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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

On $(s,t)$-Volterra Quadratic Stochastic Operators of a Bisexual Population

Journal of Applied Nonlinear Dynamics 9(4) (2020) 575--588 | DOI:10.5890/JAND.2020.12.005

U.U.Jamilov , M. Ladra

V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 81, Mirzo Ulugbek str., 100170, Tashkent, Uzbekistan Departamento of Matem'aticas & Instituto of Matem'aticas, University of Santiago de Compostela, Spain

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The authors introduce the concept of $(s,t)$-Volterra stochastic quadratic operator in a bisexual population, where each individual belongs either to the male sex group or female sex group. Under some conditions affecting the coefficients of these operators, several Lyapunov functions have been constructed so that the upper bounds for the set of limiting points of the trajectories could be obtained. This study depicts that the set of $(s,t)$-Volterra stochastic quadratic operators is a convex compact set and the extreme points of this set are found. Furthermore, $(s,t)$-Volterra stochastic quadratic operators of the aforementioned population, which have periodic trajectories, are constructed.


We thank the referees for the helpful comments and suggestions that contributed to improving this paper. The authors thank Prof. U. A. Rozikov for useful discussions.


  1. [1]  Lyubich, Y.I. (1992), Mathematical structures in population genetics, (22 vol.) of Biomathematics, Springer-Verlag, Berlin.
  2. [2]  Abdel-Gawad, H.I., Elazab, N.S., and Osman, M.S.(2013), Exact solutions of space dependent Korteweg{\textendash}de Vries equation by the extended unified method, J. Phys. Soc. Japan, 82(4), 044004.
  3. [3]  Arqub, O.A. (2018), Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space, Numer. Methods Partial Differential Eq., 34(5), 1759-1780.
  4. [4]  Arqub, O.A., and Al-Smadi, M. (2018), Atangana{\textendash}Baleanu fractional approach to the solutions of Bagley{\textendash}Torvik and Painlev{{e}} equations in Hilbert space, Chaos, Solitons $&$ Fractals, 117, 161-167.
  5. [5]  Arqub, O.A., and Al-Smadi, M. (2018), Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions, Numer. Methods Partial Differential Eq., 34(5), 1577-1597.
  6. [6]  Arqub, O.A. and Maayah, B. (2018), Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana{\textendash}Baleanu fractional operator, Chaos, Solitons $&$ Fractals, 117, 117-124.
  7. [7]  Ding, Y., Osman, M.S., and Wazwaz, A.M. (2019), Abundant complex wave solutions for the nonautonomous Fokas{\textendash}Lenells equation in presence of perturbation terms, Optik, 181, 503-513.
  8. [8]  Osman, M.S. (2019), One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada{\textendash}Kotera equation, Nonlinear Dyn., 96(2), 1491-1496.
  9. [9]  Osman, M.S., Abdel-Gawad, H.I., and Mahdy, M.E. (2018), Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results in Physics, 8, 1054-1060.
  10. [10]  Osman, M.S. and Ghanbari, B. (2018), New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach, Optik, 175, 328-333.
  11. [11]  Osman, M.S., Ghanbari, B., and Machado, J.A.T. (2019), New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity, Eur. Phys. J. Plus, 134(1), 20.
  12. [12]  Osman, M.S., Rezazadeh, H., and Eslami, M.(2019), Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity, Nonlinear Engineering, 8(1), 559-567.
  13. [13]  Osman, M.S. and Machado, J.A.T. (2018), The dynamical behavior of mixed-type soliton solutions described by (2+1)-dimensional Bogoyavlensky{\textendash}Konopelchenko equation with variable coefficients, J. Electromagnet Wave, 32(11), 1457-1464.
  14. [14]  Rezazadeh, H., Osman, M.S., Eslami, M., Mirzazadeh, M., Zhou, Q., Badri, S.A., and Korkmaz, A. (2019), Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-like equations, Nonlinear Engineering, 8(1), 224-230.
  15. [15]  Tariq, K.U., Younis, M., Rezazadeh, H., Rizvi, S.T.R., and Osman, M.S. (2018), Optical solitons with quadratic{\textendash}cubic nonlinearity and fractional temporal evolution, Modern Phys. Lett. B, 32(26), 1850317.
