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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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

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

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Universtiy Press.
  • Cano, R. J. and Borucki, M. K. (1995). Revival and identification of bacterial spores in 25- to 40-million-year-old Dominican amber. Science 268, 1060–1064.
  • Ethier, S. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
  • Hammond, A. and Sheffield, S. (2013). Power law Pólya's urn and fractional Brownian motion. To appear in Prob. Theory Relat. Fields.
  • Jacod, J. and Protter, P. (2003). Probability Essentials, 2nd edn. Springer, Berlin.
  • Kaj, I., Krone, S. M. and Lascoux, M. (2001). Coalescent theory for seed bank models. J. Appl. Prob. 38, 285–300.
  • Levin, D. A. (1990). The seed bank as a source of genetic novelty in plants. Amer. Naturalist 135, 563–572.
  • Lindvall, T. (1979). On coupling of discrete renewal processes. Z. Wahrscheinlichkeitsth. 48, 57–70.
  • Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.
  • Möhle, M. (1998). A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. Adv. Appl. Prob. 30, 493–512.
  • Tellier, A. et al. (2011). Inference of seed bank parameters in two wild tomato species using ecological and genetic data. Proc. Nat. Acad. Sci. USA 108, 17052–17057.
  • Vitalis, R., Glémin, S. and Olivieri, I. (2004). When genes go to sleep: the population genetic consequences of seed dormancy and monocarpic perenniality. Amer. Naturalist 163, 295–311.
  • Yashina, S. et al. (2012). Regeneration of whole fertile plants from 30,000-y-old fruit tissue buried in Siberian permafrost. Proc. Nat. Acad. Sci. USA 109, 4008–4013.