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We consider the reaction-diffusion competition system in the so-called  critical competition case . The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the nonexistence of  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  spreading speed ), but also provide a sharp description of the profile of the solution, thus shedding light on a new  bump phenomenon .

Lotka–Volterra competition-diffusion system: the critical competition case