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Turing patterns in a model of the CDIMA reaction-diffusion system with constant illumination. Turing patterns at higher values of b are surrounded by a uniform homogeneous state and at lower b by homogeneous bulk oscillation ͑ BO ͒ . Columns in the table illustrate transformation of Turing patterns when illumination intensity is varied. Parameters: ϭ 9, a ϭ 36. Thick solid line: Hopf bifurcation line; thick dashed line: Turing line.
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We study the resonant behavior of Turing pattern suppression in a model of the chlorine dioxide-iodine-malonic acid reaction with periodic illumination. The results of simulations based on integration of partial differential equations display resonance at the frequency of autonomous oscillations in the corresponding well stirred system. The resonan...
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... subcritical transitions to Turing patterns have been reported earlier 5,22. Figure 2 displays patterns obtained for different values of b and w using Turing patterns as initial conditions. The thick lines in 2, which correspond to the Turing and Hopf lines, indi- cate the boundaries of the Turing pattern region. ...
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... 2 displays patterns obtained for different values of b and w using Turing patterns as initial conditions. The thick lines in 2, which correspond to the Turing and Hopf lines, indi- cate the boundaries of the Turing pattern region. Figure 2 illustrates that the Turing pattern can be modified not only by varying the input concentrations parameter b) but also by changing the intensity of uniform illumination. ...
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... thick lines in 2, which correspond to the Turing and Hopf lines, indi- cate the boundaries of the Turing pattern region. Figure 2 illustrates that the Turing pattern can be modified not only by varying the input concentrations parameter b) but also by changing the intensity of uniform illumination. For example, when b is fixed at 2.5 and w is gradually increased, the Tur- ing pattern changes from hexagons to mixed hexagons and stripes, stripes, stripes-honeycombs, and pure honeycombs before stronger illumination leads to total suppression of Turing patterns. ...
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... 30 Inspired by these findings and building on our previous study, we propose a two-variable photosensitive CDIMA model to explore the effects of circularly polarized electric fields (CPEF) in combination with periodic photo-illumination. 31,32 Specifically, our objectives are twofold: first, we aim to provide a concise analysis of the parametric space where various types of symmetry-breaking spatiotemporal patterns can be generated by applying CPEF with constant photo-illumination intensity and identify the dominant pattern in that region. Second, we intend to investigate the dual action of CPEF and periodic photo-illumination in the parametric region where the dominating patterns were observed in our preceding study. ...
Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.
... The control of spatio-temporal patterns is one of the attractive topics. Physical mechanisms for this issue include thermal forcing [45] and illumination [46] by nonlinear kinetics of reactions. According to our analysis in Case II, corresponding researches can be extended to the hypergraph structure and the control time can be set in a finite interval to reduce the cost, while time-dependent diffusion parameters in Case I can provide global control to tune the emergent patterns. ...
This paper examines the Turing patterns and the spatio-temporal chaos of non-autonomous systems defined on hypergraphs. The analytical conditions for Turing instability (TI) and Benjamin-Feir instability (BFI) are obtained by linear stability analysis using new comparison principles. The comparison with pairwise interactions is presented to reveal the effect of higher-order interactions on pattern formation. In addition, numerical simulations due to different non-autonomous mechanisms, such as time-varying diffusion coefficients, time-varying reaction kinetics and time-varying diffusion coupling are provided respectively, which verifies the efficiency of theoretical results.
... Some aspects of global periodic forcings and Turing pattern formation dynamics have been demonstrated recently by the periodic illumination of the light-sensitive chlorinedioxide-iodine-malonic acid (CDIMA) reaction, which is the core part of the CIMA reaction [28][29][30][31][32][33][34] . The experiments demonstrated that periodic illumination might suppress the patterns, especially at a frequency equal to the frequency of autonomous oscillations in a well-stirred reactor 28 . ...
... Some aspects of global periodic forcings and Turing pattern formation dynamics have been demonstrated recently by the periodic illumination of the light-sensitive chlorinedioxide-iodine-malonic acid (CDIMA) reaction, which is the core part of the CIMA reaction [28][29][30][31][32][33][34] . The experiments demonstrated that periodic illumination might suppress the patterns, especially at a frequency equal to the frequency of autonomous oscillations in a well-stirred reactor 28 . The detailed exploration of the spatially resonant forcing and the non-resonant case revealed entrained and oscillating patterns 33 . ...
Turing instability is a general and straightforward mechanism of pattern formation in reaction–diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction–diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing. The generation of stationary patterns is often studied under constant experimental conditions, but in biological systems parameters such as chemical flow are not stationary. Here, the authors use experiments and numerical analyses to elucidate the mechanisms controlling Turing patterns under periodic variations in chemical feed concentration.
... However, chemical systems have an advantage such that further exploration of the spatiotemporal dynamics in the presence of external fields or internal feedback is possible. For example, photoillumination, [15][16][17][18][19][20] parametric fluctuations, [21][22][23] timedelays and delayed-feedback, [24][25][26] thermal gradient 27 have been very effective in providing a route toward symmetry-breaking leading to pattern formation. ...
... To demonstrate the effect of a constant electric field in the pattern-forming process, we choose the modified Lengyel-Epstein model, which includes the photoillumination effect and has served as an experimental paradigm of two-variable reaction-diffusion system for decades. [15][16][17]26 The corresponding dimensionless reaction-diffusion equations are the following: ...
Reaction-diffusion systems involving ionic species are susceptible to an externally applied electric field. Depending on the charges on the ionic species and the intensity of the applied electric field, diverse spatiotemporal patterns can emerge. We here considered two prototypical reaction-diffusion systems that follow activator-inhibitor kinetics: the photosensitive chlorine dioxide-iodine-malonic acid (CDIMA) reaction and the Brusselator model. By theoretical investigation and numerical simulations, we unravel how and to what extent an externally applied electric field can induce and modify the dynamics of these two systems. Our results show that both the uni- and bi-directional electric fields may induce Turing-like stationary patterns from a homogeneous uniform state resulting in horizontal, vertical, or bent stripe-like inhomogeneity in the photosensitive CDIMA system. In contrast, in the Brusselator model, for the activator and the inhibitor species having the same positive or negative charges, the externally applied electric field cannot develop any spatiotemporal instability when the diffusion coefficients are identical. However, various spatiotemporal patterns emerge for the same opposite charges of the interacting species, including moving spots and stripe-like structures, and a phenomenon of wave-splitting is observed. Moreover, the same sign and different magnitudes of the ionic charges can give rise to Turing-like stationary patterns from a homogeneous, stable, steady state depending upon the intensity of the applied electric field in the case of the Brusselator model. Our findings open the possibilities for future experiments to verify the predictions of electric field-induced various spatiotemporal instabilities in experimental reaction-diffusion systems.
... When a CDIMA Turing pattern is illuminated, the pattern in the illuminated area is knocked out, and the illuminated area goes to the light steady state (which corresponds to a significantly lower PVA-triiodide complex concentration). [43,52,53]. Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. ...
... [43,52,53]. Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. Temporal suppression by full-spectrum visible light (where the pattern is illuminated periodically) can eliminate the pattern even more quickly than constant suppression, particularly if the periodic illumination occurs at the same frequency as the chemical oscillations in the stirred CDIMA system [52,53]. ...
... Depending on the intensity of the illumination, the pattern may recover once the light is removed [46,52,53]. Temporal suppression by full-spectrum visible light (where the pattern is illuminated periodically) can eliminate the pattern even more quickly than constant suppression, particularly if the periodic illumination occurs at the same frequency as the chemical oscillations in the stirred CDIMA system [52,53]. In addition, square wave forcing (periodic on-off illumination) is more effective than sinusoidal forcing at suppressing the Turing patterns [53]. ...
In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction–diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237 , 37–72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction–diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis.
This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.
... (b) Control of spatial and spatio-temporal patterns by switching kinetics In addition to evolving the network topology and the diffusion parameters, it is also possible to control the reaction kinetics in order to modify or suppress pattern formation. Physical mechanisms for pattern control through the reaction kinetic terms include illumination [64] and thermal forcing [48]. Control of patterns using non-autonomous kinetics which switched from Figure 11. ...
Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over time. One area of active interest is pattern formation from reaction–diffusion systems, which themselves evolve over temporal networks. We derive analytical conditions for the onset of diffusive spatial and spatio-temporal pattern formation on undirected temporal networks through the Turing and Benjamin–Feir mechanisms, with the resulting pattern selection process depending strongly on the evolution of both global diffusion rates and the local structure of the underlying network. Both instability criteria are then extended to the case where the reaction–diffusion system is non-autonomous, which allows us to study pattern formation from time-varying base states. The theory we present is illustrated through a variety of numerical simulations which highlight the role of the time evolution of network topology, diffusion mechanisms and non-autonomous reaction kinetics on pattern formation or suppression. A fundamental finding is that Turing and Benjamin–Feir instabilities are generically transient rather than eternal, with dynamics on temporal networks able to transition between distinct patterns or spatio-temporal states. One may exploit this feature to generate new patterns, or even suppress undesirable patterns, over a given time interval.
