In this paper, we discuss a class of bistable reaction-diffusion systems used to model the competitive interaction of two species. The interactions are assumed to be of classic "Lotka-Volterra" type and we will consider a particular problem with relevance to applications in population dynamics: essentially, we study under what conditions the interplay of relative motility (diffusion) and competitive strength can cause waves of invasion to be halted and reversed. By establishing rigorous results concerning related degenerate and near-degenerate systems, we build a picture of the dependence of the wave speed on system parameters. Our results lead us to conjecture that this class of competition model has three "zones of response". In the central zone, varying the motility can slow, halt and reverse invasion. However, in the two outer zones, the direction of invasion is independent of the relative motility and is entirely determined by the relative competitive strengths. Furthermore, we conjecture that for a large class of competition models of the type studied here, the wave speed is an increasing function of the relative motility.