Regularity of 2-partition Poisson quadratic stochastic operator with three different parameters

Quadratic stochastic operator (QSO) was initiated by Bernstein in the early 20th century as the mathematical investigation on population genetics, involving the synthesis of Mendelian law of crossing and Galtonian law of regression. Since then, the study of QSO has been significantly developed for decades to describe dynamical systems in many areas. In this paper, we construct a Poisson quadratic stochastic operator generated by a 2-measurable partition with three different parameters defined on countable state space X={0, 1, 2, …}. The main objective of this research is to study the trajectory behaviour of such operators by reducing the dynamical systems into a one-dimensional setting corresponding to the number of measurable partitions. Some cases of 2-measurable partition generated by singleton and two points with three different defined parameters will be presented. To investigate their trajectory behaviour, we may apply the functional analysis within the measure and probability theory. We shall provide both computational and analytical results, which show that the Poisson QSO generated by a 2-measurable partition with three different parameters, i.e., λ_l≠λ₂≠λ₃ is either a regular or nonregular transformation for some parameters λ_i∈(0, ∞) for i=1,2,3.