Nonlocal dispersal equations with almost periodic dependence. I. Principal spectral theory

TL;DR

This paper develops the principal spectral theory for nonlocal dispersal operators with almost periodic dependence, focusing on top Lyapunov exponents and eigenvalues, to understand their spectral properties and effects of variations.

Contribution

It introduces new characterizations and relations of spectral quantities for nonlocal dispersal operators with almost periodic dependence.

Findings

Characterization of top Lyapunov exponents

Relations between principal eigenvalues and Lyapunov exponents

Effects of time and space variations on spectral properties

Abstract

This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In this first part of the series, we investigate the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provide various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, establish the relations between them, and study the effect of time and space variations on them. In the second part of the series, we will study the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence applying the principal spectral theory to be developed in this part.