A quasi-strictly non-volterra quadratic stochastic operator

We consider a four-parameter family of non-Volterra operators defined on the two-dimensional simplex and show that, with one exception, each such operator has a unique fixed point. Depending on the parameters, we establish the type of this fixed point. We study the set of limit points for each trajectory and show that this set can be a single point or can contain a 2-periodic trajectory.