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

Abstract 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.