Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Evolutionary Dynamics of Zero-sum Games with Degenerate Payoff Matrix and Bisexual Population

Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 43--60 | DOI:10.5890/DNC.2021.03.004

N.N. Ganikhodjaev$^{1}$, U.U. Jamilov$^{2}$ , M. Ladra$^{3}$

$^1$ Department of Computational and Theoretical Sciences, Faculty of Science, IIUM, 25200 Kuantan, Malaysia

$^2$ V.I. Romanovskiy Institute of Mathematics, 100170, Tashkent, Uzbekistan

$^3$ Department of Algebra, University of Santiago de Compostela, Spain

Download Full Text PDF



In this paper we consider the quadratic stochastic operators describing evolution of a bisexual population. We establish correlation between such operators and evolutionary games, namely demonstrate that Volterra quadratic stochastic operator with degenerate payoff matrix is non-ergodic and corresponding evolutionary game is rock-paper-scissors game. To prove this statements we study the asymptotic behavior of trajectories of the Volterra quadratic stochastic operators with the non-hyperbolic fixed points.


  1. [1]  Hofbauer, J. and Sigmund, K. (2003), Evolutionary game dynamics, { Bulletin of the American Mathematical Society}, {\bf 40}(4), 479-519.
  2. [2]  Akin, E. and Losert, V. (1984), Evolutionary dynamics of zero-sum games, { J. Math. Biology}, {\bf 20}(3), 231-258.
  3. [3]  Ganikhodjaev, N.N., Ganikhodzhaev, R.N., and Jamilov~(Zhamilov), U.U. (2015), Quadratic stochastic operators and zero-sum game dynamics, { Ergod. Th. and Dynam. Sys.}, {\bf 35}(5), 1443-1473.
  4. [4]  Ganikhodjaev, N.N., Saburov, M., and Jamilov, U.U. (2013), Mendelian and non-Mendelian quadratic operators, { Appl. Math. Inf. Sci.}, {\bf 7}(5), 1721-1729.
  5. [5]  Lyubich, Yu.I. (1992), Mathematical structures in population genetics, { Biomathematics}, {\bf 22}, Springer-Verlag.
  6. [6]  Ganikhodjaev, N.N., Jamilov, U.U., and Mukhitdinov, R.T. (2014), Non-ergodic quadratic operators of bisexual population, { Ukr. Math. Jour.}, {\bf 65}(8), 1282-1291.
  7. [7]  Bernstein, S.N. (1924), The solution of a mathematical problem related to the theory of heredity, { Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat.}, (1), 83-115 (Russian).
  8. [8]  Ganikhodzhaev, R.N. (1993), Quadratic stochastic operators, Lyapunov function and tournaments, { Acad. Sci. Sb. Math.}, {\bf 76}(2), 489-506.
  9. [9]  Ganikhodzhaev, R.N. (1994), A chart of fixed points and Lyapunov functions for a class of discrete dynamical systems, { Math. Notes}, {\bf 56}(5-6), 1125-1131.
  10. [10]  Ganikhodzhaev, R.N. (2006), Eshmamatova D.B. Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, { Vladikavkaz. Mat. Zh.}, {\bf 8}(2), 12-28 (Russian).
  11. [11]  Jenks, R.D. (1969), Quadratic Differential Systems for Interactive Population Models, { J. Diff. Eqs.}, {\bf 5}, 497-514.
  12. [12]  Ulam, S. (1960), { A collection of mathematical problems}, Interscience Publishers, New-York-London.
  13. [13]  Zakharevich, M.I. (1978), On behavior of trajectories and the ergodic hypothesis for quadratic transformations of the simplex, { Russian Math. Surveys}, {\bf3}(6), 265-266.
  14. [14]  Ganikhodjaev, N.N. and Zanin, D.V. (2004), On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex, { Russian Math.Surveys}, {\bf59}(3), 571-572,
  15. [15]  Devaney, R.L. (2003), { An Introduction to Chaotic Dynamical Systems}, Studies in Nonlinearity, Westview Press, Boulder, CO.
  16. [16]  Elaydi, S.N. (2000), { Discrete Chaos}. Chapman Hall/CRC, Boca Raton, FL.
  17. [17]  Ganikhodjaev, N.N., Jamilov, U.U., and Mukhitdinov, R.T. (2013), On Non-Ergodic Transformations on $S^3,$ Journal of Physics: Conference Series, {\bf 435}, 012005.
  18. [18]  Ganikhodzhaev, R.N., Mukhamedov, F.M., and Rozikov, U.A. (2011), Quadratic stochastic operators: Results and open problems, { Infinite Dimensional Analysis, Quantum Probability and Related Topics}, {\bf 14}(2), 279-335.
  19. [19]  Kesten, H. (1970), Quadratic transformations: A model for population growth, I { Adv. Appl. Prob.,} 2, 1-82.
  20. [20]  Lyubich, Yu.I. (1978), Basic concepts and theorems of the evolution genetics of free populations, { Russian Math. Surveys}, 26(5), 51-116