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Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources. (English) Zbl 1446.35021

Summary: We study the pattern generating mechanism of a generalized Gierer-Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer-Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature.

MSC:

35B36 Pattern formations in context of PDEs
35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
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