Global Asymptotic Stability of a Rational System

Abstract

The main goal of this paper is to investigate the global asymptotic behavior of the difference equation $x_{n+1}={\beta }_1{x}_n/\left(A_1+y_n\right)$, $y_{n+1}=\left({\beta }_2{x}_n+{\gamma }_2{y}_n\right)/\left(x_n+y_n\right)$, $n=\mathrm{0,1},2,\dots$ with ${\beta }_1,{\beta }_2,{\gamma }_2,A_1\in (0,\mathrm{\infty })$ and the initial value $(x_0,y_0)\in [0,\mathrm{\infty })\times [0,\mathrm{\infty })$ such that $x_0+y_0\ne 0$. The major conclusion shows that, in the case where , if the unique positive equilibrium $(\stackrel{-}{x},\stackrel{-}{y})$ exists, then it is globally asymptotically stable.