Turing instability in reaction–diffusion models on complex networks

Ide, Yusuke; Izuhara, Hirofumi; Machida, Takuya

doi:10.1016/j.physa.2016.03.055

Cited by 26 publications

(25 citation statements)

References 27 publications

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“… 2014 ; Ide et al. 2016 ; Nakao and Mikhailov 2010 ), and many of the aforementioned complex system geometries (Halatek and Frey 2012 ; Klünder et al. 2013 ; Halatek and Frey 2018 ).…”

Section: Introductionmentioning

confidence: 99%

Turing Patterning in Stratified Domains

et al. 2020

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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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“… 2014 ; Ide et al. 2016 ; Nakao and Mikhailov 2010 ), and many of the aforementioned complex system geometries (Halatek and Frey 2012 ; Klünder et al. 2013 ; Halatek and Frey 2018 ).…”

Section: Introductionmentioning

confidence: 99%

Turing Patterning in Stratified Domains

et al. 2020

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“…en, the Turing instability of the reaction-diffusion model defined on the complex network was studied in [19], and three types of models on the complex network were shown in the article. Numerical results showed that the uniform steady-state stability region depends on the network system's structure in the diffusion coefficient space.…”

Section: Introductionmentioning

confidence: 99%

Turing Instability of Brusselator in the Reaction-Diffusion Network

Shen

2020

Complexity

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Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

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“…Plane-wave driven Turing instability was studied on one and two dimensional lattices, where the wavefunctions and wave-numbers in continuous media are replaced by their discrete analogs corresponding to the eigenvalue and eigenvectors of the Laplacian matrix. In recent years, Turing patterns have been shown to exist in reaction-diffusion processes occurring on complex networks, where nodes in the graph are assigned an initial concentration of chemical species, and diffusion occurs along the edges connecting the nodes [12,13]. A small perturbation to the uniform state triggers the growth of Turing patterns above a critical threshold, corresponding to the ratio of the diffusion constants of the respective species.…”

Section: Introductionmentioning

confidence: 99%

“…A similar set of equations can be used if the system is not continuous but instead is composed of N independent nodes that interact via diffusive transport over m edges [12,14,22,37]. The analog of the operator ∇ 2 is now the Laplacian matrix,…”

Section: Introductionmentioning

confidence: 99%

Turing patterns mediated by network topology in homogeneous active systems

et al. 2019

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Mechanisms of pattern formation-of which the Turing instability is an archetype-constitute an important class of dynamical processes occurring in biological, ecological and chemical systems. Recently, it has been shown that the Turing instability can induce pattern formation in discrete media such as complex networks, opening up the intriguing possibility of exploring it as a generative mechanism in a plethora of socioeconomic contexts. Yet, much remains to be understood in terms of the precise connection between network topology and its role in inducing the patterns. Here, we present a general mathematical description of a two-species reaction-diffusion process occurring on different flavors of network topology. The dynamical equations are of the predator-prey class, that while traditionally used to model species population, has also been used to model competition between antagonistic ideas in social systems. We demonstrate that the Turing instability can be induced in any network topology, by tuning the diffusion of the competing species, or by altering network connectivity. The extent to which the emergent patterns reflect topological properties is determined by a complex interplay between the diffusion coefficients and the localization properties of the eigenvectors of the graph Laplacian. We find that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic. arXiv:1903.03845v1 [cond-mat.dis-nn]

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Turing instability in reaction–diffusion models on complex networks

Cited by 26 publications

References 27 publications

Turing Patterning in Stratified Domains

Turing Patterning in Stratified Domains

Turing Instability of Brusselator in the Reaction-Diffusion Network

Turing patterns mediated by network topology in homogeneous active systems

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