On a Non-Volterra Cubic Stochastic Operator
- Authors
- 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
- License
- Yellow
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.