Approximation of the generalized principal eigenvalue of cooperative nonlocal dispersal systems and applications

TL;DR

This paper develops a method to approximate the generalized principal eigenvalue of cooperative nonlocal dispersal systems, which are crucial for understanding their dynamics, especially when traditional eigenvalues are difficult to define due to non-compactness.

Contribution

It introduces a novel approximation approach for the generalized principal eigenvalue of nonlocal dispersal systems using monotonic control systems, extending the eigenvalue's applicability.

Findings

The generalized principal eigenvalue can be approximated by monotonic control systems.

The approach confirms the eigenvalue's role similar to the classical principal eigenvalue.

Provides a characterization of the eigenvalue for nonlocal dispersal systems.

Abstract

It is well known that, in the study of the dynamical properties of nonlinear evolution system with nonlocal dispersals, the principal eigenvalue of linearized system play an important role. However, due to lack of compactness, in order to obtain the existence of principal eigenvalue, certain additional conditions must be attached to the coefficients. In this paper, we approximate the generalized principal eigenvalue of nonlocal dispersal cooperative and irreducible system, which admits the Collatz-Wielandt characterization, by constructing the monotonic upper and lower control systems with principal eigenvalues; and show that the generalized principal eigenvalue plays the same role as the usual principal eigenvalue.