On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators

Ftameh Khaled,Pah Chin Hee

doi:10.13189/ms.2021.090209

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 <img src=image/13491747_01.gif>. 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 <img src=image/13491747_01.gif> and considered for a special case of the parameter Ɛ. We found that the mixing quadratic stochastic operator is a regular transformation for <img src=image/13491747_02.gif> and is a non-regular for <img src=image/13491747_03.gif>. 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 <img src=image/13491747_04.gif>, where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy <img src=image/13491747_04.gif>. We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.

Highlights

The study of quadratic stochastic operator (QSO) is rooted in the work of Bernstein [1]
We found that the mixing quadratic stochastic operator is a regular transformation for 1 < ε < 1 and is a non-regular for ε < 1
We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular

Summary

Introduction

The study of quadratic stochastic operator (QSO) is rooted in the work of Bernstein [1]. 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]. It gives significant results to both biological and mathematical areas These QSOs are defined on m −1 dimensional simplex. This probability is called P(x, y,.) ∈ S ( X , F ) for any fixed x, y ∈ X , that the heredity coefficient and denoted as P(i, j, k ) = Pij,k. From an arbitrary state x ∈ S m−1 passing to the Where λ ∈ S ( X , F ) is an arbitrary initial probability state V x in the generation, to the state measurable of m partitions of the set X and.

Three-Dimensional Mixing Geometric QSO

Case Two

Case Three

Case Four

Conclusion

Results