Qualitative study of a higher order rational difference equation

Year 2018, Volume: 47 Issue: 5, 1128 - 1143, 16.10.2018

Abdul Khaliq E.m. Elsayed

Abstract

In this paper we study the behavior of the difference equation

                     $x_{n+1}$ = $\dfrac{\alpha x_nx_{n-l}}{\beta x_{n-m}+\gamma x_{n-l}}$,$\quad n=0,1,$ $\cdots$

where the initial conditions $x_{-r}$, $x_{-r+1}$, $\cdots$ ,$x_0$ are arbitrary non zero real numbers where $r=\max\{l,m\}$ is a non-negative integer and  $\alpha$, $\beta$ and $\gamma$ are constants: Also, we obtain the solutions of some special cases of this equation. At the end we present some numerical examples to support our theoretical discussion.

Keywords

difference equation, stability, boundedness, global attractivity

References

• Abo-Zeid, R. and Cinar, C. Global behavior of the difference equation $x_{n+1}=\dfrac{A x_{n-1}}{B-Cx_nx_{n-2}}$, Bol. Soc. Paran. Mat. (3s.) 31 (1), 43-49, 2013.
• Ahmed, A. M. and Eshtewy, N. A. Basin of attraction of the recursive sequence $x_{n+1}=\dfrac{\alpha+\beta x_n+\gamma x_{n-1} }{A+B x_n+C x_{n-1}}$, J. Fract. Calc. Appl. 5 (10), 1-8, 2014.
• Amleh, A. M. Kirk, V. and Ladas, G. On the dynamics of $x_{n+1}=\dfrac{a+b x_{n-1}}{A+B x_{n-2}}$, Math. Sci. Res. Hot-Line 5, 1-15, 2001.
• Bedford, E. and Kim, K. Dynamics of rational surface automorphisms: linear fractional recurrences, J. Geo. Anal. 19 (3), 553-583, 2009.
• Belhannache, F. Touafek, N. and Abo-Zeid, R. Dynamics of a third-order rational difference equation, Bull. Math. Soc. Sci. Math. Roumanie, Tome 59 (107) (1), 13-22, 2016.
• Beverton, R. J. and Holt, S. J. On the Dynamics of Exploited Fish Populations, vol.19, Fish Invest., London, 1957.
• Cinar, C. On the positive solutions of the difference equation $x_{n+1}=\dfrac{a x_{n-1}}{1+b x_nx_{n-1}}$, Appl. Math. Comp. 156, 587-590, 2004.
• Dehghan, M. and Rastegar, N. Stability and periodic character of a third order difference equation, Math. Comput. Modelling 54 (11-12), 2560-2564, 2011.
• Din, Q. On a System of fourth-order rational difference equations, Acta Univ. Apulensis 39, 137-150, 2014.
• Din, Q. Global character of a rational difference equation, Thai J. Math. 12 (1), 55-70, 2014.
• Elabbasy, E. M. Barsoum, M. Y. and Alshawee, H. S. Behavior of solutions of a class of nonlinear rational difference equation $x_{n+1}=\alpha x_{n-k}+\dfrac{\beta x_{n-l}^\delta }{\gamma x_{n-s}^\delta}$, Electron. J. Math. Anal. Appl. 4 (2), 78-87, 2016.
• Elabbasy, E. M. El-Metwally, H. and Elsayed, E. M. Global behavior of the solutions of some difference equation, Advances in difference equation 2011, 89-100, 2011.
• El-Metwally H. and Elsayed, E. M. Solution and behavior of a third rational difference equation, Utilitas Mathematica 88, 27-42, 2012.
• EI-Metwally, H. Grove, E. A. Ladas, G. Levins, R. and Radin, M. On the difference equation $x_{n+1}=\alpha+\beta x_{n-1}e^{-x_n} $, Nonlinear Anal: Theory, Methods & Applications 47 (7), 4623- 4634, 2003.
• El-Moneam, M. A. and Alamoudy, S. O. On study of the asymptotic behavior of some rational difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 22, 157-176, 2015.
• Elsayed, E. M. Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc. 2011, Article ID 982309, 17 pages, 2011.
• Elsayed, E.M. Solutions of rational difference system of order two, Math. Comput. Modelling 55, 378-384, 2012.
• Elsayed, E. M. Behavior and expression of the solutions of some rational difference equa- tions, J. Comput. Anal. Appl. 15 (1), 73-81, 2013.
• Elsayed, E. M. Solution for systems of difference equations of rational Form of order two, Comput. Appl. Math. 33 (3), 751-765, 2014.
• Elsayed, E. M. On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath. 7 (6), 1450067, 26 pages, 2014.
• Elsayed, E. M. and Ahmed, A. M. Dynamics of a three-dimensional systems of rational difference equations, Math. Methods Appl. Sci. 39 (5), 1026-1038, 2016.
• Elsayed, E. M. and Alghamdi, A. Dynamics and global stability of higher order nonlinear difference equation, J. Comput. Anal. Appl. 21 (3), 493-503, 2016.
• Elsayed, E. M. and El-Dessoky, M. M. Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat. 42 (5), 479494, 2013.
• Elsayed E. M. and El-Metwally, H. Stability and solutions for rational recursive sequence of order three, J. Comput. Anal. Appl. 17 (2), 305-315, 2014.
• Elsayed E. M. and El-Metwally, H. Global behavior and periodicity of some difference equations, J. Comput. Anal. Appl. 19 (2), 298-309, 2015.
