Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data

TL;DR

This paper investigates the spreading speeds of competing species in a Lotka-Volterra system with exponentially decaying initial data, revealing how initial decay influences the invasion dynamics and speeds.

Contribution

It introduces a Hamilton-Jacobi comparison principle to derive explicit spreading speed formulas for systems with exponential initial decay, extending previous models.

Findings

Spreading speeds depend nonlocally on initial data.

Explicit formulas for spreading speeds are derived.

Connections to traveling wave profiles and recent spreading results are discussed.

Abstract

This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed.