Qualitative behavior of difference equation of order two and positive nonoscillatory solutions (Q744434)

Consider the two difference equations \[ \displaystyle{x_{n+1} = ax_{n-1} - {{bx_n^2 + cx_{n-1}^2}\over{1-dx_n+ex_{n-1}}}}\leqno{(1)} \] with positive coefficients \(a>0\), \(b>0\), \(c>0\), \(d>0\), \(e>0\) and arbitrary positive initial conditions \(x_{-1}>0\), \(x_0>0\), also the equation \[ \displaystyle{x_{n+1} = ax_{n-1} - {{bx_n^2 + cx_{n-1}^2}\over{1-cx_n+ x_{n-1}}}}\;,\;n\geq 0, \leqno{(2)} \] i.e., a special case of (1) with \(c=a\), \(e=1\). The authors discuss the local and global asymptotic stability of the positive equilibrium of (1), a positive nonoscillatory solution of (2), and the instability of the positive equilibrium of (1).