On the Dynamics of Rational Difference Equation with Nonlinear Terms

Our research will be focused on the local and global behavior of the following rational difference equations of the form \( x_{n + 1} = \frac{{x_{n} }}{{P\left( {x_{n} } \right)P\left( {x_{n - 1} } \right)}} \,and\, x_{n + 1} = \frac{{x_{n - 1} }}{{P\left( {x_{n} } \right)P\left( {x_{n - 1} } \right)}} \) where \({\text{P}}\left( {\text{x}} \right)\) is polynomial with nonnegative coefficients with propertiy \({\text{P}}\left( 0 \right) > 0\) with initial conditions belonging to the positive quadrant.