Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type

1992 ◽  
Vol 9 (1) ◽  
pp. 35-77 ◽  
Author(s):  
M. Kuwamura ◽  
S. -I. Ei ◽  
M. Mimura
Keyword(s):  
Reaction Diffusion ◽  
Slow Dynamics ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Inhibitor Type ◽  
Activator Inhibitor Type ◽  
Activator Inhibitor

scholarly journals Slow dynamics in reaction–diffusion systems

2016 ◽  
Vol 98 (1-2) ◽  
pp. 131-154 ◽  
Author(s):  
Marta Strani
Keyword(s):  
Reaction Diffusion ◽  
Slow Dynamics ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Linearized System ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Activator Inhibitor ◽  
Fractional Reaction

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Fahad Al Saadi ◽  
Alan Champneys
Keyword(s):  
Reaction Diffusion ◽  
Complete Sequence ◽  
Numerical Continuation ◽  
Bifurcation Diagrams ◽  
Unified Framework ◽  
Reaction Diffusion Systems ◽  
Codimension Two ◽  
Diffusion Systems ◽  
Infinite Line ◽  
Activator Inhibitor

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray–Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray–Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer–Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals Global Solutions for an -Component System of Activator-Inhibitor Type

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
S. Abdelmalek ◽  
A. Gouadria ◽  
A. Youkana
Keyword(s):  
Lyapunov Functional ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Component System ◽  
Inhibitor Type ◽  
The One ◽  
Activator Inhibitor Type ◽  
Activator Inhibitor

This paper deals with a reaction-diffusion system with fractional reactions modeling -substances into interaction following activator-inhibitor's scheme. The existence of global solutions is obtained via a judicious Lyapunov functional that generalizes the one introduced by Masuda and Takahashi.

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