  Many systems are presented using theory of nonlinear operators. A quadratic
stochastic operator (QSO) is perceived as a nonlinear operator. It has a wide
range of applications in various disciplines, such as mathematics, biology, and
other sciences. The central problem that surrounds this nonlinear operator lies
in the requirement that behavior should be studied. Nonlinear operators, even
QSO (i.e., the simplest nonlinear operator), have not been thoroughly
investigated. This study aims to present a new class of $\xi^{(as)}$-QSO
defined on 2D simplex and to classify it into 18 non-conjugate (isomorphic)
classes based on their conjugacy and the remuneration of coordinates. In
addition, the limiting points of the behavior of trajectories for four classes
defined on 2D simplex are examined.
