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

TL;DR

This paper analyzes a critical competition Lotka-Volterra reaction-diffusion system, revealing the non-existence of certain traveling waves and describing the long-term behavior and profile of solutions, including a new bump phenomenon.

Contribution

It provides a detailed analysis of the critical competition case, including phase plane analysis and large-time behavior, with new insights into solution profiles and exclusion dynamics.

Findings

No ultimately monotone traveling waves exist.

Faster species excludes slower species with known spreading speed.

Identification of a new bump phenomenon in solution profiles.

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 "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}}.