On Dynamics of ℓ-Volterra Quadratic Stochastic Operators

Abstract

We introduce a notion of -Volterra quadratic stochastic operator defined on (m − 1)dimensional simplex, where ∈ {0, 1, . . . , m}. The -Volterra operator is a Volterra operator if and only if = m. We study structure of the set of all -Volterra operators and describe their several fixed and periodic points. For m = 2 and 3, we describe behavior of trajectories of (m − 1)-Volterra operators. The paper also contains many remarks with comparisons of -Volterra operators and Volterra ones.This means that the association x 0 → x defines a map V called the evolution operator. The population evolves by starting from an arbitrary state x 0 , then passing to the state x = V (x 0 ) (in the next "generation"), then to the state x = V (V (x 0 )), 143 Int. J. Biomath. 2010.03:143-159. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. 144 U. A. Rozikov & A. Zada Int. J. Biomath. 2010.03:143-159. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only.

ON DYNAMICS OF ξS QUADRATIC STOCHASTIC OPERATORS

“…We classified ξ s -QSO into six classes. In this paper, we only investigated three classes of operators and we showed that if a population system is given by (5), then the population system has four equilibrium states, an equally distributed state is both local and global asymptotically stable, and the future of the population is predictable. Next we showed that any population system given by (6) is similar to that of population system given by (5).…”

“…By combining all previous results we can give a fascinating biological meaning of the evolution of the population system given by (5). Theorem 4.1.7 Suppose the evolution of the population system is given by (5). Then the following statements hold true.…”

“…One of such operators is quadratic stochastic operator which naturally arises in modeling of a population dynamics. 1 During many years this theory has developed and since then it has appeared in lots of papers [2][3][4][5] . In recent years it has again become of interest in connection with numerous applications to many branches of mathematics, biology and physics.…”

On orthogonality preserving quadratic stochastic operators

“…Certain other properties of such kind of operators has been studied in [15]. Some generalizations of Volterra QSO were studied in [16,22,23].…”

Classification ofξ(s)-Quadratic Stochastic Operators on 2D simplex

“…The asymptotical behavior of the QSO (even) on small dimensional simplex is complicated (see [5]). In order to solve this problem, many researchers always introduced a certain class of quadratic operators and studied their behaviors: Volterra QSO [2,10,12], ℓ-Volterra QSO [9], Non-Volterra QSO, Strictly non-Volterra QSO [13], F-QSO, Separable QSO, Quadratic doubly stochastic operators and so on. For more information, one may refer to [1].…”