Age-structured Models with Nonlocal Diffusion of Dirichlet Type, I: Principal Spectral Theory and Limiting Properties

TL;DR

This paper studies the spectral properties of age-structured models with nonlocal diffusion, establishing criteria for principal eigenvalues, defining a generalized eigenvalue, and analyzing how diffusion influences population dynamics.

Contribution

It introduces new criteria for the existence of principal eigenvalues, defines a generalized eigenvalue, and proves a maximum principle for nonlocal age-structured diffusion operators.

Findings

Criteria for principal eigenvalue existence

Definition of generalized principal eigenvalue

Strong maximum principle established

Abstract

Age-structured models with nonlocal diffusion arise naturally in describing the population dynamics of biological species and the transmission dynamics of infectious diseases in which individuals disperse nonlocally and interact each other and the age structure of individuals matters. In the first part of our series papers, we study the principal spectral theory of age-structured models with nonlocal diffusion of Dirichlet type. First, we provide two criteria on the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we define the generalized principal eigenvalue and use it to investigate the influence of diffusion rate on the principal eigenvalue. In addition, we establish the strong maximum principle for age-structured nonlocal diffusion operators. In the second part \cite{Ducrot2022Age-structuredII} we will investigate the effects of principal eigenvalues on the global dynamics of the model with monotone nonlinearity in the birth rate and show that the principal eigenvalue being zero is critical.