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

TL;DR

This paper introduces a generalized principal eigenvalue concept for time-periodic cooperative nonlocal dispersal systems, overcoming the challenge of non-existence of traditional eigenvalues due to non-compactness, and shows its significance in system analysis.

Contribution

It constructs a generalized principal eigenvalue for nonlocal dispersal systems, extending spectral theory to systems lacking traditional eigenvalues.

Findings

Established the existence of a generalized principal eigenvalue.

Proved the generalized eigenvalue's role analogous to the classical principal eigenvalue.

Provided applications demonstrating the usefulness of the generalized eigenvalue.

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.