Stability analysis of three parameters of 2-partition of three points Geometric quadratic stochastic operator - Muhammed Najmuddin, Afiqah; Hamzah, Nur Zatul Akmar (Kulliyyah of Science, IIUM, 2022)
It is known that nonlinear operators can explain a wide range of systems. A quadratic stochastic operator is a system that is related to population genetics. In the nonlinear operator theory, the study of quadratic stochastic operators still has an open problem. Examples of the finite case can be found in many papers. However, there are only several papers mentioning infinite cases. Hence, in this research, we consider the quadratic stochastic operator defined on infinite state space, Geometric quadratic stochastic operator generated by 2-partition of consecutive three points with three different parameters. In this paper, we construct the Geometric quadratic stochastic operator, investigate the trajectory behaviour and its regularity, and analyse the operator’s stability using graphical analysis. It is indicated that the Geometric quadratic stochastic operator is regular for some parameter values and non-regular for other parameter values through the convergence of the trajectory behaviour either to a unique fixed point or periodic point of period two. Furthermore, for stability, we get attracting hyperbolic fixed points as well as attracting and repelling hyperbolic periodic points. To conclude, the study of this operator is vital to understanding evolutionary phenomena or biological populations in a situation of the real world.