Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type

In this communication, we prove some 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 obtain 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, 32, 30], in which we have to overcome the new difficulties comparing to [3, 32, 30] since principal eigenvalue of a 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.

Automated summary

In this paper, we prove several limits of the principal eigenvalue for a nonlocal operator of Neumann type, which are significant in understanding the dynamics of biological populations. By classifying the dispersal rate and dispersal range based on the Ecological Stable Strategy of persistence, we obtain a complete understanding of the limits of the principal eigenvalue. This paper also solves some open problems and achieves the maximum principle for this type of operator.