Karlin-McGregor mutational occupancy problem revisited

TL;DR

This paper revisits the Karlin-McGregor mutational occupancy model, analyzing equilibrium species counts and their asymptotic behaviors based on population size, species number, and mutation probability.

Contribution

It provides a detailed analysis of the equilibrium species counts in the KMG model under various asymptotic regimes, extending previous work from 1967.

Findings

Derived formulas for equilibrium species counts

Characterized asymptotic regimes and behaviors

Enhanced understanding of mutation effects on species occupancy

Abstract

Some population is made of n individuals that can be of p possible species (or types). The update of the species abundance occupancies is from a Moran mutational model designed by Karlin and McGregor in 1967. We first study the equilibrium species counts as a function of n, p and the total mutation probability $\nu$ before considering various asymptotic regimes on n, p and $\nu$. Running title: KMG Model with Mutations.