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 non-existence 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.

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

“…In the critical competition case (i.e., a = b = 1,), Alfaro and Xiao [4] proved the non-existence of traveling waves with some monotonicity. Moreover, they studied the large time behavior of the solution of the Cauchy problem with compactly supported initial data.…”