Table 2 Number of equilibria along their qualitative behavior of the discrete-time model (4)

O

sink if \(0<\alpha _{1}<1\); never source; saddle if \(\alpha _{1}>1\); non-hyperbolic if \(\alpha _{1}=1\).

A

sink if \(\alpha _{1}\in (1,3)\) and \(0<\alpha _{2}<\frac{\alpha _{1}}{\alpha _{1}-1}\); source if \(\alpha _{1}>3\) and \(\alpha _{2}>\frac{\alpha _{1}}{\alpha _{1}-1}\); saddle if \(\alpha _{1}>3\) and \(0<\alpha _{2}<\frac{\alpha _{1}}{\alpha _{1}-1}\); non-hyperbolic if \(\alpha _{2}=\frac{\alpha _{1}}{\alpha _{1}-1}\).

B

locally asymptotically stable focus if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}<0\) and \(0<\alpha _{1}<\frac{\alpha _{2}}{\alpha _{2}-2}\);

unstable focus if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}<0\) and \(\alpha _{1}>\frac{\alpha _{2}}{\alpha _{2}-2}\);

non-hyperbolic (under which \(\kappa _{1,2}\) are a pair of complex conjugate with modulus 1) if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}<0\) and \(\alpha _{1}=\frac{\alpha _{2}}{\alpha _{2}-2}\);

locally asymptotically stable node if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}\ge 0\) and \(0<\alpha _{1}<\frac{3\alpha _{2}}{3-\alpha _{2}}\);

unstable node if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}\ge 0\) and \(\alpha _{1}>\frac{3\alpha _{2}}{3-\alpha _{2}}\);

non-hyperbolic (under which the real eigenvalues with modulus 1) if

\((\frac{2\alpha _{2}-\alpha _{1}}{\alpha _{2}} )^{2}- \frac{4\alpha _{1} (\alpha _{2}-2 )}{\alpha _{2}}\ge 0\) and \(\alpha _{1}=\frac{3\alpha _{2}}{3-\alpha _{2}}\).