Global Behavior of a System of Second-Order Rational Difference Equations

Year 2021, Volume: 4 Issue: 3, 150 - 162, 30.09.2021

Phong Mai Nam

https://doi.org/10.33434/cams.938775

Abstract

In this paper, we consider the following system of rational difference equations

$$x_{n+1}=\frac{a+x_n}{b+c{y}_n+d{x}_{n-1}}$$

, yn+1=α+ynβ+γxn+ηyn−1

, n=0,1,2,...xn+1=a+xnb+cyn+dxn−1, yn+1=α+ynβ+γxn+ηyn−1, n=0,1,2,...

where $a,b,c,d,\alpha ,\beta ,\gamma ,\eta \in (0,\infty )$ and the initial values $x_{-1},x_0,y_{-1},y_0\in (0,\infty )$. Our main aim is to investigate the local asymptotic stability and global stability of equilibrium points, and the rate of convergence of positive solutions of the system.

Keywords

Equilibrium points, local stability, global behavior, rate of convergence, positive solutions

References

• [1] M.R.S. Kulenovic ́, M. Nurkanovic ́, Global asymptotic behavior of a two dimensional system of difference equations modeling cooperation, J. Differ. Equations Appl., 9 (1) (2003), 149-159.
• [2] S. Kalabusic ́, M.R.S. Kulenovic ́, Dynamics of certain anti-competitive systems of rational difference equations in the plane, J. Difference Equ. Appl., 17 (11)(2011), 1599-1615.
• [3] Q. Din, T. F. Ibrahim, K. A. Khan, Behavior of a competitive system of second-order difference equations, Scientific World J., (2014), doi:10.1155/2014/283982.
• [4] M.N. Phong, Global behavior of a system of rational difference equations, Comm. App. Nonlinear Anal., 23(3)(2016), pp. 93-107.
• [5] M.R.S. Kulenovic ́, O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC, Boca Raton, Fla, USA, 2002.
• [6] E. Camouzis, G. Ladas, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/ CRC, Boca Raton, London, 2008.
• [7] R. P. Agarwal, Difference Equations and Inequalities, Second Ed. Dekker, New York, 1992, 2000.
• [8] L. Berg, S. Stevic ́, On some systems of difference equations, Appl. Math. Comput., 218 (2011), 1713-1718.
• [9] DZ. Burgic ́, Z. Nurkanovic ́, An example of globally asymptotically stable anti-monotonic system of rational difference equations in the plane, Sarajevo Journal of Mathematics, 5 (18) (2009), 235-245.
• [10] Q. Din, Dynamics of a discrete Lotka-Volterra model, Adv. Difference Equ., (2013), doi:10.1186/1687-1847-2013-95.
• [11] Y. Enatsu, Y. Nakata, Y. Muroya, Global stability for a class of discrete SIR epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 347-361.
• [12] V.V. Khuong, M.N. Phong, A note on boundedness, periodic nature and positive nonoscilatory solution of rational difference equation, PanAmer. Math. J., 20(2)(2010), pp. 53-65.
• [13] V.V. Khuong, M.N. Phong, A note on global behavior of solutions and positive nonoscillatory solution of a difference equation, Comm. App. Nonlinear Anal., 18(4)(2011), pp. 77-88.
• [14] V.L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic, Dordrecht, 1993.
• [15] M.R.S. Kulenovic ́, G.Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjec- tures, Chapman and Hall/ CRC, Boca Raton, London, 2001.
• [16] M. R. S. Kulenovic ́, Z. Nurkanovic ́, The rate of convergence of solution of a three dimensional linear fractional systems of difference equations, Zbornik radova PMF Tuzla - Svezak Matematika, 2 (2005), 1-6.
• [17] M. R. S. Kulenovic ́, M. Nurkanovic ́, Asymptotic behavior of a competitive system of linear fractional difference equations, Adv. Difference Equ., (2006), Art. ID 19756, 13pp.
• [18] M. Pituk, More on Poincare’s and Peron’s theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216.
• [19] S.Stevic, Boundedness and persistence of some cyclic-type systems of difference equations, Appl. Math. Lett., 56 (2016), 78-85.
• [20] S. Stevic, New class of solvable systems of difference equations, Appl. Math. Lett., 63 (2017), 137-144 .
• [21] S. Stevic, B. Iricanin, Z. Smarda, On a symmetric bilinear system of difference equations, Appl. Math. Lett., 89 (2019), 15-21.
• [22] E.Tas ̧demir, On the global asymptotic stability of a system of difference equations with quadratic terms, J. Appl. Math. Comput., 66 (2021), 423–437.