On the stationary distribution of the block counting process for population models with mutation and selection

TL;DR

This paper analyzes the stationary distribution of the block counting process in population models with mutation and selection, characterizing measures leading to geometric distributions and solving related equations for specific models.

Contribution

It characterizes measures $ extLambda$ for geometric stationary distributions and solves the linear system for the Moran model and integro-differential equations for the $ extLambda$-Wright-Fisher model.

Findings

Identified measures $ extLambda$ with geometric stationary distributions.

Solved the linear system for the Moran model.

Derived and solved integro-differential equations for specific $ extLambda$-Wright-Fisher models.

Abstract

We consider two population models subject to the evolutionary forces of selection and mutation, the Moran model and the $\Lambda$-Wright-Fisher model. In such models the block counting process traces back the number of potential ancestors of a sample of the population at present. Under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations. In this work, we first characterise the measures $\Lambda$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. Next, we solve the linear system of equations corresponding to the Moran model. For the $\Lambda$-Wright-Fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation. We solve the latter for the Kingman model and the star-shaped model.