The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

TL;DR

This paper studies the properties of the generalized principal eigenvalue for time-periodic nonlocal dispersal operators, exploring its dependence on various parameters and applying findings to nonlinear KPP equations to analyze solution stability.

Contribution

It establishes equivalence between two characterizations of the eigenvalue and investigates how it depends on frequency, dispersal rate, and spread, with applications to nonlinear equations.

Findings

Eigenvalue characterization equivalence proved

Dependence of eigenvalue on dispersal parameters analyzed

Impacts on existence and stability of solutions demonstrated

Abstract

This paper is mainly concerned with the generalised principal eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised principal eigenvalue. We further investigate the dependence of the generalised principal eigenvalue on the frequency, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.