An Example of a Globally Asymptotically Stable Anti-monotonic System of Rational Difference Equations in the Plane

• Published in: Sarajevo journal of mathematics Vol. 5; no. 2; pp. 235 - 145
• Main Authors: Burgić, Dževad, Nurkanović, Zehra
• Format: Journal Article
• Language: English
• Published: 11.06.2024
• ISSN: 1840-0655; 2233-1964
• DOI: 10.5644/SJM.05.2.07

We consider the following system of rational difference equations in the plane: $$\left\{\begin{aligned}%{rcl}x_{n+1} &= \frac{\alpha_1}{A_1+B_1 x_n+ C_1y_n} \\[0.2cm]y_{n+1} &= \frac{\alpha_2}{A_2+B_2 x_n+ C_2y_n}\end{aligned}\right. \, , \quad n=0,1,2,\ldots $$ where the parameters $\alpha_1, \alpha_2, A_1, A_2, B_1, B_2, C_1, C_2$ are positive numbers and initial conditions $x_0$ and $y_0$ are nonnegative numbers. We prove that the unique positive equilibrium of this system is globally asymptotically stable. Also, we determine the rate of convergence of a solution that converges to the equilibrium $E=(\bar{x},\bar{y})$ of this systems.   2000 Mathematics Subject Classification. 39A10, 39A11, 39A20