Abstract: We considered a platelet derived growth factor (PDGF) driven reaction-diffusion glioma mathematical model. Firstly, we gave the stability analysis of the equilibrium point for the ordinary differential system. We took the  rate m generated by chemoattractant as  the bifurcation parameter, gave the existence of the Hopf bifurcation near the positive equilibrium point, and then gave a formula to judge the stability of the periodic solution produced by the Hopf bifurcation through the gauge type theory and the central manifold theorem. Secondly, for reaction-diffusion systems, we obtained that the equilibrium point  did not occur Turing instability  when diffusion was involved. Finally, the  theoretical analysis results were verified through numerical simulation. The results show that the rate m generated by chemoattractant can be used to distinguish the types of glioma.

Key words: tumor model, reaction diffusion, Hopf bifurcation, stability