  16. [16]  Bernstein, S.N. (1942), Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Statistics, 13, 53-61.
  17. [17]  Blath, J., Jamilov(Zhamilov), U.U. and Scheutzow, M.(2014), {$(G,\mu)$}-quadratic stochastic operators, J. Difference Equ. Appl., 20(8), 1258-1267.
  18. [18]  Ganikhodjaev, N.N., Ganikhodjaev, R.N., and Jamilov, U.U. (2015), Quadratic stochastic operators and zero-sum game dynamics, Ergodic Theory Dynam. Systems, 35(5), 1443-1473.
  19. [19]  Ganikhodjaev, N.N., Jamilov, U.U., and Mukhitdinov, R.T. (2013), On non-ergodic transformations on {$S^3$}, J. Phys.: Conf. Ser., 435, 012005.
  20. [20]  Ganikhodzhaev, N.N., Zhamilov, U.U., and Mukhitdinov, R.T. (2014), Nonergodic quadratic operators for a two-sex population, Ukrainian Math. J., 65(8), 1282-1291.
  21. [21]  Ganikhodzhaev, R.N. (1993), Quadratic stochastic operators, {L}yapunov functions and tournaments, Sb. Math., 76(2), 489-506.
  22. [22]  Ganikhodzhaev, R.N. (1994), Map of fixed points and {L}yapunov functions for one class of discrete dynamical systems, Math. Notes, 56(5), 1125-1131.
  23. [23]  Ganikhodzhaev, R.N. and Eshmamatova, D.B. (2006), Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, Vladikavkaz. Mat. Zh., 8(2), 12-28.
  24. [24]  Ganikhodzhaev, R.N., Mukhamedov, F.M., and Rozikov, U.A. (2011), Quadratic stochastic operators and processes: results and open problems, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14(2), 279-335.
  25. [25]  Nagylaki, T. (1983), Evolution of a finite population under gene conversion, Proc. Nat. Acad. Sci. U.S.A., 80, 6278-6281.
  26. [26]  Nagylaki, T. (1983), Evolution of a large population under gene conversion, Proc. Nat. Acad. Sci. U.S.A., 80, 5941-5945.
  27. [27]  Rozikov, U.A. and Zhamilov, U.U. (2008), ${F}$-quadratic stochastic operators, Math. Notes, 83(3-4), 554-559.
  28. [28]  Rozikov, U.A. and Zhamilov, U.U. (2011), Volterra quadratic stochastic operators of a two-sex population, Ukrainian Math. J., 63(7), 1136-1153.
  29. [29]  Ulam, S.M. (1960), A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London.
  30. [30]  Volterra, V. (1931), Variations and fluctuations of the number of individuals in animal species living together, (in: Animal Ecology, ed. R.N. Chapman), McGraw-Hill, pp. 409-448.
  31. [31]  Zakharevich, M.I. (1978), On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russ. Math. Surv., 33(6), 265-266.
  32. [32]  Zhamilov, U.U. and Rozikov, U.A. (2009), The dynamics of strictly non-{V}olterra quadratic stochastic operators on the 2-simplex, Sb. Math., 200(9), 1339-1351.
  33. [33]  Jamilov, U.U. (2008), Regularity of {$F$}-quadratic stochastic operators, Uzbek. Mat. J., (2), 35-45.
  34. [34]  Jamilov, U.U. and Mukhitdinov, R.T. (2010), Conditional quadratic stochastic operators, Uzbek. Mat. J., (2), 31-38.
  35. [35]  Rozikov, U.A. and Zada, A. (2010), On dynamics of {$\ell$}-{V}olterra quadratic stochastic operators, Int. J. Biomath., 3(2), 143-159.
  36. [36]  Rozikov, U.A. and Zada, A. (2012), {$\ell$}-{V}olterra quadratic stochastic operators: {L}yapunov functions, trajectories, Appl. Math. Inf. Sci., 6(2), 329-335.
  37. [37]  Bessa-Gomes, C., Legendre, S., and Clobert, J. (2010), Discrete two-sex models of population dynamics: On modelling the mating function, Acta Oecologica, 36(5), 439-445.
  38. [38]  Ganikhodjaev, N.N. and Jamilov, U.U. (2015), Contracting quadratic operators of bisexual population, Appl. Math. Inf. Sci., 9(5), 2645-2650.
  39. [39]  Ganikhodjaev, N.N., Saburov, M., and Jamilov, U.U. (2013), Mendelian and non-{M}endelian quadratic operators, Appl. Math. Inf. Sci., 7(5), 1721-1729.
  40. [40]  Kesten, H. (1970), Quadratic transformations: {A} model for population growth. {I}, Advances in Appl. Probability, 2, 1-82.
  41. [41]  Ladra, M., and Rozikov, U.A. (2013), Evolution algebra of a bisexual population, J. Algebra, 378, 153-172.
  42. [42]  Tianran, Z. and Wang, W. (2005), Mathematical models of two-sex population dynamics, K\^oky\^uroku, 1432, 96-104.