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arXiv · 1401.7840

On the Random Dynamics of Volterra Quadratic Operators

U. U. Jamilov·M. Scheutzow·M. Wilke-Berenguer
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Abstract

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.

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U. U. Jamilov, M. Scheutzow, M. Wilke-Berenguer. 2014-01-30. On the Random Dynamics of Volterra Quadratic Operators. https://doi.org/10.1017/etds.2015.30

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