Abstract: We considered a class of Gierer-Meinhardt models without  diffusion term. Firstly, we studied the existence,  the stability of equilibrium points and the various bifurcation phenomena of the model. Secondly, by choosing proper bifurcation parameters, we proved that the system had saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation of codimensions 2. Finally, the theoretical results were demonstrated by numerical simulations. The results show that the rate μ influences the instability of system.

Key words: Gierer-Meinhardt model, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation