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Turing instability for a semi-discrete Gierer–Meinhardt system

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  • Mai, F.X.
  • Qin, L.J.
  • Zhang, G.

Abstract

In this paper, we investigate the spatial patterns of a Gierer–Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:5:p:2014-2022
    DOI: 10.1016/j.physa.2011.11.034
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    References listed on IDEAS

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    1. Iron, David & Ward, Michael J., 2001. "Spike pinning for the Gierer–Meinhardt model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(4), pages 419-431.
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    Cited by:

    1. Zhong, Shihong & Xia, Juandi & Liu, Biao, 2021. "Spatiotemporal dynamics analysis of a semi-discrete reaction-diffusion Mussel-Algae system with advection," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    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.
    3. 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.

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    1. 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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