# Alternatives to fittest-win and Moran processes as simple mathematical models of selection

When modeling selective sweeps as a micro-building block in models of macroevolution (not to be confused with misuses of this in creationist arguments), I use the fittest-win model of selection as a first approximation, or Moran process model when I want a more reasonable approximation.

In the fittest win model the probability for a mutant of fitness r to invade a host population of fitness 1 is 100% if r > 1 and 0 otherwise. In the Moran process model, the mutant of fitness r invades with probability $\frac{1 - r^{-1}}{1 - r^{-n}}$ for a finite population. Alternative in the limit as n goes to infinite, a mutant with r > 1 invades with probability $1 - \frac{1}{r}$ and 0 otherwise.

In general I am interested in simple models of selection of a single (or small concentration of) mutant invading an asexual host population of fitness 1 (with fitness constant and independent of frequency). Are there other common mathematical models for selection of this type?