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

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  • Wu, Ranchao
  • Zhou, Yue
  • Shao, Yan
  • Chen, Liping

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.

Suggested Citation

  • Wu, Ranchao & Zhou, Yue & Shao, Yan & Chen, Liping, 2017. "Bifurcation and Turing patterns of reaction–diffusion activator–inhibitor model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 597-610.
    • Handle: RePEc:eee:phsmap:v:482:y:2017:i:c:p:597-610
    DOI: 10.1016/j.physa.2017.04.053
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    References listed on IDEAS

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    1. Mai, F.X. & Qin, L.J. & Zhang, G., 2012. "Turing instability for a semi-discrete Gierer–Meinhardt system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2014-2022.
    2. dos S. Silva, F.A. & Viana, R.L. & Lopes, S.R., 2015. "Pattern formation and Turing instability in an activator–inhibitor system with power-law coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 487-497.
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    Cited by:

    1. Zhao, Qiuyue & Liu, Shutang & Niu, Xinglong, 2019. "Dynamic behavior analysis of a diffusive plankton model with defensive and offensive effects," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 94-102.
    2. Wang, Jinliang & Li, You & Zhong, Shihong & Hou, Xiaojie, 2019. "Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 1-17.

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