Generalized principal eigenvalue of time-periodic cooperative nonlocal dispersal systems and applications
Mingxin Wang, Lei Zhang
TL;DR
This work develops a generalized principal eigenvalue framework for time-periodic cooperative nonlocal dispersal systems, where the classical principal eigenvalue may fail to exist due to noncompactness. By constructing monotone upper and lower control systems, the authors define a generalized eigenvalue $\lambda({\mathscr L})$ as the common limit of principal eigenvalues of the control problems, show it equals the spectral bound $s({\mathscr L})$, and establish a Collatz–Wielandt-type characterization. They then apply this to threshold dynamics, proving dichotomous long-time behavior governed by the sign of $\lambda({\mathscr L})$, including the existence and stability of positive time-periodic solutions and extinction results. The framework is illustrated with a West Nile virus model featuring nonlocal dispersal and time-periodic coefficients, where persistence or elimination of disease hinges on the generalized eigenvalue. Overall, the generalized eigenvalue provides a robust tool for analyzing nonlocal, time-periodic systems in the absence of a classical principal eigenvalue, with implications for ecology and epidemiology.
Abstract
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.