Table 2 Topological classifications around fixed points of phytoplankton–zooplankton model (8)

\(F_{00}(0,0) \)

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 \(\gamma =\frac{(h\beta -2)\sqrt{1-\omega }-2h \beta \omega (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}})}{h^{2} \beta \omega (1-\omega ) (\sqrt{1-\omega }-\sqrt{\omega \nu ^{2}})}\)