Abstract
We consider a class of Volterra cubic stochastic operators. We describe the set of fixed points, the invariant sets and construct several Lyapunov functions to use them in the study of the asymptotical behavior of the given Volterra cubic stochastic operators. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.
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Acknowledgements
We thank the referees for the helpful comments and suggestions that contributed to improve this paper. This work was partially supported by a grant from the Niels Henrik Abel Board and by Ministerio de Economía y Competitividad (Spain), Grant MTM2016-79661-P (European FEDER support included). The first author (UUJ) thanks the University of Santiago de Compostela (USC), Spain, for the kind hospitality and for providing all facilities.
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Jamilov, U.U., Khamraev, A.Y. & Ladra, M. On a Volterra Cubic Stochastic Operator. Bull Math Biol 80, 319–334 (2018). https://doi.org/10.1007/s11538-017-0376-0
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DOI: https://doi.org/10.1007/s11538-017-0376-0
Keywords
- Quadratic stochastic operator
- Cubic stochastic operator
- Volterra and non-Volterra operator
- Kolmogorov system