Article

On the random dynamics of Volterra quadratic operators

Uygun JamilovInstitute of Mathematics, National University of Uzbekistan, 29, Do’rmon Yo’li str., 100125 Tashkent, Uzbekistan emailMichael ScheutzowInstitut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany emailMaite Wilke-BerenguerInstitut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany email

Ergodic Theory and Dynamical Systemsjournal2015en

ABI

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.

Topics

Stability and Controllability of Differential Equations Stochastic processes and financial applications Nonlinear Differential Equations Analysis

Identifiers

DOI: 10.1017/etds.2015.30

Citations and references

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