Global asymptotic behavior of a two-dimensional difference equation modelling competition
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 \(a=b\).
scientific article; zbMATH DE number 1880124