\(F\)-quadratic stochastic operators. (English. Russian original) Zbl 1167.92023

Math. Notes 83, No. 4, 554-559 (2008); translation from Mat. Zametki 83, No. 4, 606-612 (2008).
Motivated by applications to mathematical genetics, the authors study a special class of quadratic transformations of the unit simplex \(S^m\subset {\mathbb{R}}^{m+1}\) into itself. Using elementary methods, they prove that for each function from this class and any \(x^{(0)} \in S^m\) the trajectory \(\{ x^{(n)}\}\) is exponentially convergent to the fixed point \((1,0,...,0)\).


92D10 Genetics and epigenetics
47N60 Applications of operator theory in chemistry and life sciences
47H40 Random nonlinear operators
47H10 Fixed-point theorems
37N25 Dynamical systems in biology
Full Text: DOI arXiv


[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, 83–115 (1924).
[2] R. N. Ganikhodzhaev, ”Quadratic stochastic operators, Lyapunov functions, and tournaments,” Mat. Sb. 183(8), 119–140 (1992) [Russian Acad. Sci. Sb. Math. 76 (2), 489–506 (1993)]. · Zbl 0766.47037
[3] R. N. Ganikhodzhaev, ”On the definition fo quadratic bistochastic operators,” Uspekhi Mat. Nauk 48(4), 231–232 (1993) [Russian Math. Surveys 48 (4), 244–246 (1993)]. · Zbl 0816.15022
[4] R. N. Ganikhodzhaev, ”Map of fixed points and Lyapunov functions for a class of discrete dynamical systems,” Mat. Zametki 56(5), 40–49 (1994) [Math. Notes 56 (5), 1125–1131 (1994)]. · Zbl 0838.93062
[5] H. Kesten, ”Quadratic transformations: a model for population growth. I,” Adv. in Appl. Probab. 2(1), 1–82 (1970); ”Quadratic transformations: a model for population growth. II,” Adv. in Appl. Probab. 2 (1), 179–228 (1970). · Zbl 0328.92011
[6] Yu. I. Lyubich, Mathematical Structures in Population Genetics, in Biomathematics (Springer-Verlag, Berlin, 1992), Vol. 22. · Zbl 0593.92011
[7] R. N. Ganikhodzhaev and D. B. Éshmamatova, ”Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories,” Vladikavkaz. Mat. Zh. 8(2), 12–28 (2006) · Zbl 1313.37014
[8] R. N. Ganikhodzhaev, ”On a family of quadratic stochastic operators acting in S 2,” Dokl. Akad. Nauk UzSSR, No. 1, 3–5 (1989). · Zbl 0675.47038
[9] U. A. Rozikov and N. B. Shamsiddinov, On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure, arXiv: math/0608201v1. · Zbl 1161.37365
[10] P. R. Stein and S. M. Ulam, Nonlinear Transformations Studies on Electronic Computers, in Dissertationes Math. (Rozprawy Mat.) (Polish Acad. Sci., Warsaw, 1964), Vol. 39. · Zbl 0143.18801
[11] R. N. Ganikhodzhaev and A. I. Éshniyazov, ”Biostochastic quadratic operators,” Uzbek. Mat. Zh., No. 3, 29–34 (2004).
[12] N. N. Ganikhodzhaev and R. T. Mukhitdinov, ”On a class of quasi-Volterra operators,” Uzbek. Mat. Zh., No. 3–4, 9–12 (2003).
[13] R. N. Ganikhodzhaev and A. M. Zhuraboev, ”The set of equilibrium states of quadratic stochastic operators of type V {\(\pi\)},” Uzbek. Mat. Zh., No. 3, 23–27 (1998).
[14] R. N. Ganikhodzhaev and A. Z. Karimov, ”On the number of vertices of a polyhedron of bistochastic quadratic operators,” Uzbek. Mat. Zh., No. 6, 29–35 (1999).
[15] R. N. Ganikhodzhaev and R. É. Abdurakhmanova, ”Description of quadratic automorphisms of finitedimensional simplex,” Uzbek. Mat. Zh., No. 1, 7–16 (2002).
[16] M. I. Zakharevich, ”On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex,” Uspekhi Mat. Nauk 33(6), 207–208 (1978) [Russian Math. Surveys 33 (6), 265–266 (1978)]. · Zbl 0407.58030
[17] N. N. Ganikhodzhaev, ”On an application of the theory of Gibbs distributions in mathematical genetics,” Dokl. Ross. Akad. Nauk 372(1), 13–16 (2000) [Russian Acad. Sci. Dokl. Math. 61 (3), 321–323 (2000)]. · Zbl 1052.92040
[18] N. N. Ganikhodjaev and U. A. Rozikov, ”On quadratic stochastic operators generated by Gibbs distributions,” Regul. Chaotic Dyn. 11(4), 467–473 (2006). · Zbl 1164.37309
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.