On a family of non-Volterra quadratic operators acting on a simplex

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter $\alpha$ and study their trajectory behaviors. We showed that for any $\alpha \in [0,1)$ the trajectories of such operator converge to a fixed point. For $\alpha =1$ any trajectory of the operator converges to a periodic trajectory.