Phase portraits of the Higgins–Selkov system

In this paper we study the dynamics of the Higgins–Selkov system\begin{document}$ \begin{equation*} \dot{x} = 1-xy^\gamma, \quad\dot{y} = \alpha y(xy^{\gamma -1}-1), \end{equation*} $\end{document}where \begin{document}$ \alpha $\end{document} is a real parameter and \begin{document}$ \gamma>1 $\end{document} is an integer. We classify the phase portraits of this system for \begin{document}$ \gamma = 3, 4, 5, 6, $\end{document} in the Poincaré disc for all the values of the parameter \begin{document}$ \alpha $\end{document} . Moreover, we determine in function of the parameter \begin{document}$ \alpha $\end{document} the regions of the phase space with biological meaning.