Abstract
The purpose of this paper is to investigate a class of separable quadratic stochastic operators. Each separable quadratic stochastic operator (SQSO) depends on two quadratic matrices A and B, which have some relations. In this paper we proved that for each skew symmetric matrix A the corresponding SQSO is a linear operator. We also proved that non linear Volterra QSOs are not SQSOs. For a fixed matrix A we also discussed some properties of the set of all the corresponding matrices B of SQSOs.
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References
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).
R. L. Devaney, An introduction to chaotic dynamical system (Westviev Press, 2003).
S. N. Elaydi, Discrete chaos (Chapman Hall/CRC, 2000).
N. N. Ganikhodjaev and U. A. Rozikov, On quadratic stochastic operators generated by Gibbs distributions, Regul. Chaotic Dyn. 11(4), 467 (2006).
N. N. Ganikhodjaev, On the application of the theory of Gibbs distributions in mathematical genetics, Russian Acad. Sci. Dokl. Math. 61(3), 321 (2000).
R. N. Ganikhodzhaev, Quadratic stochastic operators, Lyapunov functions, and tournametns, Russian Acad. Sci. Sb.Math. 76(2), 489 (1993).
R. N. Ganikhodzhaev, On the definition of quadratic bistochastic operators, Russian Math. Surveys 48(4), 244 (1993).
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).
R. N. Ganikhodzhaev, Map of fixed points and Lyapunov functions for a class of discrete dynamical systems, Math. Notes. 56(5), 1125 (1994).
R. N. Ganikhodzhaev and U.A. Rozikov, Quadratic stochastic operators: results and open problems arXiv:0902.4207v2 [math.DS].
J. Hofbaver, K. Sigmund, The theory of evolution and dynamical systems (Cambridge Univ. Press, 1988).
R. D. Jenks, Quadratic differential systems for interactive population models, J.Diff. Equations 5, 497 (1969).
H. Kesten, Quadratic transformations: A model for population growth I, II, Adv. Appl. Prob. 2,1, 179 (1970).
Yu. I. Lyubich, Mathematical structures in population genetics, in Biomathematics (Springer-Verlag, 1992), vol. 22.
R. C. Robinson, An introduction to Dynamical systems: Continues and Discrete (Pearson Education, 2004).
U. A. Rozikov and S. Nazir, Separable Quadratic Stochastic Operators, Lobschevskii J. Math. 3, 215 (2010).
U. A. Rozikov and U. U. Zhamilov, On F-quadratic stochastic operators, Math. Notes. 83(4), 554 (2008).
U. A. Rozikov and A. Zada, On l-Volterra quadratic stochastic operators, Doklady Math. 79(1), 32 (2009).
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).
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Submitted by D.M. Mushtari
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Rozikov, U.A., Zada, A. On a class of separable quadratic stochastic operators. Lobachevskii J Math 32, 385–394 (2011). https://doi.org/10.1134/S1995080211040196
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DOI: https://doi.org/10.1134/S1995080211040196