Global asymptotic behavior of a two-dimensional difference equation modelling competition. (English) Zbl 1019.39006

Consider the system of difference equations \[ x_{n+1}=\frac{x_n}{a+cy_n},\quad y_{n+1}=\frac{y_n}{b+dx_n}\quad n=0,1,\dots \quad (x_0\geq 0,\;y_0\geq 0)\tag{1} \] where \(a\) and \(b\) are in \((0,1),c\) and \(d\) are positive numbers. The global asymptotic behavior of solutions of (1) is investigated. The authors show that the stable manifold of (1) separates the positive quadrant into basins of attraction of two types of asymptotic behavior. An explicit equation for the stable manifold is obtained in the case where \(a=b\).
Reviewer: Fozi Dannan (Doha)


39A11 Stability of difference equations (MSC2000)
39B05 General theory of functional equations and inequalities
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