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The Ancestral Process of Long-Range Seed Bank Models

Part of: Special processes

Published online by Cambridge University Press:  30 January 2018

Jochen Blath ,
Adrián González Casanova ,
Noemi Kurt  and
Dario Spanò
Jochen Blath*
Affiliation:
TU Berlin
Adrián González Casanova*
Affiliation:
Berlin Mathematical School
Noemi Kurt*
Affiliation:
TU Berlin
Dario Spanò*
Affiliation:
University of Warwick
*
∗ Postal address: Institut für Mathematik, TU Berlin, Sekr.MA 7-5, Strasse des 17. Juni 136, D-10623 Berlin, Germany.
∗∗ Current address: TU Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany.
∗ Postal address: Institut für Mathematik, TU Berlin, Sekr.MA 7-5, Strasse des 17. Juni 136, D-10623 Berlin, Germany.
∗∗∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

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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).

Keywords

Wright‒Fisher modelseed bankrenewal processlong-range interactionKingman coalescent

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 741 - 759
Copyright
© Applied Probability Trust 

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