Summary: We study a Lotka-Volterra type reaction-diffusion-advection system, which describes the competition for the same resources between two aquatic species undergoing different dispersal strategies, as reflected by their diffusion and/or advection rates. For the non-advective case, this problem was solved by J. Dockery et al. [J. Math. Biol. 37, No. 1, 61–83 (1998; Zbl 0921.92021)], and recently X. He and W.-M. Ni [Commun. Pure Appl. Math. 69, No. 5, 981–1014 (2016; Zbl 1338.92105)] provided a further classification on the global dynamics for a more general model. However, the key ideas developed in do not appear to work when advection terms are involved. By assuming the resource function is decreasing in the spatial variable, we establish the non-existence of co-existence steady state and perform sufficient analysis on the local stability of two semi-trivial steady states, where new techniques were introduced to overcome the difficulty caused by the non-analyticity of stationary solutions as well as the diffusion-advection type operators. Combining these two aspects with the theory of monotone dynamical systems, we finally obtain the global dynamics, which suggests that the competitive exclusion principle holds in most situations.