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We investigate the global dynamics of a Lotka-Volterra competition-diffusion-advection system for small diffusion rates in heterogenous environment. Our result suggests that the sign of $\int_{0}^{L}(m_{1}-m_{2})e^{kx}dx$ plays a significant role in understanding the global dynamics. In addition, the limiting behavior of coexistence steady state is obtained when diffusion rates of two species tend to zero meanwhile.