  The current paper {establishes} criteria for the existence of principal
eigenvalues of time periodic cooperative linear nonlocal dispersal systems with
Direchlet type, Neumann type or periodic type boundary conditions. It is shown
that such a nonlocal dispersal system has a principal eigenvalue in the
following cases: the nonlocal dispersal distance is sufficiently small; the
spatial inhomogeneity satisfies a so called vanishing condition; or the spatial
inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the
principal eigenvalue of a time periodic cooperative linear nonlocal dispersal
system (if it exists) is algebraically simple. A linear nonlocal dispersal
system may not have a principal eigenvalue. The results established in the
current paper extend those in literature for time independent or periodic
nonlocal dispersal equations to time periodic cooperative nonlocal dispersal
systems and {will} serve as {a} basic tool for the study of cooperative
nonlinear systems with nonlocal dispersal.

Analysis of PDEs