Multi-spike solutions of a hybrid reaction–transport model

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 .

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CHEMICAL PATTERNS IN SIMPLE FLOW SYSTEMS

Advances in Complex Systems ◽

10.1142/s0219525903000694 ◽

2003 ◽

Vol 06 (01) ◽

pp. 155-162 ◽

Cited By ~ 3

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.

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

Discrete and Continuous Dynamical Systems - Series S ◽

10.3934/dcdss.2021119 ◽

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

Journal of The Royal Society Interface ◽

10.1098/rsif.2019.0621 ◽

2020 ◽

Vol 17 (162) ◽

pp. 20190621 ◽

Cited By ~ 5

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.

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Turing Patterning in Stratified Domains

Bulletin of Mathematical Biology ◽

10.1007/s11538-020-00809-9 ◽

2020 ◽

Vol 82 (10) ◽

Author(s):

Andrew L. Krause ◽

Václav Klika ◽

Jacob Halatek ◽

Paul K. Grant ◽

Thomas E. Woolley ◽

...

Keyword(s):

Diffusion Processes ◽

Reaction Diffusion ◽

Turing Instability ◽

Cell Polarization ◽

The Other ◽

Modeling Framework ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Coupling Parameters ◽

Pattern Forming

Abstract Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

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