• Elsayed, E. M. and Ibrahim, T. F. Solutions and periodicity of a rational recursive sequences of order five, Bull. Malays. Math. Sci. Society 38 (1), 95-112, 2015.
• Elsayed, E. M. and Ibrahim, T. F. Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat. 44 (6), 13611390, 2015.
• Elsayed, E. M. and Khaliq, A. Global attractivity and periodicity behavior of a recursive sequence, J. Comput. Anal. Appl. 22 (2), 369-379, 2017.
• Erdogan, M. E. and Cinar, C. On the dynamics of the recursive sequence, Fasciculi Math. 50, 59-66, 2013.
• Halim, Y. Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl. 3 (1), 204-214, 2015.
• Hassan, Sk. and Chatterjee, E. Dynamics of the equation in the complex plane, Cogent Math. 2, 1-12, 2015.
• Ibrahim, T. F. Periodicity and Solution of Rational Recurrence Relation of Order Six, Appl. Math. 3 , 729-733, 2012.
• Ibrahim, T. F. Periodicity and Global Attractivity of Difference Equation of Higher Order, J. Comput. Anal. Appl. 16 , 552-564, 2014.
• Jana, D. and Elsayed, E. M. Interplay between strong Allee effect, harvesting and hydra effect of a single population discrete - time system, Int. J. Biomath. 9 (1), 1650004, 25 pages, 2016.
• Khaliq, A. and Elsayed, E. M. The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl. 9 (3), 1052-1063, 2016.
• Khan, A. Q. Din, Q. Qureshi, M. N. and Ibrahim, T. F. Global behavior of an anti- competitive system of fourth-order rational difference equations, Computational Ecology and Software 4 (1), 35-46, 2014.
• Kocic, V. L. and Ladas, G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
• Kulenovic M. R. S. and Ladas, G. Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001.
• Kulenovic, M. R. S. Ladas, G. and Sizer, W. S. On the rational recursive sequence $x_{n+1}=\dfrac{\alpha x_n+\beta x_{n-1}}{\gamma x_n+\delta x_{n-1}}$, Math. Sci Res Hot-Line 2 (5), 1-16, 1996.
• Qureshi, M. N. and Khan, A. Q. Local stability of an open-access anchovy shery model, Comput. Ecology and Software 5 (1), 48-62, 2015.
• Simsek, D. Cinar C. and Yalcinkaya, I. On the recursive sequence $x_{n+1}=\dfrac{x_{n-3}}{1+x_{n-1}}$, Int. J. Contemp. Math. Sci. 1 (10), 475-480, 2006.
• Su, Y. H. and Li, W. T. Global asymptotic stability of a second-order nonlinear difference equation, Appl. Math. Comput. 168, 981-989, 2005.
• Tollu, D. Yazlik, Y. and Taskara, N. The Solutions of Four Riccati Difference Equations Associated with Fibonacci Numbers, Balkan J. Math. 2, 163-172, 2014.
• Touafek, N. On a second order rational difference equation, Hacet. J. Math. Stat. 41 (6), 867-874, 2012.
• Touafek, N. and Elsayed, E. M. On the periodicity of some systems of nonlinear difference equations, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie Tome 55 (103) (2), 217-224, 2012.
• Touafek, N. and Elsayed, E. M. On the solutions of systems of rational difference equations, Math. Comput. Model. 55, 1987-1997, 2012.
• Touafek, N. and Haddad, N. On a mixed max-type rational system of difference equations, Electron. J. Math. Anal. Appl. 3 (1), 164 - 169, 2015.
• Wang, C. and Hu, M. On the solutions of a rational recursive sequence, J. Math. Inform. 1 (14), 25-33, 2013.
• Yalcinkaya, I. and Cinar C. On the dynamics of difference equation $x_{n+1}=\dfrac{a x_{n-k}}{b+c x_n^p} $, Fasciculi Math. 42, 141-148, 2009.
• Yalçınkaya, I. Cinar, C. and Atalay, M. On the solutions of systems of difference equations, Adv. Difference Equ. Vol. 2008, Article ID 143943, 9 pages, 2008.
• Yan, X. and Li , W. Global attractivity in the recursive sequence $x_{n+1}=\dfrac{\alpha-\beta x_n}{\gamma-x_{n-1}}$, Appl. Math. Comp. 138 (2-3), 415-423, 2003.
• Yang, X. On the global asymptotic stability of the difference equation $ x_{n+1}=\dfrac{x_{n-1}x_{n-2}+x_{n-3}+a}{x_{n-1}+x_{n-2}x_{n-3}+a}$ , Appl. Math. Comp. 171 (2), 857-861, 2005.
• Yazlik, Y., Elsayed, E. M. and Taskara, N. On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl. 16 (5), 932-941, 2014.
• Zayed, E. M. E. Qualitative behavior of the rational recursive sequence $x_{n+1}=A x_n+B x_{n-k}+\dfrac{p+x_{n-k}}{q x_n+x_{n-k}} $, Int. J. Adv. Math. 1 (1), 44-55, 2014.
• Zayed, E. M. E. and El-Moneam, M. A. On the rational recursive sequence $x_{n+1}=a x_n-\dfrac{b x_n}{c x_n-d x_{n-k}} $, Comm. Appl. Nonlinear Anal. 15, 47-57 2008.
• Zhang, Q., Zhang, W., Liu, J. and Shao, Y. On a fuzzy logistic difference equation, WSEAS Trans. Math.13, 282-290, 2014.