scholarly journals An Explicitly Solvable Nonlocal Eigenvalue Problem and the Stability of a Spike for a Sub-Diffusive Reaction-Diffusion System

2013 ◽  
Vol 8 (2) ◽  
pp. 55-87 ◽  
Author(s):  
Y. Nec ◽  
M. J. Ward
Keyword(s):  
Eigenvalue Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
The Stability ◽  
Nonlocal Eigenvalue Problem

scholarly journals A Nonlocal Eigenvalue Problem and the Stability of Spikes for Reaction–Diffusion Systems with Fractional Reaction Rates

2003 ◽  
Vol 13 (06) ◽  
pp. 1529-1543 ◽  
Author(s):  
Juncheng Wei ◽  
Matthias Winter
Keyword(s):  
Eigenvalue Problem ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Stability ◽  
Nonlocal Eigenvalue Problem ◽  
Fractional Reaction

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction–diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray–Scott system, the hypercycle of Eigen and Schuster, angiogenesis, and the generalized Gierer–Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters.

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.

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