  We study the asymptotic behavior of solutions to a monostable
integro-differential Fisher-KPP equation , that is where the standard Laplacian
is replaced by a convolution term, when the dispersal kernel is fat-tailed. We
focus on two different regimes. Firstly, we study the long time/long range
scaling limit by introducing a relevant rescaling in space and time and prove a
sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense
by means of sub-and super-solutions. Secondly, we investigate a long time/small
mutation regime for which, after identifying a relevant rescaling for the size
of mutations, we derive a Hamilton-Jacobi limit.
