scholarly journals Spot patterns in the 2‐D Schnakenberg model with localized heterogeneities

2021 ◽  
Author(s):  
Tony Wong ◽  
Michael J. Ward
Keyword(s):  
Schnakenberg Model

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)’.

scholarly journals Pattern formation of a coupled two-cell Schnakenberg model

2017 ◽  
Vol 10 (5) ◽  
pp. 1051-1062
Author(s):  
Guanqi Liu ◽  
◽  
Yuwen Wang ◽  
Keyword(s):  
Pattern Formation ◽  
Schnakenberg Model

On the Poincaré–Hill cycle map of rotational random walk: locating the stochastic limit cycle in a reversible Schnakenberg model

2009 ◽  
Vol 466 (2115) ◽  
pp. 771-788 ◽  
Author(s):  
Melissa Vellela ◽  
Hong Qian
Keyword(s):  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Quantitative Measure ◽  
Power Spectral Analysis ◽  
Chemical System ◽  
Deterministic System ◽  
Stable Limit ◽  
Novel Device ◽  
Schnakenberg Model ◽  
Cycle Map

Recent studies on stochastic oscillations mostly focus on the power spectral analysis. However, the power spectrum yields information only on the frequency of oscillation and cannot differentiate between a stable limit cycle and a stable focus. The cycle flux, introduced by Hill (Hill 1989 Free energy transduction and biochemical cycle kinetics ), is a quantitative measure of the net movement over a closed path, but it is impractical to compute for all possible cycles in systems with a large state space. Through simple examples, we introduce concepts used to quantify stochastic oscillation, such as the cycle flux, the Hill–Qian stochastic circulation and rotation number. We introduce a novel device, the Poincaré–Hill cycle map (PHCM), which combines the concept of Hill’s cycle flux with the Poincaré map from nonlinear dynamics. Applying the PHCM to a reversible extension of an oscillatory chemical system, the Schnakenberg model, reveals stable oscillations outside the Hopf bifurcation region in which the deterministic system contains a limit cycle. Bistable behaviour is found on the small volume scale with high probabilities around both the fixed point and the limit cycle. Convergence to the deterministic system is found in the thermodynamic limit.

scholarly journals The Schnakenberg model with precursors

2019 ◽  
Vol 39 (4) ◽  
pp. 1923-1955
Author(s):  
Weiwei Ao ◽  
◽  
Chao Liu
Keyword(s):  
Schnakenberg Model

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>

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