Skip to main content
SpringerLink
Log in

On Non-ergodic Volterra Cubic Stochastic Operators

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

Abstract

Let \(S^{m-1}\) be the simplex in \({{\mathbb {R}}}^m\), and \(V:S^{m-1}\rightarrow S^{m-1}\) be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^n V^k(x) \end{aligned}$$

exists for every \(x\in S^{m-1}\). It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

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. Akin, E., Losert, V.: Evolutionary dynamics of zero-sum games. J. Math. Biol. 20, 231–258 (1984)

    Article  MathSciNet  Google Scholar 

  2. Bernstein, S.N.: Solution of a mathematical problem connected with the theory of heredity. Ann. Math. Stat. 13, 53–61 (1942)

    Article  MathSciNet  Google Scholar 

  3. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Westview Press, Boulder (2003)

    MATH  Google Scholar 

  4. Elaydi, S.N.: Discrete chaos. Chapman & Hall/CRC, Boca Raton (2000)

    MATH  Google Scholar 

  5. Freedman, H.I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Ganikhodzhaev, N.N., Zanin, D.: On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex. Russ. Math. Surv. 59, 161–162 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Jamilov, U., Khamraev, A., Ladra, M.: On a Volterra cubic stochastic operator. Bull. Math. Biol. 80, 319–334 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Khamraev, A.Y.: On cubic operators of Volterra type (Russian). Uzbek. Mat. Zh. 2004(2), 79–84 (2004)

    MathSciNet  Google Scholar 

  12. Khamraev, A.Y.: A condition for the uniqueness of a fixed point for cubic operators (Russian). Uzbek. Mat. Zh. 2005(1), 79–87 (2005)

    MathSciNet  Google Scholar 

  13. Khamraev, A.Y.: On a Volterra-type cubic operator (Russian). Uzbek. Mat. Zh. 2009(3), 65–71 (2009)

    MathSciNet  Google Scholar 

  14. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Springer-Verlag, Berlin (1992)

    Book  Google Scholar 

  15. Mukhamedov, F., Embong, A.F.: On \(b\)-bistochastic quadratic stochastic operators. J. Inequal. Appl. 2015, 226 (2015)

    Article  MathSciNet  Google Scholar 

  16. Mukhamedov, F., Embong, A.F.: On non-linear Markov operators: surjectivity vs orthogonal preserving property. Linear Multilinear Algorithm 66, 2183–2190 (2018)

    Article  MathSciNet  Google Scholar 

  17. Mukhamedov, F., Embong, A.F., Rosli, A.: Orthogonal preserving and surjective cubic stochastic operators. Ann. Funct. Anal. 8, 490–501 (2017)

    Article  MathSciNet  Google Scholar 

  18. Mukhamedov, F., Jamilov, U., Pirnapasov, A.: On nonergodic uniform Lotka–Volterra operators. Math. Notes 105, 258–264 (2019)

    Article  MathSciNet  Google Scholar 

  19. Rozikov, U.A., Khamraev, A.Y.: On cubic operators defined on finite-dimensional simplices. Ukr. Math. J. 56, 1699–1711 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

The first author (FM) thanks the research grant by the UAEU, No. 31S259.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, The United Arab Emirates University, P.O. Box, 15551, Al Ain, Abu Dhabi, UAE

    Farrukh Mukhamedov

  2. Department of Computational and Theoretical Sciences, Faculty of Science, IIUM, Kuantan, Malaysia

    Chin Hee Pah & Azizi Rosli

Authors
  1. Farrukh Mukhamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chin Hee Pah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Azizi Rosli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farrukh Mukhamedov.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Mukhamedov, F., Pah, C.H. & Rosli, A. On Non-ergodic Volterra Cubic Stochastic Operators. Qual. Theory Dyn. Syst. 18, 1225–1235 (2019). https://doi.org/10.1007/s12346-019-00334-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-019-00334-8

Keywords

Search

Navigation