We study a class of bistable reaction-diffusion systems used to model two competing
species. Systems in this class possess two uniform stable steady states representing
semi-trivial solutions. Principally, we are interested in the case where the ratio of the
diffusion coefficients is small, i.e. in the
near-degenerate case. First, limiting arguments are presented to relate
solutions to such systems to those of the degenerate case where one species is assumed not
to diffuse. We then consider travelling wave solutions that connect the two stable
semi-trivial states of the non-degenerate system. Next, a general energy function for the
full system is introduced. Using this and the limiting arguments, we are able to determine
the wave direction for small diffusion coefficient ratios. The results obtained only
require knowledge of the system kinetics.