Diffusion-driven instability and bifurcation in the Lengyel–Epstein system

Yi, Fengqi; Wei, Junjie; Shi, Junping

doi:10.1016/j.nonrwa.2007.02.005

Cited by 146 publications

(72 citation statements)

References 10 publications

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“…We use r as the main bifurcation parameter. To identify possible Hopf bifurcation value r H , we recall the following necessary and sufficient condition from [2,8,9]:…”

Section: Pattern Formation In a General Degn-harrison Reaction Model 663mentioning

confidence: 99%

Pattern Formation in a General Degn-Harrison Reaction Model

Zhou¹

2017

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we study the pattern formation to a general Degn-Harrison reaction model. We show Turing instability happens by analyzing the stability of the unique positive equilibrium with respect to the PDE model and the corresponding ODE model, which indicate the existence of the non-constant steady state solutions. We also show the existence periodic solutions of the PDE model and the ODE model by using Hopf bifurcation theory. Numerical simulations are presented to verify and illustrate the theoretical results.

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“…We use r as the main bifurcation parameter. To identify possible Hopf bifurcation value r H , we recall the following necessary and sufficient condition from [2,8,9]:…”

Section: Pattern Formation In a General Degn-harrison Reaction Model 663mentioning

confidence: 99%

Pattern Formation in a General Degn-Harrison Reaction Model

Zhou¹

2017

Bulletin of the Korean Mathematical Society

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“…A complete understanding of the asymptotical behavior of solutions to Lengyel-Epstein system (3.4) or any other systems with complicated spatiotemporal dynamics is still beyond reach. Yi, Wei and Shi [44,46] showed that (3.4) also possess periodic orbits through Hopf bifurcation for certain parameters, but the dynamics is relatively simpler when a is small, or when a is large but the domain Ω is small. Jin, Shi, Wei and Yi 1) consider the global bifurcation diagrams with intervening steady state and Hopf bifurcations.…”

Section: Turing Bifurcationmentioning

confidence: 99%

Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models

Shi

2009

Front. Math. China

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“…For more results about reaction diffusion systems of activator-inhibitor type, see [19][20][21]. See [22][23][24] and the references therein for more real-world models on Turing instability. So far, most work on Turing instability of system (1) has been carried out by taking specific parameter values.…”

Section: Introductionmentioning

confidence: 99%

Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

Wang

Hou

2018

Adv Differ Equ

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We study the pattern generating mechanism of a generalized Gierer-Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer-Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature.

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Diffusion-driven instability and bifurcation in the Lengyel–Epstein system

Cited by 146 publications

References 10 publications

Pattern Formation in a General Degn-Harrison Reaction Model

Pattern Formation in a General Degn-Harrison Reaction Model

Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models

Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

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