  We perform an asymptotic analysis of models of population dynamics with a
fractional Laplacian and local or nonlocal reaction terms. The first part of
the paper is devoted to the long time/long range rescaling of the fractional
Fisher-KPP equation. This rescaling is based on the exponential speed of
propagation of the population. In particular we show that the only role of the
fractional Laplacian in determining this speed is at the initial layer where it
determines the thickness of the tails of the solutions. Next, we show that such
rescaling is also possible for models with non-local reaction terms, as
selection-mutation models. However, to obtain a more relevant qualitative
behavior for this second case, we introduce, in the second part of the paper, a
second rescaling where we assume that the diffusion steps are small. In this
way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit
which describes the asymptotic dynamics of the solutions, similarly to the case
of selection-mutation models with a classical Laplace term or an integral
kernel with thin tails. However, the rescaling introduced here is very
different from the latter cases. We extend these results to the
multidimensional case.
