A Family of Volterra Cubic Stochastic Operators

We consider a convex combination of Volterra cubic stochastic operators defined on a two-dimensional simplex depending on the parameter θ and study their trajectory behaviours. We show that at the values θ = 0.5 the trajectories change their orientation. Moreover, for θ small than 0.5 any Volterra cubic stochastic operator has the property being regular and it is non-ergodic while θ is greater than 0.5.