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We study the Fisher-KPP equation with a fractional laplacian of order{\alpha} \in (0, 1). We know that the stable state invades the unstable one atconstant speed for {\alpha} = 1, and at an exponential in time velocity for{\alpha} \in (0, 1). The transition between these two different speeds isexamined in this paper. We prove that during a time of the order - ln(1 -{\alpha}), the propagation is linear and then it is exponential.

Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations