Turing instability and pattern formation in reaction-diffusion models

Patterns are found everywhere and the past fifty years studies have advanced our understanding of the mechanisms . In this paper, we study those systems that develop temporary patterns. Special emphasis is made on Turing instabilities as one of the most common sources of pattern formation. 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 is 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. These diffusion-driven instabilities will lead to the occurrence of spatially nonhomogeneous solutions. Consequently, some pattern formations, like stripe and spots solutions, will appear. To illustrate theoretical analysis, we carry out numerical simulations. These diffusion-driven instabilities will lead to the occurrence of spatially nonhomogeneous solutions. Consequently, some pattern formations, like stripe and spots solutions, will appear. To illustrate theoretical analysis, we carry out numerical simulations.