Skip to main content
SpringerLink
Log in

Critical migration rate in the evolutionary dynamics of subdivided populations

  • Original Paper - Cross-Disciplinary Physics and Related Areas of Science and Technology
  • Published:
Journal of the Korean Physical Society Aims and scope Submit manuscript

Abstract

We investigate the evolution model which consists of subdivided populations of individual agents confined to three islands. The individuals are haploid individuals with three types. In subdivided populations, migration acts in parallel with natural selection to determine the evolutionary dynamics. We call this model the (3, 3) parallel migration-selection (ParaMiSe) model. The individuals on each island reproduce with the probability proportional to their fitness values and die at random, and migrate between two islands. The evolutionary dynamics of the (3, 3) ParaMiSe model is formulated as a master equation and can be approximated as the Fokker–Planck equation and the stochastic differential equations (SDEs). By taking the infinite population limit, we obtain the deterministic parts of the SDEs and observe that there exists a second-order continuous phase transition in the population density from the monomorphic state to the polymorphic state at the critical migration rate. We find the critical migration rate as a function of the median fitness value on the islands and predict the critical migration rate for the general (D, D) ParaMiSe model with \(D>3\). We also carry out Monte Carlo simulations using the Gillespie-type algorithm and find that the results of Monte Carlo simulations are in good agreement with the analytic predictions.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. S. Wright, Genetics 16, 97 (1931)

    Article  Google Scholar 

  2. F. Rousset, Genetic structure and selection in subdivided populations (Princeton University Press, Oxford, 2004)

    Book  Google Scholar 

  3. T. Maruyama, Genet. Res. 15, 221 (1970)

    Article  Google Scholar 

  4. T. Nagylaki, J. Math. Biol. 9, 101 (1980)

    Article  MathSciNet  Google Scholar 

  5. S. Lessard, J. Math. Biol. 59, 659 (2009)

    Article  MathSciNet  Google Scholar 

  6. F. Blanquart, S. Gandon, S.L. Nuismer, J. Evol. Biol. 25, 1351 (2012)

    Article  Google Scholar 

  7. B. Waclaw, R.J. Allen, M.R. Evans, Phys. Rev. Lett. 105, 268101 (2010)

    Article  ADS  Google Scholar 

  8. Y.-G. Kang, J.-M. Park, J. Phys. A Math. Theor. 50, 475001 (2017)

    Article  Google Scholar 

  9. Y.-G. Kang, J.-M. Park, J. Korean Phys. Soc. 71, 528 (2017)

    Article  ADS  Google Scholar 

  10. J.-M. Park, J. Korean Phys. Soc. 73, 1219 (2018)

    Article  ADS  Google Scholar 

  11. C.W. Gardiner, Handbook of stochastic methods, 4th edn. (Springer, Berlin, 2009)

    MATH  Google Scholar 

  12. J. Grasman, O.A. van Herwaarden, Asymptotic methods for the Fokker-Planck equation and the exit problem in application (Springer-Verlag, Berlin, 1999)

    Book  Google Scholar 

  13. R. Graham, T. Tel, J. Stat. Phys. 35, 729 (1984)

    Article  ADS  Google Scholar 

  14. R. Graham, T. Tel, Phys. Rev. A 31, 1109 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  15. N.G. van Kampen, Stochastic processes in physics and chemistry (Elsevier Science, Amsterdam, 1992)

    MATH  Google Scholar 

Download references

Acknowledgements

YGK was supported by a Hanshin University research grant. JMP was supported by the Catholic University of Korea research fund 2021 and by the Basic Science Research Program through the National Research Foundation of Korea (Grant No. NRF-2018R1A2B6007007).

Author information

Authors and Affiliations

  1. School of Computer Engineering, Hanshin University, Osan, 18101, Korea

    Yung-Gyung Kang

  2. Department of Physics, The Catholic University of Korea, Bucheon, 14662, Korea

    Jeong-Man Park

Authors
  1. Yung-Gyung Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeong-Man Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeong-Man Park.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kang, YG., Park, JM. Critical migration rate in the evolutionary dynamics of subdivided populations. J. Korean Phys. Soc. 80, 74–81 (2022). https://doi.org/10.1007/s40042-021-00358-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40042-021-00358-x

Keywords

Search

Navigation