Regularity of Geometric quadratic stochastic operator generated by 2-partition of infinite points

In this research, we construct a class of quadratic stochastic operator called Geometric quadratic stochastic operator generated by arbitrary 2-partition of infinite points on a countable state space , where . We also study the limiting behavior of such operator by proving the existence of the limit of the sequence through the convergence of the trajectory to a unique fixed point. It is established that such operator is a regular transformation.

Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex

We define a complementary stochastic quadratic operator on finite-dimensional space as a new sub-class of quadratic stochastic operator. We give necessary and sufficient conditions for complementary stochastic quadratic operator.

F-evolution algebra

We consider the evolution algebra of a free population generated by an F-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power-associative. We show that this algebra is not conservative, not stationary, not genetic and not train algebra, but it is a Banach algebra. The set of all derivations of the F-evolution algebra is described. We give necessary conditions for a state of the population to be a fixed point or a zero point of the F-quadratic stochastic operator which corresponds to the F-evolution algebra. We also establish upper estimate of the ?-limit set of the trajectory of the operator. For an F-evolution algebra of Volterra type we describe the full set of idempotent elements and the full set of absolute nilpotent elements.

Onξ(s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We studyξ(s)-QSO defined on 2D simplex. We first classifyξ(s)-QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.

THE ASYMPTOTIC BEHAVIOR OF TRAJECTORIES OF DISSIPATIVE QUADRATIC STOCHASTIC OPERATORS ON 2D SIMPLEX

In the present paper we study limit behavior of dissipative quadratic stochastic operators on 2D simplex. We show that dissipative quadratic stochastic operator, which is not linear, is either regular or has infinitely many fixed points. If dissipative quadratic stochastic operator is regular, it is shown that its unique fixed point is either a vertex of the simplex or the center of the face of the simplex.

ON DYNAMICS OF ℓ-VOLTERRA QUADRATIC STOCHASTIC OPERATORS

We introduce a notion of ℓ-Volterra quadratic stochastic operator 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 structure of the set of all ℓ-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 ℓ-Volterra operators and Volterra ones.