  In the present paper we consider a family of non-Volterra quadratic
stochastic operators depending on a parameter $\alpha$ and study their
trajectory behaviors. We find all fixed points for a non-Volterra quadratic
stochastic operator on a finite-dimensional simplex. We construct some Lyapunov
functions. A complete description of the set of limit points is given, and we
show that such operators have the ergodic property.
