On dynamics of Volterra and non-Volterra cubic stochastic operators

Abstract

We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.

Keywords:

• Interaction systems
• cubic stochastic operator
• Volterra cubic stochastic operator
• Lyapunov function
• trajectory

2010 Mathematics Subject Classifications:

• 82C05
• 37N25
• 92D10

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions that contributed to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).