Genealogy of a Wright-Fisher Model with Strong SeedBank Component

  • Jochen Blath
  • Bjarki Eldon
  • Adrián González Casanova
  • Noemi Kurt
Conference paper
Part of the Progress in Probability book series (PRPR, volume 69)

Abstract

We investigate the behaviour of the genealogy of a Wright-Fisher population model under the influence of a strong seedbank effect. More precisely, we consider a simple seedbank age distribution with two atoms, leading to either classical or long genealogical jumps (the latter modeling the effect of seed-dormancy). We assume that the length of these long jumps scales like a power Nβ of the population size N, thus giving rise to a ‘strong’ seedbank effect. For a certain range of β, we prove that the ancestral process of a sample of n individuals converges under a non-classical time-scaling to Kingman’s n−coalescent. Further, for a wider range of parameters, we analyze the time to the most recent common ancestor of two individuals analytically and by simulation.

Keywords

Seedbanks Wright-Fisher model Kingman’s coalescent 

Mathematics Subject Classification (2000).

Primary 60K35 Secondary 92D15 

References

  1. 1.
    N. Berestycki, Recent Progress in Coalescent Theory. Ensaios Matemáticos, vol. 16 (Sociedade brasileira de matemática, Rio de Janeiro, 2009)Google Scholar
  2. 2.
    J. Blath, A. González Casanova, N. Kurt, D. Spanò, The ancestral process of long-range seedbank models. J. Appl. Probab. 50(3), 741–759 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    J. Blath, A. González Casanova, N. Kurt, M. Wilke Berenguer, A new coalescent for seed bank models. Ann. Appl. Probab. (Accepted for publication)Google Scholar
  4. 4.
    A. González Casanova, E. Aguirre-von Wobeser, G. Espín, L. Servín-González, N. Kurt, D. Spanò, J. Blath, G. Soberón-Chávez, Strong seed-bank effects in bacterial evolution. J. Theor. Biol. 356, 62–70 (2014)CrossRefGoogle Scholar
  5. 5.
    I. Kaj, S. Krone, M. Lascoux, Coalescent theory for seed bank models. J. Appl. Probab. 38, 285–300 (2001)MathSciNetMATHGoogle Scholar
  6. 6.
    D.A. Levin, The seed bank as a source of genetic novelty in plants. Am. Nat. 135, 563–572 (1990)CrossRefGoogle Scholar
  7. 7.
    D.A. Levin, Y. Peres, E.L. Wilmer, Markov Chains and Mixing Times (AMS, Providence, 2009)Google Scholar
  8. 8.
    M. Möhle, A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. Adv. Appl. Probab. 30, 493–512 (1998)CrossRefMATHGoogle Scholar
  9. 9.
    M. Möhle, S. Sagitov, A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29(4), 1547–1562 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Pitman, Combinatorial Stochastic Processes. Lecture Notes in Mathematics, vol. 1875 (Springer, New York, 2002)Google Scholar
  11. 11.
    F. Spitzer, Principles of Random Walk, 2nd edn. (Springer, New York, 1976)CrossRefMATHGoogle Scholar
  12. 12.
    A. Tellier, S.J.Y. Laurent, H. Lainer, P. Pavlidis, W. Stephan, Inference of seed bank parameters in two wild tomato species using ecological and genetic data. PNAS 108(41), 17052–17057 (2011)CrossRefGoogle Scholar
  13. 13.
    R. Vitalis, S. Glémin, I. Oliviere, When genes got to sleep: the population genetic consequences of seed dormacy and monocarpic perenniality. Am. Nat. 163(2), 295–311 (2004)CrossRefGoogle Scholar
  14. 14.
    J. Wakeley, Coalescent Theory: An Introduction (Roberts and Company Publishers, Greenwood Village, Colorado, 2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jochen Blath
    • 1
  • Bjarki Eldon
    • 1
  • Adrián González Casanova
    • 1
  • Noemi Kurt
    • 1
  1. 1.Institut für MathematikBerlinGermany

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