Abstract
In the present paper we consider a family of non-Volterra cubic stochastic operators depending on a parameter \(\theta\) and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra cubic stochastic operator on the two-dimensional simplex. We show that if \(-1\leq\theta<0\) then any trajectory of a cubic stochastic operator converges to the center of the simplex, if \(\theta=0\) then the corresponding cubic stochastic operator is the identity map, if \(0<\theta\leq 1\) then the set of limit points of trajectories of a cubic stochastic operator of an initial point is an infinite subset of the boundary of the two-dimensional simplex.
Similar content being viewed by others
REFERENCES
S. Bernstein, ‘‘Solution of a mathematical problem connected with the theory of heredity,’’ Ann. Math. Statist. 13, 53–61 (1942).
R. R. Davronov, U. U. Jamilov (Zhamilov), and M. Ladra, ‘‘Conditional cubic stochastic operator,’’ J. Differ. Equat. Appl. 21, 1163–1170 (2015).
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity (Westview Press, Boulder, CO, 2003).
S. N. Elaydi, Discrete Chaos (Chapman and Hall/CRC, Boca Raton, FL, 2000).
N. N. Ganikhodjaev, R. N. Ganikhodjaev, and U. U. Jamilov, ‘‘Quadratic stochastic operators and zero-sum game dynamics,’’ Ergod. Theory Dyn. Syst. 35, 1443–1473 (2015).
N. N. Ganikhodzhaev, U. U. Zhamilov, and R. T. Mukhitdinov, ‘‘Nonergodic quadratic operators for a two-sex population,’’ Ukr. Math. J. 65, 1282–1291 (2014).
R. N. Ganikhodzhaev, F. M. Mukhamedov, and U. A. Rozikov, ‘‘Quadratic stochastic operators and processes: Results and open problems,’’ Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14, 279–335 (2011).
A. J. Homburg, U. U. Jamilov, and M. Scheutzow, ‘‘Asymptotics for a class of iterated random cubic operators,’’ Nonlinearity 32, 3346–3360 (2019).
U. U. Jamilov, A. Yu. Khamraev, and M. Ladra, ‘‘On a Volterra cubic stochastic operator,’’ Bull. Math. Biol. 80, 319–334 (2018).
U. U. Jamilov and M. Ladra, ‘‘On identically distributed non-Volterra cubic stochastic operator,’’ J. Appl. Nonlin. Dyn. 6, 79–90 (2017).
U. U. Jamilov and A. Reinfelds, ‘‘On constrained Volterra cubic stochastic operators,’’ J. Differ. Equat. Appl. 26, 261–274 (2020).
U. U. Jamilov and A. Reinfelds, ‘‘A family of Volterra cubic stochastic operators,’’ J. Convex Anal. 28, 19–30 (2021).
U. U. Jamilov, M. Scheutzow, and I. Vorkastner, ‘‘A prey-predator model with three interacting species.’’ arXiv: 1907.05100v1.
H. Kesten, ‘‘Quadratic transformations: A model for population growth. I,’’ Adv. Appl. Probab. 2, 1–82 (1970).
A. Yu. Khamraev, ‘‘On a Volterra type cubic operators,’’ Uzbek. Mat. Zh. 3, 65–71 (2009).
A. Yu. Khamraev, ‘‘On cubic operators of Volterra type,’’ Uzbek. Mat. Zh. 2, 79–84 (2004).
A. Yu. Khamraev, ‘‘A condition of uniqueness of fixed point for cubic operators,’’ Uzbek. Mat. Zh. 1, 79–87 (2005).
Y. I. Lyubich, Mathematical Structures in Population Genetics, Vol. 22 of Biomathematics (Springer, Berlin, 1992).
F. M. Mukhamedov, A. F. Embong, and A. Rosli, ‘‘Orthogonal preserving and surjective cubic stochastic operators,’’ Ann. Funct. Anal. 8, 490–501 (2017).
F. M. Mukhamedov, C. H. Pah, and A. Rosli, ‘‘On non-ergodic Volterra cubic stochastic operators,’’ Qual. Theory Dyn. Syst. 18, 1225–1235 (2019).
R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete (Pearson Prentice Hall, Upper Saddle River, NJ, 2004).
U. A. Rozikov and A. Yu. Khamraev, ‘‘On cubic operators, defined on the finite-dimensional simplexes,’’ Ukr. Math. J. 56, 1418–1427 (2004).
U. A. Rozikov and A. Yu. Khamraev, ‘‘On construction and a class of non-Volterra cubic stochastic operators,’’ Nonlin. Dyn. Sys. Theory 14, 92–100 (2014).
U. A. Rozikov and U. U. Zhamilov, ‘‘F-quadratic stochastic operators,’’ Math. Notes 83, 554–559 (2008).
U. A. Rozikov and U. U. Zhamilov, ‘‘Volterra quadratic stochastic operators of a two-sex population,’’ Ukr. Math. J. 63, 1136–1153 (2011).
S. M. Ulam, A Collection of Mathematical Problems, No. 8 of Interscience Tracts in Pure and Applied Mathematics (Interscience, New York, 1960).
M. I. Zakharevich, ‘‘On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex,’’ Russ. Math. Surv. 33, 265–266 (1978).
U. U. Zhamilov and U. A. Rozikov, ‘‘On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex,’’ Sb. Math. 200, 1339–1351 (2009).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by S. N. Tronin)
Rights and permissions
About this article
Cite this article
Jamilov, U.U., Kurganov, K.A. On a Non-Volterra Cubic Stochastic Operator. Lobachevskii J Math 42, 2800–2807 (2021). https://doi.org/10.1134/S1995080221120155
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080221120155