An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

TL;DR

This paper introduces a new N-barrier maximum principle for two-species reaction-diffusion systems, enabling analysis of traveling wave solutions and their existence in multi-species models.

Contribution

It develops a novel N-barrier maximum principle for Lotka-Volterra systems and applies it to establish conditions for traveling wave solutions in three-species systems.

Findings

Established a new maximum principle for two-species systems

Proved existence and nonexistence of traveling waves under certain conditions

Constructed explicit wave solutions using tanh functions

Abstract

By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain conditions, the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function provided that the parameters satisfy certain conditions.