$$F_{00}(0,0)$$ never sink; source if $$\gamma >\frac{2}{h \omega }$$; saddle if $$0<\gamma <\frac{2}{h \omega }$$; non-hyperbolic if $$\gamma =\frac{2}{h \omega }$$
$$F_{P0}(1,0)$$ sink if $$\frac{2(\nu ^{2}+1)}{\gamma (\omega (\nu ^{2}+1)-1)}< h <\frac{2}{\beta }$$; source if $$\frac{2(\nu ^{2}+1)}{\gamma (\omega (\nu ^{2}+1)-1)}>h >\frac{2}{\beta }$$; saddle if $$h>\max \{ \frac{2}{\beta }, \frac{2(\nu ^{2}+1)}{\gamma (\omega (\nu ^{2}+1)-1)} \}$$; non-hyperbolic if $$h=\frac{2}{\beta }$$ or $$h= \frac{2(\nu ^{2}+1)}{\gamma (\omega (\nu ^{2}+1)-1)}$$
$$F^{+}_{PZ} (\sqrt{\frac{\omega \nu ^{2}}{1-\omega }},\frac{\beta \nu (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{\sqrt{\omega }(1-\omega )} )$$ stable focus if $$\gamma <\frac{\sqrt{1-\omega }-2\omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{2h\omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}$$; unstable focus if $$\gamma >\frac{\sqrt{1-\omega }-2\omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{2h\omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}$$; non-hyperbolic if $$~\gamma =\frac{\sqrt{1-\omega }-2\omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{2h\omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}$$; stable node if $$\gamma <\frac{(h\beta -2)\sqrt{1-\omega }-2h \beta \omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{h^{2} \beta \omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}$$; unstable node if $$\gamma >\frac{(h\beta -2)\sqrt{1-\omega }-2h \beta \omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}{h^{2} \beta \omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}} )}$$; non-hyperbolic if $$\gamma =\frac{(h\beta -2)\sqrt{1-\omega }-2h \beta \omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}})}{h^{2} \beta \omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}})}$$