Skip to main content
SpringerLink
Log in

Generalized Stepping Stone Model with Ξ-resampling Mechanism

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A generalized stepping stone model with Ξ-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman’s coalescent is replaced by Ξ-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued Ξ-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berg, C., Christensen, J., Ressel, P.: Harmonic Analysis on Semigroups, Springer-Verlag, New York, 1984

    Book  MATH  Google Scholar 

  2. Birkner, M., Blath, J., Möhle, M., et al.: A modified lookdown construction for the Xi—Fleming—Viot with mutation and populations with recurrent bottlenecks process. ALEA Lat. Am. J. Probab. Math. Stat., 6, 25–61 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Donnelly, P., Kurtz, T. G.: Particle representations for measure-valued population models. Ann. Probab., 27, 166–205 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Evans, S. N.: Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. H. Poincaré Probab. Statist., 33, 339–358 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng, S., Schmuland, B., Vaillancourt, J., et al.: Reversibility of interacting Fleming—Viot processes with mutation selection, and recombination. Canad. J. Math., 63, 104–122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fleming, W. H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J., 28, 817–843 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Handa, K.: A measure-valued diffusion process describing the stepping stone model with infinitely many alleles. Stochastic Process. Appl., 36, 269–296 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kermany, A. R. R., Hickey, D. A., Zhou, X.: Joint stationary moments of a two-island diffusion model of population subdivision. Theoretical Population Biology, 74, 226–232 (2008)

    Article  MATH  Google Scholar 

  9. Li, Z., Shiga, T., Yao, L.: A reversibility problem for Fleming—Viot processes. Electron. Comm. Probab., 4, 65–76 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, Z., Liu, H., Xiong, J., et al.: The reversibility and an SPDE for the generalized Fleming—Viot processes with mutation. Stochastic Process. Appl., 123, 4129–4155 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Notohara, M.: The coalescent and the genealogical process in geographically structured population. J. Math. Biol., 29, 59–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pitman, J.: Coalescents with multiple collisions. Ann. Probab., 27, 1870–1902 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sagitov, S.: The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab., 36, 1116–1125 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sagitov, S.: Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab., 40, 839–854 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schweinsberg, J.: Coalescents with simultaneous multiple collisions. Electron. J. Probab., 5, 1–50 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shiga, T.: An interacting system in population genetics. J. Math. Kyoto Univ., 20, 213–242 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Shiga, T.: An interacting system in population genetics. II. J. Math. Kyoto Univ., 20, 723–733 (1980)

    MathSciNet  MATH  Google Scholar 

  18. Shiga, T.: Continuous time multi-allelic stepping stone models in population genetcis. J. Math. Kyoto Univ., 22, 1–40 (1982)

    MathSciNet  MATH  Google Scholar 

  19. Shiga, T., Uchiyama, K.: Stationary states and their stability of the stepping stone model involving mutation and selection. Probab. Theory Relat. Fields, 73, 87–117 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, P. R. China

    Hui Li Liu

  2. Department of Mathematics and Statistics, Concordia University, Montreal, H3G1M8, Canada

    Xiao Wen Zhou

Authors
  1. Hui Li Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao Wen Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Li Liu.

Additional information

H. Liu’s research is supported by NSF of Hebei Province (Grant No. A2019205299), Hebei Education Department (Grant No. QN2019073), NSFC (Grant No. 11501164) and HNU (Grant No. L2019Z01). X. Zhou’s research is supported by Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2016-06704) and National Science Foundation of China (Grant No. 11771018)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H.L., Zhou, X.W. Generalized Stepping Stone Model with Ξ-resampling Mechanism. Acta. Math. Sin.-English Ser. 38, 1998–2018 (2022). https://doi.org/10.1007/s10114-022-1092-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-1092-8

Keywords

MR(2010) Subject Classification

Search

Navigation