Global Asymptotic Stability of a Rational System

Abstract EN

The main goal of this paper is to investigate the global asymptotic behavior of the difference equation x n + 1 = β 1 x n / A 1 + y n , y n + 1 = β 2 x n + γ 2 y n / x n + y n , n = 0,1 , 2 , … with β 1 , β 2 , γ 2 , A 1 ∈ ( 0 , ∞ ) and the initial value ( x 0 , y 0 ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) such that x 0 + y 0 ≠ 0 .

The major conclusion shows that, in the case where γ 2 < β 2 , if the unique positive equilibrium ( x - , y - ) exists, then it is globally asymptotically stable.