Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments

Abstract

In this paper, we mainly study a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates. One interesting feature of the model is that the boundary condition at the downstream end represents a net loss of individuals, which is tuned by a parameter b to measure the magnitude of the loss. When the upstream end has the no-flux condition, Lou and Zhou (2015) [11] have confirmed that large diffusion rate is more favorable when 0≤b<1. Here we consider the case where the upstream end has the free-flow condition, which means that the upstream end is linked to a lake. We firstly investigate the corresponding single species model. Here we establish the existence and uniqueness of positive steady states. Then for the two-species model, we find that the parameter b can be regarded as a bifurcation parameter. Precisely, when 0≤b<1, large diffusion rate is more favorable while when b>1, small diffusion rate is selected (if exists). When b=1, the system is degenerate in the sense that there is a compact global attractor consisting of a continuum of steady states.