Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays

TL;DR

This paper investigates traveling wave solutions in delayed reaction-diffusion systems, especially Lotka-Volterra models, establishing existence, nonexistence, and asymptotic behaviors using fixed point theorems and contraction techniques.

Contribution

It introduces a novel approach combining Schauder's fixed point theorem and contracting rectangles to analyze traveling waves in delayed systems, including nonmonotone solutions.

Findings

Existence of traveling wave solutions under certain conditions

Nonexistence results for specific parameter regimes

Large delays may not affect solutions if self-limitation is present

Abstract

This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka-Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.