Abstract
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 \lq\lq faster'' species excludes the \lq\lq 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}}.