  We consider the reaction-diffusion competition system in the so-called {\it
critical competition case}. The associated ODE system then admits infinitely
many equilibria, which makes the analysis intricate. We first prove the
non-existence of {\it ultimately monotone} traveling waves by applying the
phase plane analysis. Next, we study the large time behavior of the solution of
the Cauchy problem with a compactly supported initial datum. We not only reveal
that the "faster" species excludes the "slower" one (with a known {\it
spreading speed}), but also provide a sharp description of the profile of the
solution, thus shedding light on a new {\it{bump phenomenon}}.
