Selection Mechanisms, Stationary Distributions, and Reversibility in Multiallelic Moran Models

The Moran process with selection and recurrent mutation is a classical model in population genetics, yet how the placement of selection within the update rule shapes the stationary distribution has received little attention.

We study a finite, well-mixed haploid population of constant size $n$ with $m$ labeled alleles, parent-independent mutation, and allele-specific fitnesses.

Within this common framework we compare three Moran update kernels that differ only in the stage at which selection acts: during reproduction, when the offspring copies one of two sampled parents (Scheme~I); through fitness-biased mate choice, followed by neutral copying (Scheme~II); and at death, so that fitter individuals are less likely to be replaced (Scheme~III).

Although all three favor fitter alleles, they define different Markov chains.

For two alleles, each scheme reduces to a birth-death chain and admits an exact stationary law, but the three laws differ.

For $m\ge 3$, the placement of selection becomes decisive: Schemes~I and~II are generally nonreversible when fitnesses are unequal, so no detailed-balance product form exists, whereas Scheme~III remains reversible for every $m$ and has a closed stationary distribution -- a Dirichlet-multinomial core modified by an explicit fitness factor.

We further show that all three mechanisms can act simultaneously in the two-allele case without losing exact solvability, and we derive weak-selection expansions that make explicit how small fitness differences tilt the neutral beta-binomial and Dirichlet-multinomial benchmarks.

Together, these results clarify when neutral stationary structure survives the introduction of selection and when multiallelic Moran dynamics become genuinely nonreversible