Abstract
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.
Highlights
Pattern formation can be induced by uneven diffusions
Chemical compounds can interact with each other and spread in space in some ways, which result in heterogeneous spatial patterns of chemical compound or morphogen concentration [2]
Reaction–diffusion equations and systems can characterize a substantial number of pattern-related biology phenomena
Summary
Pattern formation can be induced by uneven diffusions. It was first discovered by Turing [1] in the 1950s. In [9], the authors investigated the same system but with saturated activator production under Neumann boundary conditions in the interval (0, π) and showed the existence of Turing instabilities of the positive spatial homogeneous equilibrium and homogenous periodic solution. They found that there are at least two limit cycles. When r = T, s = u, and ρ = ρ , system (1) is called the general form of activator–inhibitor system with common sources [3], which was studied in [18] They obtained a precise parametric condition for the presence of Turing instability. From the simulations we see spot and stripe spatial patterns
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
