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On the random dynamics of Volterra quadratic operators

Jamilov, U. U.; Scheutzow, Michael; Wilke-Berenguer, M.

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.
Published in: Ergodic theory and dynamical systems, 10.1017/etds.2015.30, Cambridge University Press
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  • 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.