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Regulating spatiotemporal dynamics for a delay Gierer–Meinhardt model

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  • He, Haoming
  • Xiao, Min
  • He, Jiajin
  • Zheng, Weixing

Abstract

As a classical activator-inhibitor system with diffusion effects, the Gierer–Meinhardt (GM) model has received a considerable attention in recent years. Discussions on dynamic behaviors of the GM model are well underway. However, we still lack the means to further detect the bifurcation direction and the stability of bifurcated periodic solutions as a Hopf bifurcation appears in the space–time evolution. Besides, control strategies for regulating the space-time dynamics to achieve the desired biological patterns are also neglected. These are the tasks that this paper attempts to tackle. Through Hopf bifurcation and Turing instability theorems, it is demonstrated that by adjusting the controller parameters appropriately, the occurrence of the Hopf bifurcation can be delayed and the threshold of Turing instability can also be broadened. Go a step further, we have deliberated the bifurcation direction of the controlled GM model under the influence of self-diffusion coefficients. As a result, the control effect is verified by comparison in numerical simulations.

Suggested Citation

  • He, Haoming & Xiao, Min & He, Jiajin & Zheng, Weixing, 2024. "Regulating spatiotemporal dynamics for a delay Gierer–Meinhardt model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 637(C).
    • Handle: RePEc:eee:phsmap:v:637:y:2024:i:c:s0378437124001110
    DOI: 10.1016/j.physa.2024.129603
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    References listed on IDEAS

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    1. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    2. Duan, Moran & Chang, Lili & Jin, Zhen, 2019. "Turing patterns of an SI epidemic model with cross-diffusion on complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
    3. Si, Lingzhi & Xiao, Min & Wang, Zhengxin & Huang, Chengdai & Cheng, Zunsgui & Tao, Binbin & Xu, Fengyu, 2019. "Dynamic optimal control at Hopf bifurcation of a Newman–Watts model of small-world networks via a new PD1n scheme," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 532(C).
    4. 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.
    5. Mukherjee, Nayana & Banerjee, Malay, 2022. "Hunting cooperation among slowly diffusing specialist predators can induce stationary Turing patterns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 599(C).
    6. 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.
    7. Liu, Haicheng & Ge, Bin, 2022. "Turing instability of periodic solutions for the Gierer–Meinhardt model with cross-diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    8. 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.
    9. Ruizhi Yang & Xiao Zhao & Yong An, 2022. "Dynamical Analysis of a Delayed Diffusive Predator–Prey Model with Additional Food Provided and Anti-Predator Behavior," Mathematics, MDPI, vol. 10(3), pages 1-18, January.
    10. Yao, Shao-Wen & Ma, Zhan-Ping & Yue, Jia-Long, 2018. "Bistability and Turing pattern induced by cross fraction diffusion in a predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 982-988.
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