Denote by CH, CSH, CQH, and CSQH the planar cubic homogeneous, cubic semi-homogeneous, cubic quasi-homogeneous and cubic semiquasi-homogeneous differential systems, respectively. The problems on limit cycles and global dynamics of these systems have been solved for CH, and partially for CSH. This paper studies the same problems for CQH and CSQH. We prove that CQH have no limit cycles and CSQH can have at most one limit cycle with the limit cycle realizable. Moreover, we classify all the global phase portraits of CSQH.

<p style='text-indent:20px;'>In this paper, we consider the model of the tumor and immune system, proposed by Han, He and Kuang (Discrete and Contin. Dyn. Syst. Ser. S 13:2347–2363, 2020). We give the necessary and sufficient conditions for the model to have no tumor equilibrium, one, two or three tumor equilibria respectively. Moreover, we prove that the model has no periodic solution, and give its global dynamics in the first quadrant. We find that the magnitude of the tumor reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> relative to 1 can describe the final state of tumor and the level of the immune system. If <inline-formula><tex-math id="M2">\begin{document}$ R_0\leq1 $\end{document}</tex-math></inline-formula>, the tumor will be cleaned out by immune system in the end. However, if <inline-formula><tex-math id="M3">\begin{document}$ R_0&gt;1 $\end{document}</tex-math></inline-formula>, the model appears monostable or bistable, i.e., the tumor coexists with the immune system and they are ultimately in a stable steady state.</p>