scholarly journals Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Weihua Jiang ◽  
Xun Cao ◽  
Chuncheng Wang
Keyword(s):  
Bounded Domain ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Pattern Formations

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Acharya ◽  
N. Fonseka ◽  
R. Shivaji
Keyword(s):  
Steady State ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Normal Derivative ◽  
Positive Parameter ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Increasing Functions

AbstractWe study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ { − Δ u = λ f ( v ) ; Ω , − Δ v = λ g ( u ) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where $\lambda >0$ λ > 0 is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ R N $(N > 1)$ ( N > 1 ) with smooth boundary ∂Ω, or $\Omega =(0,1)$ Ω = ( 0 , 1 ) , $\frac{\partial z}{\partial \eta }$ ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ f ( 0 ) = 0 = g ( 0 ) and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ lim s → ∞ f ( M g ( s ) ) s = 0 for all $M>0$ M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

scholarly journals The space spectral interpolation collocation method for reaction-diffusion systems

2021 ◽  
pp. 22-22
Author(s):  
Xiao-Li Zhang ◽  
Wei Zhang ◽  
Yu-Lan Wang ◽  
Ting-Ting Ban
Keyword(s):  
Collocation Method ◽  
Complex Dynamics ◽  
Solution Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spectral Interpolation ◽  
Fractal Calculus ◽  
Pattern Formations ◽  
Good Agreement

A space spectral interpolation collocation method is proposed to study nonlinear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.

scholarly journals Turing Patterning in Stratified Domains

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.

scholarly journals Analysis of stochastic sensitivity of Turing patterns in distributed reaction-diffusion systems

2020 ◽  
Vol 55 ◽  
pp. 155-163
Author(s):  
A.P. Kolinichenko ◽  
L.B. Ryashko
Keyword(s):  
Diffusion Coefficients ◽  
Stochastic Dynamics ◽  
Random Noise ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Stochastic Sensitivity ◽  
Diffusion Systems ◽  
The Mean

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed.

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