Criteria for the Existence of Principal Eigenvalues of Time Periodic Cooperative Linear Systems with Nonlocal Dispersal

TL;DR

This paper provides criteria for the existence of principal eigenvalues in time periodic cooperative linear nonlocal dispersal systems, extending previous results and offering foundational tools for nonlinear system analysis.

Contribution

It establishes new criteria for principal eigenvalues in time periodic nonlocal systems, including conditions on dispersal distance and spatial inhomogeneity.

Findings

Principal eigenvalues exist when dispersal distance is small.

Principal eigenvalues are algebraically simple if they exist.

Nonlocal dispersal systems may lack principal eigenvalues without specific conditions.

Abstract

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.