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On the random dynamics of Volterra quadratic operators

Published online by Cambridge University Press:  21 July 2015

U. U. JAMILOV ,
M. SCHEUTZOW  and
M. WILKE-BERENGUER
U. U. JAMILOV
Affiliation:
Institute of Mathematics, National University of Uzbekistan, 29, Do’rmon Yo’li str., 100125 Tashkent, Uzbekistan email jamilovu@yandex.ru
M. SCHEUTZOW
Affiliation:
Institut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany email ms@math.tu-berlin.de, wilkeber@math.tu-berlin.de
M. WILKE-BERENGUER
Affiliation:
Institut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany email ms@math.tu-berlin.de, wilkeber@math.tu-berlin.de
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 228 - 243
Copyright
© Cambridge University Press, 2015 

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