The explicit solution to the initial–boundary value problem of Gierer–Meinhardt model

An, Xiaowei; He, Zhen; Song, Xianfa

doi:10.1016/j.aml.2017.12.024

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…Ghergu [13] has considered steady state solutions in the Gierer-Meinhardt system with Dirichlet boundary condition. An et al [14] have studied the explicit solution to the initial-boundary value problem of Gierer-Meinhardt model under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

Gong

Wang

2020

Discrete Dynamics in Nature and Society

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The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns.

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Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

Gong

Wang

2020

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

show abstract

Regulating spatiotemporal dynamics for a delay Gierer–Meinhardt model

He,

Xiao,

et al. 2024

Physica A: Statistical Mechanics and its Applications

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Turing–Hopf Bifurcation in Diffusive Gierer–Meinhardt Model

Yang

2022

Int. J. Bifurcation Chaos

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Gierer–Meinhardt system is a molecularly plausible model to describe pattern formation. When gene expression time delay is added, the behavior of the Gierer–Meinhardt model profoundly changes. In this paper, we study the delayed reaction–diffusion Gierer–Meinhardt system with Neumann boundary condition. Necessary and sufficient conditions for the occurrence of Turing instability, Hopf bifurcation and Turing–Hopf bifurcation deduced by diffusion and gene expression time delay are obtained through linear stability analysis and root distribution of the characteristic equation with two transcendental terms. With the aid of the normal form Turing–Hopf bifurcation and numerical simulations, we theoretically and numerically obtain the expected solutions including stable spatially inhomogeneous steady states, stable spatially homogeneous periodic orbit and stable spatially inhomogeneous periodic orbit from Turing–Hopf bifurcation.

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The explicit solution to the initial–boundary value problem of Gierer–Meinhardt model

Cited by 3 publications

References 12 publications

Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

Regulating spatiotemporal dynamics for a delay Gierer–Meinhardt model

Turing–Hopf Bifurcation in Diffusive Gierer–Meinhardt Model

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