# Fixation probabilities in evolutionary dynamics under weak selection

## TL;DR

This paper develops a mathematical framework to approximate the fixation probability of mutants under weak selection in diverse evolutionary models, simplifying complex calculations and providing new insights for dynamics on graphs.

## Contribution

It introduces a perturbation expansion method for fixation probabilities applicable to arbitrary spatial structures and initial conditions under weak selection.

## Key findings

- Fixation probability can be approximated with polynomial complexity under weak selection.
- The method applies broadly to stochastic models with fixed population size and structure.
- New results are obtained for evolutionary dynamics on graphs.

## Abstract

In evolutionary dynamics, a key measure of a mutant trait's success is the probability that it takes over the population given some initial mutant-appearance distribution. This "fixation probability" is difficult to compute in general, as it depends on the mutation's effect on the organism as well as the population's spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation's effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant's fixation probability, from an arbitrary initial configuration of mutant and resident types. Our results apply to a broad class of stochastic evolutionary models, in which the size and spatial structure are arbitrary (but fixed). The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. We conclude by applying these methods to obtain new results for evolutionary dynamics on graphs.