scholarly journals The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guido Schneider ◽  
Matthias Winter
Keyword(s):  
Numerical Simulation ◽  
Analytical Approach ◽  
Reaction Diffusion ◽  
Short Wave ◽  
Reaction Diffusion Systems ◽  
Long Wave ◽  
Diffusion Systems ◽  
Schnakenberg Model ◽  
Pattern Forming ◽  
Amplitude System

<p style='text-indent:20px;'>We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.</p>

scholarly journals Stabilizing a homoclinic stripe

2018 ◽  
Vol 376 (2135) ◽  
pp. 20180110 ◽  
Author(s):  
Theodore Kolokolnikov ◽  
Michael Ward ◽  
Justin Tzou ◽  
Juncheng Wei
Keyword(s):  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Vegetation Patterns ◽  
Theme Issue ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Schnakenberg Model ◽  
Break Up ◽  
Variable Substrate

For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

CHEMICAL PATTERNS IN SIMPLE FLOW SYSTEMS

2003 ◽  
Vol 06 (01) ◽  
pp. 155-162 ◽  
Author(s):  
ANNETTE TAYLOR
Keyword(s):  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Stationary Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Differential Flow ◽  
Chemical Waves ◽  
Pattern Forming ◽  
Simple Flow ◽  
Concentration Patterns

The addition of flow to chemical reaction-diffusion systems provides robust pattern-forming mechanisms which are expected to occur in a wide variety of natural and artificial systems. Experiments demonstrating some of these mechanisms are presented here, including the differential-flow-induced chemical instability (DIFICI), which gives rise to traveling chemical waves, and flow-distributed oscillations (FDO), which produce stationary concentration patterns.

scholarly journals Multi-spike solutions of a hybrid reaction–transport model

2021 ◽  
Vol 477 (2247) ◽  
pp. 20200829
Author(s):  
P. C. Bressloff
Keyword(s):  
Transport Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Strongly Nonlinear ◽  
Diffusion Systems ◽  
Spike Solution ◽  
Classical Pattern ◽  
Final Steady State ◽  
Reaction Transport ◽  
Pattern Forming

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit D a  ≪  D h , where D a and D h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans . We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α  → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α  → ∞ with a graphical construction for finite α .

scholarly journals From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ

2020 ◽  
Vol 17 (162) ◽  
pp. 20190621 ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Thomas E. Woolley ◽  
Eamonn A. Gaffney
Keyword(s):  
Symmetry Breaking ◽  
Heterogeneous Media ◽  
Heterogeneous Reaction ◽  
Reaction Diffusion ◽  
Spatial Patterning ◽  
Instability Mechanism ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Pattern Forming

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction–diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction–diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing’s original thesis to a far wider and more realistic class of systems.

Analytical approach to stationary wall solutions in bistable reaction-diffusion systems

1993 ◽  
Vol 176 (3-4) ◽  
pp. 207-212 ◽  
Author(s):  
R. Dohmen ◽  
F.-J. Niedernostheide ◽  
H. Willebrand ◽  
H.-G. Purwins
Keyword(s):  
Analytical Approach ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Stationary Wall ◽  
Diffusion Systems
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