  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.
