scholarly journals A rigorous reduction of the L2-stability of the solutions to a nonlinear binary reaction–diffusion system of PDE's to the stability of the solutions to a linear binary system of ODE's

2006 ◽  
Vol 319 (2) ◽  
pp. 377-397 ◽  
Author(s):  
S. Rionero
Keyword(s):  
Binary System ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
The Stability ◽  
Binary Reaction

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

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Lianchao Gu ◽  
Peiliang Gong ◽  
Hongqing Wang
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Turing Patterns ◽  
System Parameters ◽  
System Equation ◽  
Representative Model ◽  
The Stability

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.

scholarly journals POSITIVE SOLUTIONS BIFURCATING FROM ZERO SOLUTION IN A PREDATOR-PREY REACTION–DIFFUSION SYSTEM

2011 ◽  
Vol 16 (4) ◽  
pp. 558-568 ◽  
Author(s):  
Cun-Hua Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Reduction Method ◽  
Reaction Diffusion ◽  
Zero Solution ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Prey ◽  
The Stability

An elliptic system subject to the homogeneous Dirichlet boundary con- dition denoting the steady-state system of a two-species predator-prey reaction– diffusion system with the modified Leslie–Gower and Holling-type II schemes is con- sidered. By using the Lyapunov–Schmidt reduction method, the bifurcation of the positive solution from the trivial solution is demonstrated and the approximated ex- pressions of the positive solutions around the bifurcation point are also given accord- ing to the implicit function theorem. Finally, by applying the linearized method, the stability of the bifurcating positive solution is also investigated. The results obtained in the present paper improved the existing ones.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

scholarly journals The human EDAR 370V/A polymorphism affects tooth root morphology potentially through the modification of a reaction–diffusion system

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Keiichi Kataoka ◽  
Hironori Fujita ◽  
Mutsumi Isa ◽  
Shimpei Gotoh ◽  
Akira Arasaki ◽  
...  
Keyword(s):  
Molecular Mechanisms ◽  
Computational Study ◽  
Reaction Diffusion ◽  
Human Migration ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Tooth Root ◽  
Computed Tomography Images ◽  
Diffusion Dynamics ◽  
Root Morphogenesis

AbstractMorphological variations in human teeth have long been recognized and, in particular, the spatial and temporal distribution of two patterns of dental features in Asia, i.e., Sinodonty and Sundadonty, have contributed to our understanding of the human migration history. However, the molecular mechanisms underlying such dental variations have not yet been completely elucidated. Recent studies have clarified that a nonsynonymous variant in the ectodysplasin A receptor gene (EDAR370V/A; rs3827760) contributes to crown traits related to Sinodonty. In this study, we examined the association between theEDARpolymorphism and tooth root traits by using computed tomography images and identified that the effects of theEDARvariant on the number and shape of roots differed depending on the tooth type. In addition, to better understand tooth root morphogenesis, a computational analysis for patterns of tooth roots was performed, assuming a reaction–diffusion system. The computational study suggested that the complicated effects of theEDARpolymorphism could be explained when it is considered that EDAR modifies the syntheses of multiple related molecules working in the reaction–diffusion dynamics. In this study, we shed light on the molecular mechanisms of tooth root morphogenesis, which are less understood in comparison to those of tooth crown morphogenesis.

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