A nonlocal diffusion single population model in advective environment

TL;DR

This paper investigates a nonlocal reaction-diffusion-advection model for freshwater organisms in rivers, analyzing how advection influences species persistence or extinction through mathematical and numerical methods.

Contribution

It establishes well-posedness, analyzes eigenvalues related to species survival, and reveals that high advection rates lead to extinction, providing sharp criteria and numerical validation.

Findings

Large advection causes species extinction.

Existence and stability of stationary solutions depend on eigenvalues.

Numerical simulations confirm theoretical results.

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 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.