On the random dynamics of Volterra quadratic operators. (English) Zbl 1408.37090

Summary: 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.


37H10 Generation, random and stochastic difference and differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
60H25 Random operators and equations (aspects of stochastic analysis)
47H40 Random nonlinear operators
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