Bifurcation and Turing patterns of reaction–diffusion activator–inhibitor model

Ranchao Wu, Yue Zhou, Yan Shao and Liping Chen

Physica A: Statistical Mechanics and its Applications, 2017, vol. 482, issue C, 597-610

Abstract: Gierer–Meinhardt system is one of prototypical pattern formation models. Turing instability could induce various patterns in this system. Hopf bifurcation analysis and its direction are performed on such diffusive model in this paper, by employing normal form and center manifold reduction. The effects of diffusion on the stability of equilibrium point and the bifurcated limit cycle from Hopf bifurcation are investigated. It is found that under some conditions, diffusion-driven instability, i.e, Turing instability, about the equilibrium point and the bifurcated limit cycle will happen, which are stable without diffusion. Those diffusion-driven instabilities will lead to the occurrence of spatially nonhomogeneous solutions. As a result, some patterns, like stripe and spike solutions, will form. The explicit criteria about the stability and instability of the equilibrium point and the limit cycle in the system are derived, which could be readily applied. Further, numerical simulations are presented to illustrate theoretical analysis.

Keywords: Hopf bifurcation; Turing instability; Pattern; Activator–inhibitor model (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:482:y:2017:i:c:p:597-610

DOI: 10.1016/j.physa.2017.04.053

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