The Dynamics of Non-homogeneous Markov Chains Associated with the b-bistochastic Quadratic Stochastic Operators on 1-dimensional Simplex
DOI:
https://doi.org/10.11113/mjfas.v22n1.4643Keywords:
b-order majorization, hyperbolicity, rate of convergence, Dobrushin ergodicity coefficient, weak ergodicityAbstract
This research investigates the dynamics of a quadratic stochastic operator (QSO), namely the b-bistochastic QSO defined on a 1-dimensional simplex as well as the dynamics of a non-homogeneous Markov chain (NHMC) associated with the said QSO. The QSO was first constructed on a 1-dimensional simplex and a fixed-point analysis was performed onto the constructed QSO. The limiting behavior of the QSO was also studied in order to find its rate of convergence. The QSO was then associated with an NHMC to determine its ergodicity by using an ergodicity coefficient called the Dobrushin ergodicity coefficient. The QSO was found to converge to its attracting fixed point at a constant rate of convergence, and the NHMC associated with the QSO was found to be weakly ergodic.
References
Freedman, H. I., & Waltman, P. (1984). Persistence in models of three interacting predator–prey populations. Mathematical Biosciences, 68, 213–231. https://doi.org/10.1016/0025-5564(84)90007-3.
Bernstein, S. (1942). Solution of a mathematical problem connected with the theory of heredity. The Annals of Mathematical Statistics, 13, 53–61https://doi.org/10.1214/aoms/1177731605.
Mukhamedov, F., & Ganikhodjaev, N. (2015). Quantum quadratic operators and processors. Springer. https://doi.org/10.1007/978-3-662-44983-3.
Lyubich, Y. I. (1992). Mathematical structures in population genetics. Springer. https://doi.org/10.1007/978-3-642-76233-1
Jarne, P., & Charlesworth, D. (1993). The evolution of the selfing rate in functionally hermaphrodite plants and animals. Annual Review of Ecology and Systematics, 24(1), 441–466. https://doi.org/10.1146/annurev.es.24.110193.002301.
Mukhamedova, F., & Mukhamedov, F. (2025). Stability and robustness of kinetochore dynamics under sudden perturbations and stochastic influences. Scientific Reports, 15, 14883. https://doi.org/10.1038/s41598-025-14883-4.
Jamilov, U., Mukhamedov, F., & Mukhamedova, F. (2023). Discrete time model of sexual systems. Heliyon, 9, e17913. https://doi.org/10.1016/j.heliyon.2023.e17913.
Ganikhodzhaev, R., Mukhamedov, F., & Rozikov, U. (2011). Quadratic stochastic operators and processes: Results and open problems. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(2), 279–335. https://doi.org/10.1142/S0219025711004391
Mukhamedov, F., & Embong, A. F. (2015). On b-bistochastic quadratic stochastic operators. Journal of Inequalities and Applications, 2015, 226. https://doi.org/10.1186/s13660-015-0732-2.
Alligood, K. T., Sauer, T. D., & Yorke, J. A. (2000). Chaos: An introduction to dynamical systems. Springer. https://doi.org/10.1007/978-1-4612-1146-2.
Mukhamedov, F., & Embong, A. F. (2018). On stable b-bistochastic quadratic stochastic operators and associated non-homogeneous Markov chains. Linear and Multilinear Algebra, 66(1), 1–21. https://doi.org/10.1080/03081087.2017.1325904
Jamilov, U., Mukhamedov, F., Mukhamedova, F., & Souissi, A. (2024). Dynamics of nonlinear stochastic operators and associated Markov measure. Mathematical Methods in the Applied Sciences, 47(17), 13300–13312. DOI: https://doi.org/10.1002/mma.10458.
Souissi, A., & Mukhamedov, F. (2024). Nonlinear stochastic operators and associated inhomogeneous entangled quantum Markov chains. Journal of Nonlinear Mathematical Physics, 31(1), 11–25. https://doi.org/10.1515/jnmp-2024-0002.
Souissi, A., Mukhamedov, F., El Gheteb, S., Rhaima, M., & Mukhamedova, F. (2024). Entangled hidden elephant random walk model. Chaos, Solitons & Fractals, 186, 115252. https://doi.org/10.1016/j.chaos.2024.115252
Sarymsakov, T. A., & Ganikhodjaev, N. N. (1991). Central limit theorem for quadratic chains. Uzbek Mathematical Journal, 1, 57–64.
Ganikhodjaev, N. N. (2023). Central limit theorem for inhomogeneous Markov chains generated by quadratic stochastic operators. Uzbek Mathematical Journal, 67(1), 24–30.
Zada, A., & Shah, S. (2017). On dynamics of quadratic stochastic operators: A survey. Surveys in Mathematics and Its Applications, 12, 117–164.
Embong, A. F., & Rosli, N. A. M. (2023). The fixed point of two-dimensional b-bistochastic quadratic stochastic operators and their dynamics. AIP Conference Proceedings, 2880, 030051. https://doi.org/10.1063/5.0152816.
Saburov, M., & Yusof, N. A. (2018). On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex. Methods of Functional Analysis and Topology, 24(3), 255–264.
Ahlberg, D., de la Riva, D., & Griffiths, S. (2021). On the rate of convergence in quenched Voronoi percolation. Electronic Journal of Probability, 26, 1–26. https://doi.org/10.1214/21-EJP665.
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Copyright (c) 2025 Abdurrahman Azman, Wan Nur Fairuz Alwani Wan Rozali, Farrukh Mukhamedov
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