Principal Spectral Theory of time-periodic nonlocal dispersal operators of Neumann type

TL;DR

This paper analyzes the principal eigenvalue limits of nonlocal Neumann operators with time-periodic coefficients, providing insights into biological population dynamics and solving open problems related to dispersal strategies.

Contribution

It establishes the limits of the principal eigenvalue for nonlocal Neumann operators under various dispersal parameters, addressing open questions and overcoming challenges due to non-monotonicity.

Findings

Limits of principal eigenvalue with respect to dispersal rate and range

Complete characterization of eigenvalue behavior in ecological contexts

Maximum principle established for nonlocal Neumann operators

Abstract

In this communication, we prove some important limits of the principal eigenvalue for nonlocal operator of Neumann type with respect to the parameters, which are significant in the understanding of dynamics of biological populations. We obtained a complete picture about limits of the principal eigenvalue in term of the large and small dispersal rate and dispersal range classified by "Ecological Stable Strategy" of persistence. This solves some open problems remainning in the series of work [3, 28, 29], in which we have to overcome the new difficulties comparing to [3, 28, 29] since principal eigenvalue of nonlocal Neumann operator is not monotone with respect to the domain. The maximum principle for this type of operator is also achieved in this paper.