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On the random dynamics of Volterra quadratic operators
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 37 / Issue 1 / February 2017
- Published online by Cambridge University Press:
- 21 July 2015, pp. 228-243
- Print publication:
- February 2017
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- Article
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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.
