The notion of generalized principal eigenvalue was first introduced in the celebrated work of Berestycki-Nirenberg-Varadhan [5] and since then it has become an important basic of the theory of partial differential equations because of its usefulness in the study of the existence/nonexistence, uniqueness of positive solutions, maximum principle and long time dynamics of second order parabolic equations. Later, in the pioneering studies, Berestycki et al. [3, 4] defined and studied the qualitative properties for the generalized the principal eigenvalue for nonlocal operators and it has received a lot of attention of the community from theory to application. In that spirit, the current work, which is motivated from the study of mathematical modeling the dynamics of infectious diseases in [27, 46, 55, 56], is concerned with the investigation of the asymptotic behavior of the generalized the principal eigenvalue. First, we provide a sharp criterion, based on Lax-Milgram theorem and different from the approach of [3], for the existence of the principal eigenvalue for a nonlocal cooperative system with inhomogeneous coefficients and a counterexample for nonexistence of the principal eigenvalue. Second, we analyze the asymptotic properties of the generalized principal eigenvalue with respect to the dispersal rate and dispersal range. Our work gives a substantial contribution, besides [3, 7, 24, 48] on the existence, simplicity and asymptotic properties of the generalized principal eigenvalue for nonlocal cooperative systems with inhomogeneous coefficients and provides a fundamental step to tackle other problems in the studies of semiwave, spreading speed and age structure in nonlocal dispersal cooperative systems as investigated in [22, 23, 34, 44, 55, 58]
