamic behavior of this system into two or three scenarios, that is, (i) both populations go extinct; (ii) the predator can not invade and the prey survives in the long run; (iii) the predator can invade successfully when rare and it will coexist permanently with the prey. Specially, the predator can invade successfully when rare if both the mortality rate of the predator and the advection rate are suitably small. Furthermore, by the global bifurcation theory and some auxiliary techniques, the existence and uniqueness of coexistence steady states of this system are established. Finally, by means of numerical simulations, the effects of diffusion on the dynamics of this system are investigated. The numerical results show that the random dispersals of both populations favor the invasion of the predator.