Principal eigenvalue theory of nonlocal dispersal cooperative systems in unbounded domains and applications

This paper is concerned with the theory of principal eigenvalue of a class of nonlocal dispersal cooperative systems in unbounded domains. We begin by establishing a new Harnack inequality for positive solutions of the nonlocal dispersal system. Using this inequality, we derive a sufficient condition for the existence of principal eigenvalues in unbounded domains. Additionally, we construct a matrix-valued sequence that approximates the nonlocal dispersal system and meets the sufficient condition. We then investigate the relationships among various definitions of generalized principal eigenvalues in both bounded and unbounded domains. A detailed analysis is conducted on how the principal eigenvalue impacts the effectiveness of the maximum principle. Finally, we apply our principal eigenvalue theory to examine the nonlocal dispersal cooperative system in unbounded domains and obtain the global dynamics of two vector-host epidemic models.