... The framework developed in this paper may also be applied to problems with time-dependent reaction kinetics, on either growing or static domains. In particular, temporal oscillations have been employed in photosensitive reactions to control Turing patterns, and in some cases eliminate them (Dolnik et al. 2001;Horváth et al. 1999;Wang et al. 2006). Spatiotemporal forcing has also been used to mimic domain growth in such systems (Konow et al. 2019;Míguez et al. 2006Míguez et al. , 2005Rüdiger et al. 2003). ...
The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.
... This is well known as Turing instability or diffusion-driven instability. Nevertheless, it is hard to achieve such a large difference of diffusivities between two interacting chemical species in chemical reactions in real experiments; a large number of studies have been reported so far in this context to investigate pattern formation processes both theoretically and experimentally [6][7][8][9][10][11]. Efforts had been made in literature to overcome the stringent criteria of equal diffusivities to generate spatiotemporal instabilities by externally perturbing the chemical reaction-diffusion systems using differential flow [12], electric fields and magnetic fields [13][14][15][16][17], photoillumination [18,19], or by including timedelay and time-delayed feedbacks [20,21] or fluctuations in different forms [22][23][24][25][26][27]. In the same line, the problem of equal diffusion coefficients has been addressed in reaction-diffusion * pghosh@tifrh.res.in ...
Thermodiffusion or thermophoresis or Soret effect, i.e., mass-transport induced by thermal gradient, has immense application in segregation of species in two or multicomponent gaseous, liquid, or colloidal mixtures. Here, we show that an external thermal gradient can be effectively utilized in creation and modification of patterns in spatially extended systems. We consider Brusselator and chlorine-dioxide iodine malonic acid (CDIMA) reaction-diffusion systems, which follow activator-inhibitor kinetics subjected to an external thermal gradient. We find that the conspicuous interaction of emergent thermodiffusive flux with reaction kinetics and diffusion can lead to various spatiotemporal instabilities in these two models. Specifically, our result reveals formation of Turing-like spatial patterns even for equal diffusivities of the activator and inhibitor components in the Brusselator model under the influence of differential thermodiffusion, whereas formation of such stationary patterns in the CDIMA system from a homogenous stable steady state, which is also stable under differential diffusion, requires the same sign and magnitude of Soret coefficients. However, with equal diffusivities of the components of the CDIMA system and without starch in the medium, our result identifies formation of drifting spiral waves which finally disappears at longer times under the influence of thermodiffusion. We also show formation of propagating patterns of spotlike or stripelike heterogeneity in both the model systems under appropriate conditions. Our study provides a route to pattern formation beyond Turing space and reveals remarkable influence of thermodiffusion to modify the pattern types just by employing an external thermal gradient which also opens up the possibility to set up new related experiments.
... Spatiotemporal pattern formation is an all-important and ubiquitous self-organization phenomenon in nature [1]. Processes in Physics, Chemistry, Biology, and even Social Sciences, display the formation of organized structures arising from the interaction of individual constituents [2][3][4][5][6]. In this regard, Alan Turing's pioneering studies on pattern formation stands out among various mechanisms [7]. ...
Reaction-diffusion schemes are widely used to model and interpret phenomena in various fields. In that context, phenomena driven by Turing instabilities are particularly relevant to describe patterning in a number of biological processes. While the conditions that determine the appearance of Turing patterns and their wavelength can be easily obtained by a linear stability analysis, the estimation of pattern amplitudes requires cumbersome calculations due to non-linear terms. Here we introduce an expansion method that makes possible to obtain analytical, approximated, solutions of the pattern amplitudes. We check and illustrate the reliability of this methodology with results obtained from numerical simulations.
... The framework developed in this paper may also be applied to problems with time-dependent reaction kinetics, on either growing or static domains. In particular, temporal oscillations have been employed in photosensitive reactions to control Turing patterns, and in some cases eliminate them (Dolnik et al. 2001;Horváth et al. 1999;Wang et al. 2006). Spatiotemporal forcing has also been used to mimic domain growth in such systems (Konow et al. 2019;Míguez et al. 2006Míguez et al. , 2005Rüdiger et al. 2003). ...
The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities.
