  The propagation of a beneficial mutation in a spatially extended population
is usually studied using the phenomenological stochastic Fisher-Kolmogorov
(SFKPP) equation. We derive here an individual based, stochastic model founded
on the spatial Moran process where fluctuations are treated exactly. At high
selection pressure, the results of this model are different from the classical
FKPP. At small selection pressure, the front behavior can be mapped into a
Brownian motion with drift, the properties of which can be derived from
microscopic parameters of the Moran model. Finally, we show that the diffusion
coefficient and the noise amplitude of SFKPP are not independent parameters but
are both determined by the dispersal kernel of individuals.
