Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system

2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Stephan Hausberg ◽  
Matthias Röger
Keyword(s):  
Surface Reaction ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Diffusion Limit ◽  
Fast Diffusion ◽  
Well Posedness ◽  
Bulk Surface

scholarly journals A bulk-surface reaction-diffusion system for cell polarization

2020 ◽  
Vol 22 (1) ◽  
pp. 85-117
Author(s):  
Barbara Niethammer ◽  
Matthias Röger ◽  
Juan Velázquez
Keyword(s):  
Surface Reaction ◽  
Reaction Diffusion ◽  
Cell Polarization ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Bulk Surface

scholarly journals Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140546 ◽  
Author(s):  
Anotida Madzvamuse ◽  
Andy H. W. Chung ◽  
Chandrasekhar Venkataraman
Keyword(s):  
Surface Reaction ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Reaction Diffusion Equations ◽  
Diffusion System ◽  
Diffusion Equations ◽  
Surface Dynamics ◽  
Bulk Reaction ◽  
Bulk Surface ◽  
Type Boundary

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.

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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