Theory and Modern Applications

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

E.P Corresponding behavior
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 $$\alpha _{1}=\frac{3\alpha _{2}}{3-\alpha _{2}}$$.