# Global and local behavior of a class of $\xi^{(s)}$-QSO

Volume 10, Issue 9, pp 4834--4845
Publication Date: September 17, 2017 Submission Date: September 17, 2016
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### Authors

Abdelwahab Alsarayreh - Institute of Engineering Mathematics, University Malaysia, Perlis Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia. Izzat Qaralleh - Department of mathematics, Faculty of Science, Tafila Technical University, P. O. Box, 179, 66110, Tafila, Jordan. Muhammad Zaini Ahmad - Institute of Engineering Mathematics, University Malaysia, Perlis Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia. Basma Al-Shutnawi - Department of mathematics, Faculty of Science, Tafila Technical University, P. O. Box, 179, 66110, Tafila, Jordan. Saba Al-Kaseasbeh - Department of mathematics, Faculty of Science, Tafila Technical University, P. O. Box, 179, 66110, Tafila, Jordan.

### Abstract

A quadratic stochastic operator (QSO) describes the time evolution of different species in biology. The main problem with regard to a nonlinear operator is to study its behavior. This has not been studied in depth; even QSOs, which are the simplest nonlinear operators, have not been studied thoroughly. This paper investigates the global behavior of an operator taken from $\xi^{(s)}$-QSO when the parameter $a=\frac{1}{2}$. Moreover, we study the local behavior of this operator at each value of $a,$ where $0< a < 1$.

### Share and Cite

##### ISRP Style

Abdelwahab Alsarayreh, Izzat Qaralleh, Muhammad Zaini Ahmad, Basma Al-Shutnawi, Saba Al-Kaseasbeh, Global and local behavior of a class of $\xi^{(s)}$-QSO, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4834--4845

##### AMA Style

Alsarayreh Abdelwahab, Qaralleh Izzat, Ahmad Muhammad Zaini, Al-Shutnawi Basma, Al-Kaseasbeh Saba, Global and local behavior of a class of $\xi^{(s)}$-QSO. J. Nonlinear Sci. Appl. (2017); 10(9):4834--4845

##### Chicago/Turabian Style

Alsarayreh, Abdelwahab, Qaralleh, Izzat, Ahmad, Muhammad Zaini, Al-Shutnawi, Basma, Al-Kaseasbeh, Saba. "Global and local behavior of a class of $\xi^{(s)}$-QSO." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4834--4845

### Keywords

• local behavior
• global behavior.

•  37E99
•  37N25
•  39B82
•  47H60
•  92D25

### References

• [1] S. Bernstein , Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Statistics, 13 (1942), 53–61.

• [2] A. Dohtani , Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math., 52 (1992), 1707– 1721.

• [3] O. Galor , Discrete dynamical systems, Springer, Berlin (2007)

• [4] R. N. Ganikhodzhaev, A family of quadratic stochastic operators that act in $S^2$, (Russian); Dokl. Akad. Nauk UzSSR, 1 (1989), 3–5.

• [5] R. N. Ganikhodzhaev, Quadratic stochastic operators, Lyapunov functions and tournaments, (Russian); translated from Mat. Sb., 183 (1992), 119–140, Russian Acad. Sci. Sb. Math., 76 (1992), 489–506.

• [6] R. N. Ganikhodzhaev , A chart of fixed points and Lyapunov functions for a class of discrete dynamical systems, (Russian); translated from Mat. Zametki, 56 (1994), 40–49, Math. Notes, 56 (1994), 1125–1131.

• [7] N. N. Ganikhodzhaev, An application of the theory of Gibbs distributions to mathematical genetics, Doklady Math., 61 (2000), 321–323.

• [8] R. N. Ganikhodzhaev, R. È. Abdirakhmanova , Description of quadratic automorphisms of a finite-dimensional simplex, (Russian); Uzbek. Mat. Zh., 1 (2002), 7–16.

• [9] R. N. Ganikhodzhaev, A. M. Dzhurabaev , The set of equilibrium states of quadratic stochastic operators of type $V_\pi$, (Russian); Uzbek. Mat. Zh., 3 (1998), 23–27.

• [10] R. N. Ganikhodzhaev, D. B. Èshmamatova, Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories , (Russian); Vladikavkaz. Mat. Zh., 8 (2006), 12–28.

• [11] R. Ganikhodzhaev, F. Mukhamedov, U. Rozikov, Quadratic stochastic operators and processes: results and open problems, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14 (2011), 279–335.

