We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.