Skip to main content
SpringerLink
Log in

On a Volterra Cubic Stochastic Operator

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We consider a class of Volterra cubic stochastic operators. We describe the set of fixed points, the invariant sets and construct several Lyapunov functions to use them in the study of the asymptotical behavior of the given Volterra cubic stochastic operators. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

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

Similar content being viewed by others

Notes

  1. http://www.bbc.com/news/health-31594856.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bernstein SN (1942) Solution of a mathematical problem connected with the theory of heredity. Ann Math Stat 13:53–61

    Article  MathSciNet  MATH  Google Scholar 

  • Chauvet E, Paullet JE, Privite JP, Walls Z (2002) A Lotka-Volterra three-species food chain. Math Mag 75(4):243–255

    Article  MathSciNet  MATH  Google Scholar 

  • Davronov RR, Jamilov UU, Ladra M (2015) Conditional cubic stochastic operator. J Differ Equ Appl 21(12):1163–1170

    Article  MathSciNet  MATH  Google Scholar 

  • Devaney RL (2003) An introduction to chaotic dynamical systems, studies in nonlinearity. Westview Press, Boulder

    MATH  Google Scholar 

  • Elaydi SN (2000) Discrete chaos. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Freedman HI, Waltman P (1977) Mathematical analysis of some three-species food-chain models. Math Biosci 33(3–4):257–276

    Article  MathSciNet  MATH  Google Scholar 

  • Freedman HI, Waltman P (1984) Persistence in models of three interacting predator-prey populations. Math Biosci 68(2):213–231

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ganikhodzhaev RN (1994) Map of fixed points and Lyapunov functions for one class of discrete dynamical systems. Math Notes 56(5):1125–1131

    Article  MathSciNet  MATH  Google Scholar 

  • Ganikhodzhaev RN, Eshmamatova DB (2006) Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories. Vladikavkazskii Mat Zh 8(2):12–28 ( in Russian)

  • Ganikhodzhaev RN, Mukhamedov FM, Rozikov UA (2011) Quadratic stochastic operators and processes: results and open problems, Infin. Dimens Anal Quantum Probab Relat Top 14(2):279–335

    Article  MathSciNet  MATH  Google Scholar 

  • Ganikhodzhaev NN, Zhamilov UU, Mukhitdinov RT (2014) Nonergodic quadratic operators for a two-sex population. Ukr Math J 65(8):1282–1291

    Article  MathSciNet  MATH  Google Scholar 

  • Ganikhodjaev NN, Ganikhodjaev RN, Jamilov UU (2015) Quadratic stochastic operators and zero-sum game dynamics. Ergod Theory Dyn Syst 35(5):1443–1473

    Article  MathSciNet  MATH  Google Scholar 

  • Jenks RD (1969) Quadratic differential systems for interactive population models. J Differ Equ 5:497–514

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Khamraev AY (2004) On cubic operators of Volterra type (Russian). Uzbek Mat Zh 2004(2):79–84

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Khamraev AY (2009) On a Volterra-type cubic operator (Russian). Uzbek Mat Zh 2009(3):65–71

    MathSciNet  Google Scholar 

  • Lyubich YI (1992) Mathematical structures in population genetics. In: Biomathematics, vol 22 (translated from the 1983 Russian original by D. Vulis and A, Karpov). Springer, Berlin

  • Robinson RC (2012) An introduction to dynamical systems—continuous and discrete. In: Pure and Applied Undergraduate Texts, 2nd edn, vol. 19. American Mathematical Society, Providence

  • Rozikov UA, Khamraev AY (2004) On cubic operators defined on finite-dimensional simplices. Ukr Math J 56(10):1699–1711

    Article  MathSciNet  MATH  Google Scholar 

  • Rozikov UA, Khamraev AY (2014) On construction and a class of non-Volterra cubic stochastic operators. Nonlinear Dyn Syst Theory 14(1):92–100

    MathSciNet  MATH  Google Scholar 

  • Rozikov UA, Zhamilov UU (2008) \({F}\)-quadratic stochastic operators. Math Notes 83(3–4):554–559

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ulam SM (1960) A collection of mathematical problems, interscience tracts in pure and applied mathematics, vol 8. Interscience Publishers, New York

    Google Scholar 

  • Volterra V (1931) Variations and fluctuations of the number of individuals in animal species living together. In: Chapman RN (ed) Animal Ecology. McGrawHill, New York, p 409–448

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

    Article  MATH  Google Scholar 

  • Zhamilov UU, Rozikov UA (2009) The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex. Sb Math 200(9):1339–1351

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for the helpful comments and suggestions that contributed to improve this paper. This work was partially supported by a grant from the Niels Henrik Abel Board and by Ministerio de Economía y Competitividad (Spain), Grant MTM2016-79661-P (European FEDER support included). The first author (UUJ) thanks the University of Santiago de Compostela (USC), Spain, for the kind hospitality and for providing all facilities.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan, 100170

    U. U. Jamilov

  2. Kharshi State University, Kharshi, Uzbekistan, 180100

    A. Yu. Khamraev

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

    M. Ladra

Authors
  1. U. U. Jamilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Khamraev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Ladra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to U. U. Jamilov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jamilov, U.U., Khamraev, A.Y. & Ladra, M. On a Volterra Cubic Stochastic Operator. Bull Math Biol 80, 319–334 (2018). https://doi.org/10.1007/s11538-017-0376-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-017-0376-0

Keywords

Mathematics Subject Classification

Search

Navigation