On a Lotka-Volterra competition system: diffusion vs advection

Abstract

We study a two-species Lotka-Volterra competition system in one-dimensional advective environments, where two species are supposed to be identical except their diffusion and advection rates. We obtain the following useful observations: (1) if two species drift along the same direction and one competitor takes both smaller diffusion and advection rates, then it will win the competition provided its ratio of advection and diffusion rate is still smaller; while this ratio relation is reversed, either one can become the winner; (2) if two species drift along the same direction and one competitor takes smaller advection rate but larger diffusion rate, then it will always win; (3) if two species drift along opposite directions, then they will coexist finally, regardless of the size of diffusion and advection rates. Our main approach is to establish several non-existence results of co-existence steady state, where we employed Cauchy-Kowalevski theory and developed completely new techniques.