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

Abstract

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, it was investigated several classes of QSO. In this paper, we study ξ (s) -QSO defined on 2D simplex. We first classify ξ (s) -QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.Mathematics Subject Classification 2010: 37E99; 37N25; 39B82, 47H60, 92D25. Key words: Quadratic stochastic operator; ℓ−Volterra quadratic stochastic operator; ξ (s) −quadratic stochastic operator; permuted ℓ−Volterra quadratic stochastic operator; dynamics10 = {V 11 , V 12 }, K 11 = {V 19 , V 31 }, K 12 = {V 20 , V 33 }, K 13 = {V 21 , V 32 }, K 14 = {V 22 , V 34 }, K 15 = {V 23 , V 36 }, K 16 = {V 24 , V 35 }, K 17 = {V 25 }, K 18 = {V 26 , V 27 }, K 19 = {V 28 }, K 20 = {V 29 , V 30 }.

On orthogonality preserving quadratic stochastic operators

“…From the result of [28,32] one can conclude that even dynamics of Volterra QSO is very complicated. We note that if α, β, γ ∈ {0, 1} then dynamics of operators taken from the classes K 1 ,K 2 were investigated in [17,18]. In [5] certain general properties of dynamics of permuted Volterra QSO were studied.…”

“…The difficulty of the problem depends on the given cubic matrix (P ijk ) m i,j,k=1 . An asymptotic behavior of the QSO even on the small dimensional simplex is complicated [19,28,32].…”

On Lebesgue Nonlinear Transformations

“…The analytic theory of stochastic processes generated by quadratic operators was established in [7,30]. A fixed point set and an omega limiting set of quadratic stochastic operators defined on the finite dimensional simplex were deeply studied in [13,14,15,16,20,21,22]. Ergodicity and chaotic dynamics of quadratic stochastic operators on the finite dimensional simplex were studied in the papers [8,11,12,23,24,25,26,32].…”

Classification of a new subclass of ξ(as)-QSO and its dynamics

“…Note that even in the small dimensional simplexes, this is a tricky job [3,5,12,21,25]. Since there is no general theory for these operators, it is natural to look first at their subclasses.…”

“…Since there is no general theory for these operators, it is natural to look first at their subclasses. The main problem has been solved for example for the subclass of Volterra operators [1,6,9,14], -Volter-QSO [12,19,20], bistochastic QSO [7,10,22], etc.. However, all these classes together would not cover a set of all QSOs.…”