On the dynamics of non-Volterra quadratic operators corresponding to permutations. (English) Zbl 1541.37099
Summary: The present paper deals to a family of non-Volterra quadratic stochastic operators depending on a parameter and a permutation and their trajectory behaviours. We find all fixed and periodic points for such non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We proved that the \(\omega\)- limit set of any trajectory of such non-Volterra quadratic stochastic operator consists of either single point or finite points.
MSC:
| 37N25 | Dynamical systems in biology |
| 37H12 | Random iteration |
| 47B80 | Random linear operators |
| 47H40 | Random nonlinear operators |
| 92D25 | Population dynamics (general) |
Keywords:
quadratic stochastic operator; Volterra operator; non-Volterra operator; trajectory; simplexReferences:
| [1] | Bernstein, S., Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Stat.13(1) (1942), pp. 53-61. · Zbl 0063.00333 |
| [2] | Blath, J., Jamilov (Zhamilov), U.U., and Scheutzow, M., \( (G,\mu ) \) -quadratic stochastic operators, J. Differ. Equ. Appl.20(8) (2014), pp. 1258-1267. · Zbl 1303.37036 |
| [3] | Devaney, R.L., An Introduction to Chaotic Dynamical Systems, , Westview Press, Boulder, CO, 2003. reprint of the second (1989) edition. · Zbl 1025.37001 |
| [4] | Ganikhodzhaev, R.N., Quadratic stochastic operators, Lyapunov functions and tournaments, Sb. Math.76(2) (1993), pp. 489-506. · Zbl 0791.47048 |
| [5] | Ganikhodzhaev, R.N., Map of fixed points and Lyapunov functions for one class of discrete dynamical systems, Math. Notes56(5) (1994), pp. 1125-1131. · Zbl 0838.93062 |
| [6] | Ganikhodzhaev, R.N. and Eshmamatova, D.B., Quadratic automorphisms of a simplex and asymptotic behaviour of their trajectories, Vladikavkaz Math. Zh.8(2) (2006), pp. 12-28. (Russian). · Zbl 1313.37014 |
| [7] | Ganikhodzhaev, R., Mukhamedov, F., and Rozikov, U., Quadratic stochastic operators and processes: Results and open problems, Infin. Dimens. Anal. Quan. Probab. Relat. Top.14(2) (2011), pp. 279-335. · Zbl 1242.60067 |
| [8] | Ganikhodjaev, N.N., Ganikhodjaev, R.N., and Jamilov, U.U., Quadratic stochastic operators and zero-sum game dynamics, Ergod. Theory Dyn. Syst.35(5) (2015), pp. 1443-1473. · Zbl 1352.37026 |
| [9] | Ganikhodjaev, N.N., Pah, C.H., and Rozikov, U.A., Dynamics of quadratic operators generated by China’s five elements philosophy, J. Differ. Equ. Appl.27(8) (2021), pp. 1173-1192. · Zbl 1482.37045 |
| [10] | Jamilov, U.U., On symmetric strictly non-Volterra quadratic stochastic operators, Disc. Nonlin. Comp.5(3) (2016), pp. 263-283. · Zbl 1349.92123 |
| [11] | Jamilov, U.U. and Khudoyberdiev, Kh. O., An \((\alpha, \beta )\) quadratic stochastic operator acting in \(S^2 \), J. Appl. Nonlinear. Dyn.11(4) (2022), pp. 777-788. · Zbl 1497.37056 |
| [12] | Jamilov, U.U., Ladra, M., and Mukhitdinov, R.T., On the equiprobable strictly non-Volterra quadratic stochastic operators, Qual. Theory Dyn. Syst.16(3) (2017), pp. 645-655. · Zbl 1390.37090 |
| [13] | Jamilov, U.U., Scheutzow, M., and Wilke-Berenguer, M., On the random dynamics of Volterra quadratic operators, Ergodic Theory Dyn. Syst.37(1) (2017), pp. 228-243. · Zbl 1408.37090 |
| [14] | Jamilov, U.U., Khudoyberdiev, Kh. O., and Ladra, M., Quadratic operators corresponding to permutations, Stoch. Anal. Appl.38(5) (2020), pp. 929-938. · Zbl 1466.37044 |
| [15] | Jenks, R.D., Quadratic differential systems for interactive population models, J. Differ. Equ.5(3) (1969), pp. 497-514. · Zbl 0186.15801 |
| [16] | Kesten, H., Quadratic transformations: A model for population growth. I, Adv. Appl. Probab.2(1) (1970), pp. 1-82. · Zbl 0328.92011 |
| [17] | Khamraev, A.Yu., On the dynamics of a quasi strictly non-Volterra quadratic stochastic operator, Ukr. Math. J.71(8) (2020), pp. 1273-1281. · Zbl 07622915 |
| [18] | Lyubich, Y.I., Iterations of quadratic maps, in Math. Econ. Func. Anal. M. Nauka, 1974, pp. 109-138. (Russian). |
| [19] | Lyubich, Y.I., Mathematical Structures in Population Genetics, , Springer-Verlag, Berlin, 1992. · Zbl 0747.92019 |
| [20] | Mukhamedov, F. and Ganikhodjaev, N., Quantum Quadratic Operators and Processes, Springer, Berlin, 2015. · Zbl 1335.60002 |
| [21] | Mukhamedov, F. and Taha, M.H.M., On Volterra and orthogonality preserving quadratic stochastic operators, Miskolc. Math. Notes17(1) (2016), pp. 457-470. · Zbl 1389.37030 |
| [22] | Rozikov, U.A. and Zhamilov, U., F-quadratic stochastic operators, Math. Notes83(3-4) (2008), pp. 554-559. · Zbl 1167.92023 |
| [23] | Ulam, S.M., A Collection of Mathematical Problems, , Interscience Publishers, New York-London, 1960. · Zbl 0086.24101 |
| [24] | Zhamilov, U.U. and Rozikov, U.A., On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex, Sb. Math.200(9) (2009), pp. 1339-1351. · Zbl 1194.47077 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
