  A quadratic stochastic operator (in short 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, it was investigated
several classes of QSO. In this paper, we study $\xi^{(s)}$--QSO defined on 2D
simplex. We first classify $\xi^{(s)}$--QSO into 20 non-conjugate classes.
Further, we investigate the dynamics of three classes of such operators.
