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

Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread