<p style='text-indent:20px;'>In this paper we classify the phase portraits in the Poincaré disc of the Selkov model for the glycolysis process</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \dot{x}\, = \, -x+a y+x^2 y, \quad \dot{y} \, = \, b-a y-x^2 y, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in function of its parameters <inline-formula><tex-math id="M1">\begin{document}$ a, b \in \mathbb{R} $\end{document}</tex-math></inline-formula>. In particular we determine the regions in the parameter plane with biological meaning, i.e. with <inline-formula><tex-math id="M2">\begin{document}$ a $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ x $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ y $\end{document}</tex-math></inline-formula> positive.</p>

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The problem of distinguish between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincaré disc of the centers of this family having degree 2, 4 and 6.

Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).