On the stationary distribution of the block counting process for population models with mutation and selection. (English) Zbl 1415.92145

Cordero, F.; Möhle, M.

J. Math. Anal. Appl. 474, No. 2, 1049-1081 (2019).

Summary: 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.

MSC:

92D25	Population dynamics (general)
92D15	Problems related to evolution

Keywords:

Kingman coalescent; star-shaped coalescent; Bolthausen-Sznitman coalescent; Wright-Fisher model; Moran model

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