摘要：The Gierer-Meinhardt model is a prototypical and significant activator-inhibitor model of the reaction-diffusion system. This paper focuses on the Turing instabilities analysis and pattern formation in the general Gierer-Meinhardt model. Based on the analysis of eigenvalues of the eigenpolynomial, we derive the existence and stability of the positive equilibrium. By using center manifold theory, the critical value and type of Hopf bifurcation are obtained. The effects of diffusions on the stability of the equilibrium and the bifurcated limit cycle are studied by employing normal form and center manifold reduction. The results show that the equilibrium undergoes a supercritical Hopf bifurcation. If the diffusion coefficients of the two species are sufficiently different, the stable equilibrium and the limit cycle will occur Turing instability, respectively. Moreover, we perform the numerical simulations for the derived results, which states clearly that the Turing patterns are either spot or stripe patterns.
