On the random dynamics of Volterra quadratic operators

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.