Competition in periodic media: II -- Segregative limit of pulsating fronts and "Unity is not Strength"-type result

TL;DR

This paper studies the limiting behavior of pulsating fronts in periodic media for two competing species as competition intensifies, revealing segregation phenomena, characterizing front solutions, and analyzing their speeds.

Contribution

It extends previous work by analyzing the segregative limits and speed signs of pulsating fronts in space-periodic media under strong competition.

Findings

Existence of segregated stationary equilibria in null speed cases.

Full characterization and convergence of pulsating fronts in non-null speed cases.

Explicit conditions determining which species invades based on motility and competitiveness.

Abstract

This paper is concerned with the limit, as the interspecific competition rate goes to infinity, of pulsating front solutions in space-periodic media for a bistable two-species competition--diffusion Lotka--Volterra system. We distinguish two important cases: null asymptotic speed and non-null as-ymptotic speed. In the former case, we show the existence of a segregated stationary equilibrium. In the latter case, we are able to uniquely characterize the segregated pulsating front, and thus full convergence is proved. The segregated pulsating front solves an interesting free boundary problem. We also investigate the sign of the speed as a function of the parameters of the competitive system. We are able to determine it in full generality, with explicit conditions depending on the various parameters of the problem. In particular, if one species is sufficiently more motile or competitive than the other, then it is the invader. This is an extension of our previous work in space-homogeneous media.