Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications

TL;DR

This paper develops spectral theory for nonlocal dispersal operators with time periodic indefinite weights, establishing conditions for principal spectrum points and eigenvalues, with applications to biological models.

Contribution

It introduces new criteria for the existence and bounds of principal spectrum points and eigenvalues in nonlocal dispersal operators with indefinite weights.

Findings

Necessary and sufficient conditions for positive principal spectrum points.

Upper bounds for principal spectrum points.

Applications to nonlinear biological models.

Abstract

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.