Qualitative Behavior of a Rational Difference Equation

Abstract

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n-k}}{bcx_{n-k}^rx_{n-(2k+1)}^s}, \quad n=0,1,\ldots\]

where \(a, b, c, d, e\) are nonnegative parameters, initial conditions are nonnegative real numbers, \(k\) is a nonnegative integer, and \(r, s \geq 1\).