We investigate the asymptotic behavior of the principal eigenvalue of an elliptic operator as the coefficient of the advection term approaches infinity. As a biological application, a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is studied. The two species are assumed to be identical except their dispersal strategies: both species disperse by random movement and advection along environmental gradients, but one species has stronger biased movement than the other one. It is shown that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species will coexist; if both species have such strong biased movements, the species with the stronger biased movement will go to extinct. These results provide a new mechanism for the coexistence of competing species, and they also suggest that an intermediate biased movement rate may be evolutionary stable.