A Nonlocal Diffusion Single Population Model in Advective Environment

ABSTRACT 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 dynamics of the 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 key roles. We then demonstrate the existence, nonexistence, uniqueness, and stability of nontrivial stationary solutions by analyzing the principal eigenvalue of the integro‐differential operator, particularly focusing on its monotonicity with respect to the advection rate. This analysis helps us understand the long‐term dynamics of solutions and derive sharp criteria for persistence or extinction. Furthermore, we study the limiting behavior of solutions with respect to the advection rate and find that a sufficiently large advection rate leads to species extinction. Lastly, numerical simulations support our theoretical results.