Global Bifurcation and Structure of Turing Patterns in the 1-D Lengyel?Epstein Model

Jang, Jaeduck; Ni, Wei Ming; Tang, Moxun

doi:10.1007/s10884-004-2782-x

Cited by 104 publications

(42 citation statements)

References 19 publications

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“…Moreover, we described ideas which connect differential equations with cellular automata. More mathematical analysis of the Lengyel-Epstein model may be found in [7]. General theory about Turing patterns and their analysis are explained in [12].…”

Section: Discussionmentioning

confidence: 99%

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems

Suzuki

2011

IIS

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In 1952, A. M. Turing proposed the notion of ''diffusion-driven instability'' in his attempt of modeling biological pattern formation. Following his ingenious idea, many reaction-diffusion systems have been proposed later on. On the other hand, Turing patterns can be explained by some cellular automata. Cellular automata are theoretical models which consist of a regular grid of cells, and they exhibit the complex behavior from quite simple rules. In this paper, we describe the mathematical properties of reaction-diffusion systems modeling pattern formation, in particular, Turing patterns. Moreover, we explain ideas which connect differential equations with cellular automata.

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Section: Discussionmentioning

confidence: 99%

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems

Suzuki

2011

IIS

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“…China. 2 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, P.R. China.…”

Section: Competing Interestsmentioning

confidence: 99%

Spatiotemporal patterns in the Lengyel-Epstein reaction-diffusion model

Dong

Zhang

2016

Adv Differ Equ

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In this paper, we continue the mathematical study started in (Jang et al. in J. Dyn. Differ. Equ. 16:297-320, 2004; Ni and Tang in Trans. Am. Math. Soc. 357:3953-3969, 2005) on the analytic aspects of the Lengyel-Epstein reaction-diffusion system. First, we further analyze the fundamental properties of nonconstant positive solutions. On the other hand, we continue to consider the effect of the diffusion coefficient d. We obtain another nonexistence result for the case of large d by the implicit function theory, and investigate the direction of bifurcation solutions from (u * , v * ). These results promote the Turing patterns arising from the Lengyel-Epstein reaction-diffusion system.

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“…In fact, in Ref. [18], it was showed that all the bifurcating branches are unbounded in positive ddirection but bounded in (u, v) norm with the a priori estimates. 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.…”

Section: Turing Bifurcationmentioning

confidence: 99%