On the dynamics of the nonlinear rational difference equation $x_{n+1}=\frac{\alpha x_{n-m}+\delta x_n}{\beta +\gamma x_{n-k}{x}_{n-l}\left(x_{n-k}+x_{n-l}\right)}$

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $x_{n+1}=\frac{\alpha x_{n-m}+\delta x_n}{\beta +\gamma x_{n-k}{x}_{n-l}\left(x_{n-k}+x_{n-l}\right)}$, $n=0,1,2,...$ where the parameters $\alpha ,\beta ,\gamma ,\delta \in (0,\mathrm{\infty })$, while $m,k,l$ are positive integers, such that $m<k<l.$ The initial conditions $x_{-m},...,x_{-k},...,x_{-l},...,x_{-1},...,x_0$ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.