Global Dynamics of an Ecosystem in Open Advective Environments

Abstract

This paper deals with a kind of reaction–diffusion–advection model which depicts a predator–prey ecosystem in rivers or streams. We obtain a complete classification on the dynamical behavior of the system in the parameter space of the predator’s mortality rate [Formula: see text] and the prey’s intrinsic growth rate [Formula: see text]. More precisely, both species fail to survive when [Formula: see text] is small. With the increase of [Formula: see text], there exist two critical values [Formula: see text] and [Formula: see text] such that both species can coexist in the long run when [Formula: see text] and [Formula: see text], otherwise the prey survives alone. Finally, with the aid of numerical simulations, we investigate the effects of diffusion and advection on the global dynamics of the system by computing these two critical values. Numerical simulations illustrate that the diffusion of both predators and prey would benefit the invasion of the predators, and the advection would be unfavorable to the survival of the predators. Moreover, numerical simulations also suggest that the unique positive steady state is globally asymptotically stable among all non-negative and nontrivial initial data.