Global behavior of a two-dimensional monotone difference system

Abstract

We investigate global behavior of solutions of a nonlinear difference system $$ x_{n+1} = px_n (1 + y_n),\quad y_{n+1} = qy_n (1 + x_n),\quad n = 0, 1, 2, \dots, $$ where parameters $p$, $q$ and initial values $x_0$, $y_0$ are positive. We give sufficient conditions for every solution of the system to be unbounded and sufficient conditions for the global stable manifold of the positive equilibrium to exist, which is a unbounded separatrix for the system. Some related conjectures are also given.