A quadratic stochastic operator (Qso) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. In this paper, we study the fri)-Qso defined on 2D simplex. We first classify 4-(a)-QS0 into 2 non-conjugate classes. Further, we investigate the dynamics of these classes of such operators.