Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices

Let $P$ denote normalized nearest-neighbor averaging on $\mathbb{Z}^d$ and set $A=I-P$. We consider $\partial_tu+A^su=f(u),t>0, x\in\mathbb{Z}^d, 00$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixe