The authors study the large-time behavior of a two-species competition model in a spatially heterogeneous environment and investigate the influence of dispersal strategy on the competition. In particular, they allow one species to exhibit a random dispersal movement while the second species is constrained to a non-spatial movement dynamic. The authors show that there is a nonnegative threshold number (possibly equal to infinity and determined by both species' local intrinsic growth rates) such that the species adopting the random dispersal strategy persists uniformly in space if its diffusion rate is kept below this number while it goes extinct if its diffusion rate is greater than this number. Furthermore, in the critical case that the diffusing species' movement rate equals this number, exactly two scenarios are possible: the non-diffusing species wins the competition if it has a sink area; otherwise, both species coexist.