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We perform an asymptotic analysis of models of population dynamics with afractional Laplacian and local or nonlocal reaction terms. The first part ofthe paper is devoted to the long time/long range rescaling of the fractionalFisher-KPP equation. This rescaling is based on the exponential speed ofpropagation of the population. In particular we show that the only role of thefractional Laplacian in determining this speed is at the initial layer where itdetermines the thickness of the tails of the solutions. Next, we show that suchrescaling is also possible for models with non-local reaction terms, asselection-mutation models. However, to obtain a more relevant qualitativebehavior for this second case, we introduce, in the second part of the paper, asecond rescaling where we assume that the diffusion steps are small. In thisway, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limitwhich describes the asymptotic dynamics of the solutions, similarly to the caseof selection-mutation models with a classical Laplace term or an integralkernel with thin tails. However, the rescaling introduced here is verydifferent from the latter cases. We extend these results to themultidimensional case.

Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity