Turing and Hopf bifurcation of Gierer-Meinhardt activator-substrate model

Abstract

Gierer-Meinhardt model acts as one of prototypical reaction diffusion systems describing pattern formation phenomena in natural events. Bifurcation analysis, including theoretical and numerical analysis, is carried out on the Gierer-Meinhardt activator-substrate model. The effects of diffusion on the stability of equilibrium points and the bifurcated limit cycle from Hopf bifurcation are investigated. It shows that under some conditions, diffusion-driven instability, i.e, the Turing instability, about the equilibrium point will occur, which is stable without diffusion. While once the diffusive effects are present, the bifurcated limit cycle, which is the spatially homogeneous periodic solution and stable without the presence of diffusion, will become unstable. These diffusion-driven instabilities will lead to the occurrence of spatially nonhomogeneous solutions. Consequently, some pattern formations, like stripe and spike solutions, will appear. To understand the Turing and Hopf bifurcation in the system, we use dynamical techniques, such as stability theory, normal form and center manifold theory. To illustrate theoretical analysis, we carry out numerical simulations.

Keywords

• Pattern formation
• Gierer-Meinhardt model
• Turing instability
• Hopf bifurcation