Quadratic stochastic operator (QSO) is a non-linear operator that describes the time evolution of different species in biology. The main concern with regard to this operator is to study its behavior. Such a behavior has not been studied in comprehensively; even QSOs, which are the simplest non-linear operators, have not been studied thoroughly. In this paper, we consider two classes of QSO on 2D simplex, namely, -QSO and -QSO. First, we study -QSO and classify it into 3 non-conjugate classes. Further, we investigate the dynamics of classes of such operators. Second, we study -QSO and classify it into 6 non-conjugate classes. Further, we investigate the dynamics of a class of such an operator.

fixed point, limiting point, quadratic stochastic operator.