Generalized principal eigenvalue of time-periodic cooperative nonlocal dispersal systems and applications

It is well known that, in the study of the dynamical properties of nonlinear reaction-diffusion systems, the sign of the principal eigenvalue of the linearized system plays an important role. However, for the nonlocal dispersal systems, due to the lack of compactness, the essential spectrum appear, and the principal eigenvalue may not exist. In this paper, by constructing monotonic upper and lower control systems, we obtain the generalized principal eigenvalue of the cooperative irreducible system and demonstrate that this generalized principal eigenvalue plays the same role as the usual principal eigenvalue.