On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators

It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space . We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space and considered for a special case of the parameter Ɛ. We found that the mixing quadratic stochastic operator is a regular transformation for and is a non-regular for . Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter Ɛ by the parameter , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy . We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.

Abstract It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space X . We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space X and considered for a special case of the parameter ε . We found that the mixing quadratic stochastic operator is a regular transformation for 1 1 2 4 ε < < and is a non-regular for 1 4 ε < . Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter ε by the parameter 6 r , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy 6 r . We conclude this paper by comparing with

Introduction
The study of quadratic stochastic operator (QSO) is rooted in the work of Bernstein [1]. Also, the study triggered the idea of heredity in QSO where it is also known as "evolutionary operator". At present, QSO can be reinterpreted as an operator describing the dynamics of gene frequencies for a set of given laws of heredity in mathematical population genetics [2][3][4]. Therefore, it gives significant results to both biological and mathematical areas. These QSOs are defined on X P X , where ( ) P X is a power set of , X that is the set of all subset of . X For arbitrary ( , ) i j X X ∈ × we specify a discrete probability ( , , ); 1, 2,... P i j k k = , that is ( , , ) 0 P i j k ≥ for any k and 1 ( , , ) 1.
and so on. Moreover, it can be redefined on the set of all probability measures on a measurable state space. For example, a QSO generated by a family of pure geometric distributions was studied in [5,6], while QSO generated by pure Poisson distributions was studied in [7,8], and QSO generated by Gaussian distributions was studied in [9][10][11], and QSO generated by Lebesgue distributions was studied in [12,13] respectively. Some of applications of such QSO were established to generate algebra named as Rock-Paper-Scissor algebra [14]. Transitions of blood groups on population based on QSO were investigated in [14]. There are many other applications of QSO, e.g., evolutionary games theory [15][16][17], gene conversion theory [3,4] and statistical physics [18,19].
One of the main problems in the theory of QSO is to study the asymptotic behaviour of the trajectories 1  ( , , ) ( , , ) , , We consider a nonlinear transformation called quadratic stochastic operator (QSO), is an arbitrary initial probability measurable of m partitions of the set X and For any measure We select a family { : , 1, 2,..., } ij i j m µ = of pure probability measures on ( , ) X F and define probability measure ( , , ) P x y A as follows: Then for arbitrary where A F ∈ is an arbitrary measurable set.   . exists.
Note that the limit point is a fixed point of a QSOW . Thus, the fixed points of QSO describe limit or long-run behaviour the trajectories of any initial point. Limiting behaviour of trajectories and the fixed points of QSO play important role in many applied problems [8,9,13,14]. Recall ergodic hypothesis for quadratic stochastic operators [16]. Note that a regular QSO W is ergodic, however in general, ergodicity does not imply regularity. For the counter examples on finite dimensional QSO, one can refer [1,8,9,10,13,14,15,17].
In 1-dimensional case, i.e., 2 m = the behaviour of the iterations was found to be rather simple in [14,15].
For 2-dimensional case, a quadratic stochastic operator W on 1 S has the following form It is evident that a function in term of x in (8) In order to avoid the analysis of particularities, by considering that 1 a < and 0. c > Then, the following statements are valid.

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On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators Theorem 1 [20]. A fixed point of the transformation (8) is unique and belongs to the open interval (0,1).
We consider the discriminant of the quadratic equation. 2 ( 2 ) 2( ) 0 a b c x b c x c − + + − + = (15) to investigate the local characteristic of the fixed point: We construct the family of mixture geometric nonlinear transformations defined on the countable sample space of nonnegative integers generated by 3-partitions.

r i j A A A A P r r i j A A A A P r r i j A A A A
where 1, 2, 3. l = It is true that the limiting behaviour of the recurrence equation in (19) is fully determined by limiting behaviour of recurrence equations in (20).
The recurrence equations in (13) 1. ; ; , r r r r r r = = = then the QSO W was investigated in [15], where the authors proved the regularity of such operator.
To avoid the complexity of study of the regularity of W we consider the QSOW with . x It was studied by Zakharevich [21], where the author investigated ergodicity and regularity of such operators.
Next, we study the regularity of QSO W in terms of . ε

Regularity
In this section, the general form of the QSO W has the form: We consider the measure.
The QSO W has four fixed points namely, It is easy to see for we have that. According to above calculations, we have proved the following theorem. , The QSO W has four fixed points namely,