On a Non-Volterra Cubic Stochastic Operator

In the present paper we consider a family of non-Volterra cubic stochastic operators depending on a parameter $$\theta$$ and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra cubic stochastic operator on the two-dimensional simplex. We show that if $$-1\leq\theta<0$$ then any trajectory of a cubic stochastic operator converges to the center of the simplex, if $$\theta=0$$ then the corresponding cubic stochastic operator is the identity map, if $$0<\theta\leq 1$$ then the set of limit points of trajectories of a cubic stochastic operator of an initial point is an infinite subset of the boundary of the two-dimensional simplex.

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