On the role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis

In this paper, we study the asymptotic behavior as $\varepsilon\to0^+$ of solutions $u_\varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ \frac{1}{\varepsilon^m}\int_{\mathbb{R}^N}J_\varepsilon(x-y)(u_\varepsilon(y)-u_\varepsilon(x))\mathrm{d}y+u_\varepsilon(x)(a(x)-u_\varepsilon(x))=0\text{ in }\mathbb{R}^N, $$where $\varepsilon>0$ and $0\leq m