Fisher waves: An individual-based stochastic model

The propagation of a beneficial mutation in a spatially extended population is usually studied using the phenomenological stochastic Fisher-Kolmogorov-Petrovsky-Piscounov (SFKPP) equation. We derive here an individual-based, stochastic model founded on the spatial Moran process where fluctuations are treated exactly. The mean-field approximation of this model leads to an equation that is different from the phenomenological FKPP equation. At small selection pressure, the front behavior can be mapped into a Brownian motion with drift, the properties of which can be derived from the microscopic parameters of the Moran model. Finally, we generalize the model to take into account dispersal kernels beyond migration to nearest neighbors. We show how the effective population size (which controls the noise amplitude) and the diffusion coefficient can both be computed from the dispersal kernel.