Dynamical analysis of a competition‐diffusion‐advection system with Dirichlet boundary conditions

Abstract

In this paper, we consider the dynamical behaviors of a Lotka–Volterra competition‐diffusion‐advection system with Dirichlet boundary conditions in a one‐dimensional homogeneous environment, where the two competing species have the same growth and advection rates but different random dispersal rates. Firstly, We determine the existence and uniqueness of positive steady states for the single species model by investigating the properties of the corresponding principal eigenvalues. Secondly, some conditions are provided to assert the globally asymptotical stability of semi‐steady states for the two species competitive model. The results suggest that: (a) the species will prevail whenever it exists if the diffusion rate of the other species is small or large enough; (b) the species with much faster diffusion rate will win in the long run if the diffusion rates of the two species fall into some particular ranges. Finally, we show that if the appropriate diffusion rates are selected, then the two species will coexist. Our results indicate that both “competition exclusion principle” and “co‐existence” may happen depending on the different dispersal strategies.