Let S be the simplex in R , and V: S → S be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ S. It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonalpreserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the 2dimensional simplex.

In present paper, we study dynamical systems generated by a class of Lotka -Volterra operators. It is proved that such class of operator has non-ergodic property.

It is well known that any classical Lotka-Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

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