• [12] N. N. Ganikhodzhaev, R. T. Mukhitdinov , On a class of measures corresponding to quadratic operators, (Russian); Dokl. Akad. Nauk Rep. Uzb., 3 (1995), 3–6.

• [13] N. N. Ganikhodzhaev, U. A. Rozikov , On quadratic stochastic operators generated by Gibbs distributions , Regul. Chaotic Dyn., 11 (2006), 467–473.

• [14] J. Hofbauer, V. Hutson, W. Jansen , Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553–570.

• [15] J. Hofbauer, K. Sigmund , The theory of evolution and dynamical systems, Mathematical aspects of selection, Translated from the German, London Mathematical Society Student Texts, Cambridge University Press, Cambridge (1988)

• [16] R. D. Jenks , Quadratic differential systems for interactive population models, J. Differential Equations, 5 (1969), 497–514.

• [17] H. Kesten , Quadratic transformations: A model for population growth, II, Advances in Appl. Probability, 2 (1970), 179–228.

• [18] S.-T. Li, D.-M. Li, G.-K. Qu , On stability and chaos of discrete population model for a single-species with harvesting, J. Harbin Univ. Sci. Tech., 6 (2006), 021.

• [19] A. J. Lotka , Undamped oscillations derived from the law of mass action, J. Amer. Chem. Soc., 42 (1920), 1595–1599.

• [20] Y. I. Lyubich , Mathematical structures in population genetics, Translated from the 1983 Russian original by D. Vulis and A. Karpov, Biomathematics, Springer-Verlag, Berlin (1992)

• [21] F. Mukhamedov, A. H. M. Jamal , On $\xi^s$-quadratic stochastic operators in 2-dimensional simplex, Proc. of the 6th IMTGT Inter. Conf. on Math., Stat. and Its Appl., Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia, (2010), 159–172.

• [22] F. Mukhamedov, M. Saburov, A. H. M. Jamal , On dynamics of $\xi^s$-quadratic stochastic operators, Inter. Jour. Modern Phys., Conf. Ser., 9 (2012), 299–307.

• [23] F. Mukhamedov, M. Saburov, I. Qaralleh, Classification of $\xi(s)$-quadratic stochastic operators on 2D simplex, J. Phys., Conf. Ser., 435 (2013), 8 pages.

• [24] F. Mukhamedov, M. Saburov, I. Qaralleh , On $\xi^(s)$-quadratic stochastic operators on two-dimensional simplex and their behavior, Abstr. Appl. Anal., 2013 (2013), 12 pages.

• [25] M. Plank, Hamiltonian structures for the n-dimensional Lotka-Volterra equations, J. Math. Phys., 36 (1995), 3520–3543.

• [26] U. A. Rozikov, N. B. Shamsiddinov, On non-Volterra quadratic stochastic operators generated by a product measure, Stoch. Anal. Appl., 27 (2009), 353–362.

• [27] U. A. Rozikov, A. Zada , On dynamics of $\ell$-Volterra quadratic stochastic operators, Int. J. Biomath., 3 (2010), 143–159.

• [28] U. A. Rozikov, A. Zada, $\ell$-Volterra quadratic stochastic operators: Lyapunov functions, trajectories, Appl. Math. Inf. Sci., 6 (2012), 329–335.

• [29] U. A. Rozikov, U. U. Zhamilov , F-quadratic stochastic operators, (Russian); translated from Mat. Zametki, 83 (2008), 606–612, Math. Notes, 83 (2008), 554–559.

• [30] M. Saburov , Some strange properties of quadratic stochastic Volterra operators, World Appl. Sci. J., 21 (2013), 94–97.

• [31] P. R. Stein, S. M. Ulam, Non-linear transformation studies on electronic computers, Rozprawy Mat., 39 (1964), 66 pages.

• [32] F. E. Udwadia, N. Raju, Some global properties of a pair of coupled maps: quasi-symmetry, periodicity, and synchronicity, Phys. D, 111 (1998), 16–26.

• [33] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York (1964)

• [34] V. Volterra , Lois de fluctuation de la population de plusieurs espèces coexistant dans le même milieu, Association Franc. Lyon., 1926 (1927), 96–98.

• [35] M. I. Zaharevič, The behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, (Russian); Uspekhi Mat. Nauk, 33 (1978), 207–208.

• [36] U. U. Zhamilov, U. A. Rozikov , On the dynamics of strictly non-Volterra quadratic stochastic operators on a twodimensional simplex, (Russian); translated from Mat. Sb., 200 (2009), 81–94, Sb. Math., 200 (2009), 1339–1351.