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On a Non-Volterra Cubic Stochastic Operator

Authors
  • Jamilov, U. U.1, 2, 3
  • Kurganov, K. A.4
  • 1 V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 100174, Uzbekistan , Tashkent (Uzbekistan)
  • 2 Akfa University, Tashkent, 100095, Uzbekistan , Tashkent (Uzbekistan)
  • 3 National University of Uzbekistan, Tashkent, 100174, Uzbekistan , Tashkent (Uzbekistan)
  • 4 Faculty of Mathematics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan , Tashkent (Uzbekistan)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Dec 13, 2021
Volume
42
Issue
12
Pages
2800–2807
Identifiers
DOI: 10.1134/S1995080221120155
Source
Springer Nature
Keywords
Disciplines
  • Article
License
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Abstract

AbstractIn the present paper we consider a family of non-Volterra cubic stochastic operators depending on a parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra cubic stochastic operator on the two-dimensional simplex. We show that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1\leq\theta<0$$\end{document} then any trajectory of a cubic stochastic operator converges to the center of the simplex, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta=0$$\end{document} then the corresponding cubic stochastic operator is the identity map, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\theta\leq 1$$\end{document} then the set of limit points of trajectories of a cubic stochastic operator of an initial point is an infinite subset of the boundary of the two-dimensional simplex.

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