  This paper is devoted to a nonlocal reaction-diffusion-advection model that
describes the spatial dynamics of freshwater organisms in a river with a
directional motion. Our goal is to investigate how the advection rate affects
the dynamic behaviors of species. We first establish the well-posedness of
global solutions, where the regularized problem containing a viscosity term and
the re-established maximum principle play an important role. And we then show
the existence/nonexistence, uniqueness, and stability of nontrivial stationary
solutions by analyzing the principal eigenvalue of integro-differential
operator (especially studying the monotonicity of the principal eigenvalue with
respect to the advection rate), which enables us to understand the longtime
behaviors of solutions and obtain the sharp criteria for persistence or
extinction. Furthermore, we study the limiting behaviors of solutions with
respect to the advection rate and find that the sufficiently large directional
motion will cause species extinction in all situations. Lastly, the numerical
simulations verify our theoretical proofs.
