On the random dynamics of Volterra quadratic operators Article Swipe

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Uygun Jamilov , Michael Scheutzow , Maite Wilke-Berenguer ·

YOU? · · 2015 · Open Access · · DOI: https://doi.org/10.1017/etds.2015.30 · OA: W2166184028

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$ . 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 $S^{m-1}$ , 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.