ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 12 (2017), 117 -- 164
This work is licensed under a Creative Commons Attribution 4.0 International License.
ON DYNAMICS OF QUADRATIC STOCHASTIC OPERATORS: A SURVEY
Akbar Zada and Syed Omar Shah
Abstract. We discuss the notion of Volterra, l-Volterra and separable quadratic stochastic operators defined on (m-1)-dimensional simplex, where l∈{0,1,...,m}. The l-Volterra operator is a Volterra operator if and only if l=m. We study the structure of the set of all Volterra and l-Volterra operators and describe their several fixed and periodic points. For m=2 and m=3 we describe behavior of trajectories of (m-1)-Volterra operators. We also mention many remarks with comparisons of l-Volterra operators and Volterra ones. Also we discuss the dynamics of separable quadratic stochastic operators.
2010 Mathematics Subject Classification: 47J35 (37H10; 60H15; 37N25; 92D25)
Keywords: Quadratic stochastic operator; fixed point; trajectory; Volterra and non-Volterra operators; simplex.
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Akbar Zada,
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
e-mail: zadababo@yahoo.com, akbarzada@uop.edu.pk
Syed Omar Shah (Corresponding Author),
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
e-mail: omarshah89@yahoo.com, omarshahstd@uop.edu.pk