The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application

TL;DR

This paper analyzes the principal eigenvalue of an age-structured operator with diffusion and advection, exploring its asymptotic behavior and applying findings to ecological models to understand species distribution.

Contribution

It provides a novel qualitative analysis of the eigenvalue problem with nonlocal terms, focusing on asymptotic behaviors related to advection and diffusion.

Findings

Asymptotic behavior of eigenvalues for large advection

Impact of diffusion rates on spatial distribution

Construction of super- and sub-solutions for nonlocal problems

Abstract

In this paper, we investigate an eigenvalue problem associated with an age-structured operator incorporating random diffusion and advection. Our primary focus is on examining the asymptotic behaviors of the principal eigenvalue with respect to large advection and small or large diffusion rates. We subsequently apply these results to a nonlinear age-structured model, providing a better understanding of how diffusion and advection influence the spatial distribution of species. Among other ingredients, our approach involves constructing various types of super- and sub-solutions to tackle the novel challenges posed by the nonlocal terms in the problems under consideration.