of (resp. , ). As in the random dispersal case, the spectral theory for the eigenvalue problems , , and will play an important role in the study of positive solutions of , , and ).

However, little is known about the eigenvalue problems of nonlocal dispersal operators with (time periodic) indefinite weight functions. The objective of this paper is to investigate the principal spectrum points /eigenvalues (if exists) of the nonlocal time-periodic weighted eigenvalue problems with Dirichlet type boundary condition , Neumann type boundary condition and spatial periodic type boundary condition , respectively. Note that, unlike random dispersal operators, a nonlocal dispersal operator may not have a principal eigenvalue (see [@Cov; @ShZh0] for examples). We hence first introduce the notion of the principal spectrum points of , and , which are the natural generalization of notion of the principal eigenvalue of , , and . Moreover, due to the lack of compactness of nonlocal operators, the Krein-Rutman’s theorem is not applicable, and there is in general no variational characterization for time-periodic nonlocal dispersal operators. We overcome the difficulties by developing various new techniques, and prove the necessary and sufficient conditions for the existence of positive principal spectrum points of 