  This paper studies the effects of the dispersal spread, which characterizes
the dispersal range, on nonlocal diffusion equations with the nonlocal
dispersal operator
$\frac{1}{\sigma^{m}}\int_{\Omega}J_{\sigma}(x-y)(u(y,t)-u(x,t))dy$ and Neumann
boundary condition in the spatial heterogeneity environment. More precisely, we
are mainly concerned with asymptotic behaviors of generalised principal
eigenvalue to the nonlocal dispersal operator, positive stationary solutions
and solutions to the nonlocal diffusion KPP equation in both large and small
dispersal spread. For large dispersal spread, we show that their asymptotic
behaviors are unitary with respect to the cost parameter $m\in[0,\infty)$.
However, small dispersal spread can lead to different asymptotic behaviors as
the cost parameter $m$ is in a different range. In particular, for the case
$m=0$, we should point out that asymptotic properties for the nonlocal
diffusion equation with Neumann boundary condition are different from those for
the nonlocal diffusion equation with Dirichlet boundary condition.
