Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{% \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$

Abstract

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation ${x_{n+1}}=% \frac{{\alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{% {\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$, $% n=0,1,2,...$ where the parameters $\alpha ,\beta ,\gamma ,\delta ,{\eta },{% \sigma }\in (0,\infty )$, while $m,k,l$ are positive integers, such that $% m