Journal of Applied Probability

The ancestral process of long-range seed bank models

Jochen Blath, Adrián González Casanova, Noemi Kurt, and Dario Spanò

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Abstract

We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright‒Fisher model, as well as a seed bank model with bounded age distribution considered in Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that, for age distributions with finite mean, the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Furthermore, we present a construction of the forward-in-time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced in Kaj, Krone and Lascoux (2001) as well as on a paper by Hammond and Sheffield (2013).

Article information

Source
J. Appl. Probab. Volume 50, Number 3 (2013), 741-759.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1378401233

Digital Object Identifier
doi:10.1239/jap/1378401233

Mathematical Reviews number (MathSciNet)
MR3102512

Zentralblatt MATH identifier
1301.92053

Subjects
Primary: 92D15: Problems related to evolution
Secondary: 60K05: Renewal theory

Keywords
Wright‒Fisher model seed bank renewal process long-range interaction Kingman coalescent

Citation

Blath, Jochen; González Casanova, Adrián; Kurt, Noemi; Spanò, Dario. The ancestral process of long-range seed bank models. J. Appl. Probab. 50 (2013), no. 3, 741--759. doi:10.1239/jap/1378401233. https://projecteuclid.org/euclid.jap/1378401233.


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Journal of Applied Probability

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