# Fisher Waves: an individual based stochastic model

## TL;DR

This paper introduces an individual-based stochastic model for the spread of beneficial mutations in spatial populations, revealing differences from classical models at high selection and linking parameters to dispersal behavior.

## Contribution

It develops a stochastic model based on the spatial Moran process that explicitly treats fluctuations and connects model parameters to dispersal kernels.

## Key findings

- At high selection, the model differs from classical FKPP predictions.
- At low selection, front behavior resembles Brownian motion with drift.
- Diffusion and noise are determined by dispersal kernel properties.

## Abstract

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.