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author:
- |
    Wenxian Shen\
    Department of Mathematics and Statistics\
    Auburn University\
    Auburn, AL 36849, U.S.A.\
    \
    Xiaoxia Xie\
    Department of Applied Mathematics\
    Illinois Institute of Technology\
    Chicago, IL 60616, U.S.A.
title: Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
---

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[**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.

[**Key words.**]{} Nonlocal dispersal operator, time periodic weight function, principal spectrum point, principal eigenvalue, KPP equations

[**Mathematics subject classification.**]{} 45C05, 45M05, 45M20, 47G10, 92D25.

Introduction
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Random dispersal operators, such as $u(t, \cdot)\mapsto \Delta u(t, \cdot)$ with proper boundary condition, are often adopted in modeling dissipative systems in applied sciences when the organisms 