Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system

In this paper, we study a classical two species Lotka-Volterra competition-diffusion-advection system, where the diffusion and advection rates of two competitors are supposed to be proportional. By employing the principal spectral theory, we first establish a key a priori estimate on the co-existence (positive) steady state, which is a powerful tool to link the local and global dynamics. We then further present a complete classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems. Lastly, we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics. This work gives a positive answer to the conjecture proposed by Lou et al. in [34] by considering a more general model under certain conditions, and also, can be seen as a further development of He and Ni [19] for competition-diffusion system, where we bring new ingredients in the arguments to overcome the difficulty caused by the involvement of advection.