On \(\xi^{(s)}\)-quadratic stochastic operators on two-dimensional simplex and their behavior. (English) Zbl 1470.47031

Summary: A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We study \(\xi^{(s)}\)-QSO defined on 2D simplex. We first classify \(\xi^{(s)}\)-QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.


47D07 Markov semigroups and applications to diffusion processes
92D25 Population dynamics (general)
Full Text: DOI arXiv


[1] Bernstein, S., Solution of a mathematical problem connected with the theory of heredity, Annals of Mathematical Statistics, 13, 53-61, (1942) · Zbl 0063.00333
[2] Hofbauer, J.; Hutson, V.; Jansen, W., Coexistence for systems governed by difference equations of Lotka-Volterra type, Journal of Mathematical Biology, 25, 5, 553-570, (1987) · Zbl 0638.92019
[3] Hofbauer, J.; Sigmund, K., The Theory of Evolution and Dynamical Systems. The Theory of Evolution and Dynamical Systems, Mathematical Aspects of Selection, (1988), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0678.92010
[4] Li, S.-T.; Li, D.-M.; Qu, G.-K., On stability and chaos of discrete population model for a singlespecies with harvesting, Harbin University of Science and Technology, 6, article 021, (2006)
[5] Lotka, A. J., Undamped oscillations derived from the law of mass action, Journal of the American Chemical Society, 42, 1595-1599, (1920)
[6] Lyubich, Y. I., Mathematical Structures in Population Genetics, (1992), Berlin, Germany: Springer, Berlin, Germany
[7] Volterra, V., Lois de fluctuation de la population de plusieurs espèces coexistant dans le même milieu, (1926), Lyon, France: Association Française pour l’Avancement des Sciences, Lyon, France · JFM 52.0452.01
[8] Plank, M.; Losert, V., Hamiltonian structures for the n-dimensional Lotka-Volterra equations, Journal of Mathematical Physics, 36, 7, 3520-3534, (1995) · Zbl 0842.34012
[9] Udwadia, F. E.; Raju, N., Some global properties of a pair of coupled maps: quasi-symmetry, periodicity, and synchronicity, Physica D, 111, 1–4, 16-26, (1998) · Zbl 0932.37014
[10] Kesten, H., Quadratic transformations: a model for population growth. I, II, Advances in Applied Probability, 2, 1-82, 179–228, (1970) · Zbl 0328.92011
[11] Ulam, S. M., Problems in Modern Mathematics, (1964), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0137.24201
[12] Ganikhodzhaev, R.; Mukhamedov, F.; Rozikov, U., Quadratic stochastic operators and processes: results and open problems, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14, 2, 279-335, (2011) · Zbl 1242.60067
[13] Ganikhodzhaev, R. N., A family of quadratic stochastic operators that act in S, Doklady Akademii Nauk UzSSR, 1, 3-5, (1989) · Zbl 0675.47038
[14] Saburov, M., Some strange properties of quadratic stochastic Volterra operators, World Applied Sciences Journal, 21, 94-97, (2013)
[15] Stein, P. R.; Ulam, S. M., Non-Linear Transformation Studies on Electronic Computers, (1962), Los Alamos, NM, USA: Los Alamos National Laboratory, Los Alamos, NM, USA · Zbl 0143.18801
[16] Zakharevich, M. I., The behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russian Mathematical Surveys, 33, 207-208, (1978) · Zbl 0407.58030
[17] Ganikhodzhaev, R. N., Quadratic stochastic operators, Lyapunov functions and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76, 2, 489-506, (1993) · Zbl 0791.47048
[18] Ganikhodzhaev, R. N., A chart of fixed points and Lyapunov functions for a class of discrete dynamical systems, Mathematical Notes, 56, 5, 1125-1131, (1994) · Zbl 0838.93062
[19] Ganikhodzhaev, R. N.; Èshmamatova, D. B., Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, Vladikavkaz Mathematical Journal, 8, 2, 12-28, (2006) · Zbl 1313.37014
[20] Jenks, R. D., Quadratic differential systems for interactive population models, Journal of Differential Equations, 5, 497-514, (1969) · Zbl 0186.15801
[21] Ganikhodzhaev, R. N.; Dzhurabaev, A. M., The set of equilibrium states of quadratic stochastic operators of type \(V_\pi\), Uzbek Mathematical Journal, 3, 23-27, (1998)
[22] Ganikhodzhaev, R. N.; Abdirakhmanova, R. È., Description of quadratic automorphisms of a finite-dimensional simplex, Uzbek Mathematical Journal, 1, 7-16, (2002)
[23] Ganikhodzhaev, N. N.; Mukhitdinov, R. T., On a class of measures corresponding to quadratic operators, Doklady Akademii Nauk Respubliki Uzbekistan, 3, 3-6, (1995)
[24] Rozikov, U. A.; Zada, A., On \(\ell\)-Volterra quadratic stochastic operators, International Journal of Biomathematics, 3, 2, 143-159, (2010) · Zbl 1342.92199
[25] Rozikov, U. A.; Zada, A., \(\ell\)-Volterra quadratic stochastic operators: Lyapunov functions, trajectories, Applied Mathematics & Information Sciences, 6, 2, 329-335, (2012)
[26] Rozikov, U. A.; Zhamilov, U. U., On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex, Sbornik, 200, 9, 81-94, (2009) · Zbl 1194.47077
[27] Rozikov, U. A.; Zhamilov, U. U., F-quadratic stochastic operators, Mathematical Notes, 83, 3-4, 554-559, (2008) · Zbl 1167.92023
[28] Ganikhodzhaev, N. N.; Rozikov, U. A., On quadratic stochastic operators generated by Gibbs distributions, Regular & Chaotic Dynamics, 11, 4, 467-473, (2006) · Zbl 1164.37309
[29] Ganikhodjaev, N. N., An application of the theory of Gibbs distributions to mathematical genetics, Doklady Mathematics, 61, 321-323, (2000)
[30] Rozikov, U. A.; Shamsiddinov, N. B., On non-Volterra quadratic stochastic operators generated by a product measure, Stochastic Analysis and Applications, 27, 2, 353-362, (2009) · Zbl 1161.37365
[31] Mukhamedov, F.; Jamal, A. H. M., On \(\xi^s\)-quadratic stochastic operators in 2-dimensional simplex, Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and Its Application (ICMSA ’10), Universiti Tunku Abdul Rahman
[32] Mukhamedov, F.; Saburov, M.; Jamal, A. H. M., On dynamics of \(\xi^s\)-quadratic stochastic operators, International Journal of Modern Physics: Conference Series, 9, 299-307, (2012)
[33] Dohtani, A., Occurrence of chaos in higher-dimensional discrete-time systems, SIAM Journal on Applied Mathematics, 52, 6, 1707-1721, (1992) · Zbl 0774.93049
[34] Ganikhodzhaev, R. N.; Karimov, A. Z., Mappings generated by a cyclic permutation of the components of Volterra quadratic stochastic operators whose coefficients are equal in absolute magnitude, Uzbek Mathematical Journal, 4, 16-21, (2000)
[35] Mukhamedov, F.; Qaralleh, I.; Rozali, W. N. F. A. W., On \(\xi^a\)-quadratic stochastic operators on 2-D simplex
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.