The dynamics of piecewise-continuous Volterra QSO on S2
In this article, we study a class of piecewise-continuous quadratic stochastic operators (QSOs) of
Volterra type defined on the two-dimensional simplex S2. The operator is defined by two Volterra
operators V1 and V2 acting on disjoint subsets of the simplex. We describe the fixed points of
these operators for various parameter configurations and analyze the asymptotic behavior of their
trajectories. In particular, we prove that for specific parameter choices, the limit set of any interior
point of the simplex lies on the boundary, and in the symmetric case a = b = c 6= 0, all trajectories
converge to the vertex (1; 0; 0).
1. Abdurakhimova Sh. B., Rozikov U. A., Dynamical system of a quadratic stochastic operator with two
discontinuity points. Math. Notes., V.111, No.5, (2022), 676-687.
2. Ganikhodzhaev R. N., Quadratic stochastic operators, Lyapunov functions, and tournaments. Sb. Math.,
76(2),(1993), 489-506.
3. Khamraev A. Yu., Jumayev J. N., Dynamics of a cubic stochastic operator with one discontinuity point.
Uzbek Mathematical Journal, Vol.69, Iss.1, (2025), 79-86.
4. Rozikov U. A., Usmonov J. B., Dynamics of a population with two equal dominated species. Qual.Theory
Dyn.Syst., 19, 62 (2020).
5. Usmonov J. B., On a two dimensional dynamical system generated by the foor function. Uzbek
Mathematical Journal, 2 (2019), 127-134.
6. Usmonov J. B., On dynamics of a discontinuous Volterra operator. Uzbek Mathematical Journal, Vol.65,
Iss.2, (2021), 164-173.
7. Usmonov J. B., Kodirova M. A., A quadratic stochastic operator with variable coefficients. Bulletin of
the Institute of Mathematics, Iss.3, (2020), 98-107.
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