On the random dynamics of Volterra quadratic operators

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex Sm-1. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex Sm-1, implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.DFG, GSC 14, Berlin Mathematical Schoo

A family of non-Volterra quadratic operators corresponding to permutations

In the present paper we consider a family of non-Volterra quadraticstochastic operators depending on a parameter $\alpha$ and study theirtrajectory behaviors. We find all fixed points for a non-Volterra quadraticstochastic operator on a finite-dimensional simplex. We construct some Lyapunovfunctions. A complete description of the set of limit points is given, and weshow that such operators have the ergodic property.Comment: 9 page

Contracting Quadratic Operators of Bisexual population

In this paper we find a sufficient condition under which the operator of bisexual population is contraction and show that this condition is not necessary