Global Dynamics of a Lotka--Volterra Competition-Diffusion System in Advective Heterogeneous Environments
De Tang, Yuming Chen
Abstract
This work is a continuation and extension of a recent one by Tang and Chen [Global dynamics of a Lotka--Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 269 (2020), pp. 1465--1483]. In that work, we studied a Lotka--Volterra competition-diffusion model in a one-dimensional advective homogeneous environment, where the downstream end has a net loss of individuals measured by $b$. The global dynamics is completely characterized by $b$. In this paper, we consider the case where the environment is heterogeneous and $b\ge 1$. We first provide necessary and sufficient conditions on persistence of the corresponding single species model. Then for the two-species model, after analyzing the linear stability of the semitrivial steady states (if they exist) and excluding the existence of coexistence steady states (with highly nontrivial arguments), we apply the theory of monotone dynamical systems to find that the species with slower diffusion rate (if it persists) is always selected. This is different from the situation that $b=1$ is a bifurcation value when the environment is homogeneous. We leave the challenging case where $b<1$ for future work.