Skip to main content
SpringerLink
Log in

On a Family of Non-Volterra Quadratic Operators Acting on a Simplex

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter \(\alpha \) and study their trajectory behaviours. We showed that for any \(\alpha \in [0,1)\) the trajectories of such operator converge to a fixed point. For \(\alpha =1\) any trajectory of the operator converges to a periodic trajectory.

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. Bernstein, S.: The solution of a mathematical problem related to the theory of heredity, Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat. 1, 83–115 (1924)

    Google Scholar 

  2. Devaney, R. L.: An introduction to chaotic dynamical systems. In: Studies in Nonlinearity. Westview Press, Boulder. Reprint of the second (1989) edition (2003)

  3. Ganikhodjaev, N.N., Ganikhodjaev, R.N., Jamilov, U.U.: Quadratic stochastic operators and zero-sum game dynamics. Ergod. Theory Dyn. Syst. 35(5), 1443–1473 (2015)

    Article  MathSciNet  Google Scholar 

  4. Ganikhodjaev, N.N., Saburov, M., Nawi, A.M.: Mutation and chaos in nonlinear models of heredity. Sci. World J. (2014). https://doi.org/10.1155/2014/835069

    Article  Google Scholar 

  5. Ganikhodzhaev, N.N., Zanin, D.V.: On a necessary condition for the ergodicity of quadratic operators defined on a two-dimensional simplex. Russ. Math. Surv. 59(3), 571–572 (2004)

    Article  MathSciNet  Google Scholar 

  6. Ganikhodzhaev, R.N.: A family of quadratic stochastic operators that act in \(S^2\). Dokl. Akad. Nauk UzSSR 1, 3–5 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Ganikhodzhaev, R.N.: Quadratic stochastic operators, Lyapunov functions and tournaments. Sb. Math. 76(2), 489–506 (1993)

    Article  MathSciNet  Google Scholar 

  8. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(2), 279–335 (2011)

    Article  MathSciNet  Google Scholar 

  9. Jamilov, U.U.: Quadratic stochastic operators corresponding to graphs. Lobachevskii J. Math. 34(2), 148–151 (2013)

    Article  MathSciNet  Google Scholar 

  10. Jamilov, U.U.: On a family of strictly non-Volterra quadratic stochastic operators. J. Phys. Conf. Ser. 697, 012013 (2016)

    Article  Google Scholar 

  11. Jamilov, U.U.: On symmetric strictly non-Volterra quadratic stochastic operators. Discontin. Nonlinear Complex 5(3), 263–283 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Jamilov, U.U., Ladra, M.: Non-ergodicity of uniform quadratic stochastic operators. Qual. Theory Dyn. Syst. 15(1), 257–271 (2016)

    Article  MathSciNet  Google Scholar 

  13. Jamilov, U.U., Ladra, M., Mukhitdinov, R.T.: On the equiprobable strictly non-Volterra quadratic stochastic operators. Qual. Theory Dyn. Syst. 16(3), 645–655 (2017)

    Article  MathSciNet  Google Scholar 

  14. Jamilov, U.U., Scheutzow, M., Wilke-Berenguer, M.: On the random dynamics of Volterra quadratic operators. Ergod. Theory Dyn. Syst. 37(1), 228–243 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kesten, H.: Quadratic transformations: a model for population growth: I. Adv. Appl. Probab. 2, 1–82 (1970)

    Article  MathSciNet  Google Scholar 

  16. Khamraev, AYu.: On the dynamics of a quasistrictly non-Volterra quadratic stochastic operator. Ukr. Math. J. 71(8), 1116–1122 (2019)

    MathSciNet  Google Scholar 

  17. Lyubich, Y.I.: Mathematical structures in population genetics. In: Biomathematics, vol. 22. Springer, Berlin (1992)

  18. Rozikov, U.A., Zhamilov, U.: F-quadratic stochastic operators. Math. Notes 83(3–4), 554–559 (2008)

    Article  MathSciNet  Google Scholar 

  19. Rozikov, U.A., Zhamilov, U.U.: Volterra quadratic stochastic operators of a two-sex population. Ukr. Math. J. 63(7), 1136–1153 (2011)

    Article  MathSciNet  Google Scholar 

  20. Saburov, M.: A class of nonergodic Lotka-Volterra operators. Math. Notes 97(5), 759–763 (2015)

    Article  MathSciNet  Google Scholar 

  21. Ulam, S. M.: A collection of mathematical problems. In: Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience Publishers, New York (1960)

  22. Vallander, S.S.: On the limit behavior of sequences of iterations of some quadratic transformations. Dokl. Akad. Nauk SSSR 202(3), 515–517 (1972). (in Russian)

    MathSciNet  Google Scholar 

  23. Vallander, S. S.: On “ergodic properties of a family of quadratic stochastic operators” in rings and modules. In: Limit Theorems of Probability Theory, 1, pp. 153–157. Izd. Leningrad. Gos. Univ., Leningrad (1986) (in Russian)

  24. Vallander, S.S.: Change in the orbit orientation for a certain family of stochastic quadratic mappings. Vest. St. Petersb. Univ. Math. 46(4), 187–188 (2013)

    Article  MathSciNet  Google Scholar 

  25. Zakharevich, M.I.: On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex. Russ. Math. Surv. 33(6), 265–266 (1978)

    Article  Google Scholar 

  26. Zhamilov, U.U., Rozikov, U.A.: On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex. Sb. Mat. 200(9), 1339–1351 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referee for the helpful comments and suggestions that contributed to the improvement of this paper. This work was partially supported by a grant from the IMU-CDC. The first author (UJ) thanks the University of Santiago de Compostela (USC), Spain, for the kind hospitality and for providing all facilities. The authors were partially supported by Agencia Estatal de Investigación (Spain), Grant MTM2016-79661-P and by Xunta de Galicia, Grant ED431C 2019/10 (European FEDER support included, UE).

Author information

Authors and Affiliations

  1. V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b, University Str., Tashkent, Uzbekistan, 100174

    Uygun Jamilov

  2. National University of Uzbekistan, 4, University Str., Tashkent, Uzbekistan, 100174

    Uygun Jamilov

  3. University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Manuel Ladra

Authors
  1. Uygun Jamilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel Ladra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uygun Jamilov.

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

Jamilov, U., Ladra, M. On a Family of Non-Volterra Quadratic Operators Acting on a Simplex. Qual. Theory Dyn. Syst. 19, 95 (2020). https://doi.org/10.1007/s12346-020-00433-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-020-00433-x

Keywords

Mathematics Subject Classification

Search

Navigation