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Separable quadratic stochastic operators. (English) Zbl 1304.37033

The aim of the paper is to classify the set of separable stochastic quadratic operators and mainly to study properties of nonlinear ones. The authors describe some Lyapunov functions of such nonlinear operators and obtain upper estimates for the set of \(\omega\)-limit points of trajectories. The results of the paper can be applied in mathematical biology in order to recognize the asymptotic behaviour of the trajectories of evolution operators associated to population dynamics models.

MSC:

37H10 Generation, random and stochastic difference and differential equations
37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems
92D25 Population dynamics (general)
47B80 Random linear operators
49N10 Linear-quadratic optimal control problems
15A63 Quadratic and bilinear forms, inner products
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References:

[1] S. N. Bernstein, The solution of a mathematical problem related to the theory of heredity, Uchen. Zapiski Nauchno-Issled. Kafedry Ukr. Otd. Matem. 1, 83 (1924).
[2] R. L. Devaney, An introduction to chaotic dynamical system (Westview Press, 2003). · Zbl 1025.37001
[3] N. N. Ganikhodjaev and U. A. Rozikov, On quadratic stochastic operators generated by Gibbs distributions, Regul. Chaotic Dyn. 11(4), 467 (2006). · Zbl 1164.37309 · doi:10.1070/RD2006v011n04ABEH000364
[4] N. N. Ganikhodjaev, On the application of the theory of Gibbs distributions in mathematical genetics, Russian Acad. Sci. Dokl. Math. 61(3), 321 (2000).
[5] R. N. Ganikhodzhaev, Quadratic stochastic operators, Lyapunov functions, and tournaments, Russian Acad. Sci. Sb. Math. 76(2), 489 (1993). · Zbl 0791.47048
[6] R. N. Ganikhodzhaev, On the definition of quadratic bistochastic operators, Russian Math. Surveys 48(4), 244 (1993). · Zbl 0816.15022 · doi:10.1070/RM1993v048n04ABEH001058
[7] R. N. Ganikhodzhaev and D. B. Eshmamatova, Quadratic automorphisms of a simplex and the asymptotic behaviour of their trajectories, Vladikavkaz.Math. Zh. 8(2), 12 (2006). · Zbl 1313.37014
[8] R. N. Ganikhodzhaev, Map of fixed points and Lyapunov functions for a class of discrete dynamical systems, Math. Notes. 56(5), 1125 (1994). · Zbl 0838.93062 · doi:10.1007/BF02274660
[9] R. N. Ganikhodzhaev and U. A. Rozikov, Quadratic stochastic operators: results and open problems arXiv: 0902.4207v2 [math.DS]
[10] H. W. Kuhn and A. W. Tucker, (eds), Linear inequalities and related systems, Annal. Math. Stud. Princeton (Univ. Press., 1985). · Zbl 0103.13003
[11] L.-P. Pang, E. Spedicato, Z.-Q. Xia, and W. Wang, A method for solving the system of linear equations and linear inequalities, Math. Comp. Model. 46, 823 (2007). · Zbl 1136.65063 · doi:10.1016/j.mcm.2006.12.007
[12] U. A. Rozikov and U. U. Zhamilov, On F-quadratic stochastic operators, Math. Notes. 83(4), 554 (2008). · Zbl 1167.92023 · doi:10.1134/S0001434608030280
[13] U. A. Rozikov and A. Zada, On F-Volterra quadratic stochastic operators, Doklady Math. 79(1), 324 (2009). · Zbl 1263.47073 · doi:10.1134/S1064562409010104
[14] U. A. Rozikov and N. B. Shamsiddinov, On non-Volterra quadratic stochastic operators generated by a product measure, Stoch, Anal. Appl. 27(2), 353 (2009). · Zbl 1161.37365 · doi:10.1080/07362990802678994
[15] A. N. Shiryaev, Probability, 2nd Ed. (Springer, 1996).
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