Traveling Wave Solutions for Lotka-Volterra System Re-Visited

TL;DR

This paper introduces a novel monotone iteration method to establish the existence, uniqueness, and stability of traveling wave solutions for the classical Lotka-Volterra system across various wave speeds.

Contribution

It develops a new approach using smooth lower- and upper-solutions to explicitly determine minimal wave speed and asymptotic behavior, extending understanding of wave solutions in Lotka-Volterra models.

Findings

Existence of traveling wave solutions for a range of wave speeds.

Explicit calculation of the minimal wave speed.

Stability analysis of non-critical wave solutions.

Abstract

Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique modulo a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of the linearized operator in exponentially weighted Banach spaces.