Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems

TL;DR

This paper explores the spectral properties and asymptotic behavior of nonlocal dispersal systems with partial degeneracy, providing conditions for principal eigenvalues and analyzing their limits as diffusion coefficients vary.

Contribution

It introduces new sufficient conditions for the existence of principal eigenvalues in partially degenerate systems and studies their asymptotic behavior under varying diffusion coefficients.

Findings

Established conditions for principal eigenvalue existence in degenerate systems

Identified threshold behavior of spectral bound as diffusion coefficients grow large

Applied results to viral diffusion models to understand reproductive ratios

Abstract

The purpose of this paper is to investigate the principal spectral theory and asymptotic behavior of the spectral bound for cooperative nonlocal dispersal systems, specifically focusing on the case where partial diffusion coefficients are zero, referred to as the partially degenerate case. We propose two sufficient conditions that ensure the existence of the principal eigenvalue in these partially degenerate systems. Additionally, we study the asymptotic behavior of the spectral bound for nonlocal dispersal operators with small and large diffusion coefficients, considering both non-degenerate and partially degenerate cases. Notably, we find a threshold-type result as the diffusion coefficients tend towards infinity in the partially degenerate case. Finally, we apply these findings to discuss the asymptotic behavior of the basic reproduction ratio in a viral diffusion model.