# 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?

## 1 Answer

I may suggest to give a look to game theory based model and their generalization from homogenous spatial diatribution to network like model. An easy review can be found in Martin A. Nowak's Evolutionary Dynamics (Harvard press 2006).

In particular you can see the classical Moran process as an evolutionary graph game going on a complete graph with identical weights. Changing the topology of the graph can give different results, though it is not necessary (and there're results showing in which case a graph behave just as a Moran process).

Graph related behaviour are, i.e., suppressing or amplifier of selection...