On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure

Abstract

In this paper we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure µ being defined on the set of all configurations which are given on a graph G. We show that if µ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators.

On a class of separable quadratic stochastic operators

“…This construction depends on the probability measure μ which is given on a fixed graph G. In [5], it is proved that the QSO resulting from this construction is of Volterra type if and only if the graph G is connected. A construction of QSOs involving a general finite graph and probability measure μ (here μ is the product of measures defined on maximal subgraphs of the graph G) and yielding a class of non-Volterra operator was described in [19]. It was shown that if μ is given as the product of probability measures, then the corresponding non-Volterra operator can be studied by N Volterra operators (where N is the number of connected components of the graph).…”

“…Since then, QSOs have been intensively investigated, see ( [1][2][3][4][5][6][7][8][9][10], [17][18][19]. The recent advances of the theory of the QSOs interact with many branches of mathematics particularly with Graph theory, Probability theory, Spectral theory, Measure theory, Algebra, Biology and Physics.…”

On Dynamics of ℓ-Volterra Quadratic Stochastic Operators

“…In [21] using the construction of QSO for the general finite graph and probability measure µ (here µ is product of measures defined on maximal subgraphs of the graph G) a class of non-Volterra QSOs is described.…”

Volterra quadratic stochastic operators of a two-sex population

“…The theory of quadratic stochastic operators has been developed for more than 85 years, and many works related to this theory have been published (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). In recent years, there has been a considerable growth of interest in this theory due to its numerous applications to problems of mathematics, biology, and physics.…”