Population Genetics: Classical Wright-Fisher Model and Its Extensions

Abstract

This paper presents a mathematical analysis of the Wright-Fisher model, a fundamental framework in population genetics that describes allele frequency dynamics in finite populations. We begin with the classical neutral model, deriving exact expressions for the mean and variance of allele counts over time and establishing conditions for fixation and loss under genetic drift alone. We then extend the analysis to incorporate bidirectional mutation, proving that mutation prevents absorption and instead produces a stationary beta distribution which parameters depend on population size and mutation rates. Through computational simulations, we validate theoretical predictions for fixation probabilities, absorption times, and equilibrium distributions.