Summary: This paper concerns the global existence and asymptotic behavior of solutions to some reaction-diffusion-advection models for two competing species, where the species have the same population dynamics but different dispersal strategies. When one species possesses a combination of random dispersal and directed movement upward along its fitness gradient whereas the other species adopts random dispersal, the global existence of smooth solutions to the quasilinear parabolic system is established. When one species adopts the fitness-dependent dispersal but the other species does not disperse at all, we show the global existence of weak solutions to the degenerate parabolic-ODE system and further describe the asymptotic behavior of these weak solutions. In particular, we show that in the latter case the total population density approaches the so-called ideal free distribution in an appropriate sense.