Turing Patterns, Spatial Bistability, and Front Instabilities in a Reaction−Diffusion System

Long-living transients in ecological models: Recent progress, new challenges, and open questions

Traditionally, mathematical models in ecology placed an emphasis on asymptotic, long-term dynamics. However, a large number of recent studies highlighted the importance of transient dynamics in ecological and eco-evolutionary systems, in particular 'long transients' that can last for hundreds of generations or even longer. Many models as well as empirical studies indicated that a system can function for a long time in a certain state or regime (a 'metastable regime') but later exhibits an abrupt transition to another regime not preceded by any parameter change (or following the change that occurred long before the transition). This scenario where tipping occurs without any apparent source of a regime shift is also referred to as 'metastability'. Despite considerable evidence of the presence of long transients in real-world systems as well as models, until recently research into long-living transients in ecology has remained in its infancy, largely lacking systematisation. Within the past decade, however, substantial progress has been made in creating a unifying theory of long transients in deterministic as well as stochastic systems. This has considerably accelerated further studies on long transients, in particular on those characterised by more complicated patterns and/or underlying mechanisms. The main goal of this review is to provide an overview of recent research on long transients and related regime shifts in models of ecological dynamics. We pay special attention to the role of environmental stochasticity, the effect of multiple timescales (slow-fast systems), transient spatial patterns, and relation between transients and spatial synchronisation. We also discuss current challenges and open questions in understanding transients with applications to ecosystems dynamics.

Reaction diffusion system prediction based on convolutional neural network

The reaction-diffusion system is naturally used in chemistry to represent substances reacting and diffusing over the spatial domain. Its solution illustrates the underlying process of a chemical reaction and displays diverse spatial patterns of the substances. Numerical methods like finite element method (FEM) are widely used to derive the approximate solution for the reaction-diffusion system. However, these methods require long computation time and huge computation resources when the system becomes complex. In this paper, we study the physics of a two-dimensional one-component reaction-diffusion system by using machine learning. An encoder-decoder based convolutional neural network (CNN) is designed and trained to directly predict the concentration distribution, bypassing the expensive FEM calculation process. Different simulation parameters, boundary conditions, geometry configurations and time are considered as the input features of the proposed learning model. In particular, the trained CNN model manages to learn the time-dependent behaviour of the reaction-diffusion system through the input time feature. Thus, the model is capable of providing concentration prediction at certain time directly with high test accuracy (mean relative error <3.04%) and 300 times faster than the traditional FEM. Our CNN-based learning model provides a rapid and accurate tool for predicting the concentration distribution of the reaction-diffusion system.

Tuning Electrochemical Bistability by Surface Area Blocking in the Cathodic Deposition of Copper

We report herein a precise control of the electrochemical bistability induced by surface area changes during the cathodic deposition of copper. Small additions of 1,10-phenanthroline (Phen) in the reaction media present an inhibiting effect on the global rate mainly due to the adsorption of protonated Phen. The increase of its concentration favors a shrinkage of the bifurcation (saddle-node) diagram and shifts it to less negative potentials. The dynamic instability is verified by impedance measurements, and a negative impedance is clearly found. We calculated the apparent molar mass of the adsorbents using in situ gravimetric monitoring in the electrochemical experiments, and the results indicate that mass changes occur mainly due to the reduction of copper from bivalent ions dissolved in the reaction media. Importantly, the adsorption of protonated Phen molecules does not show a considerable contribution in mass variations but prevents the formation of a copper course grained morphology over the surface. Imaging analysis indicates finer nodulations at the lower branch compared to the upper branch in the bistability domain. On the basis of these observations, a kinetic mechanism is proposed and a good agreement is obtained between the apparent molar mass extracted from experiments and the theoretical values. Altogether, our results contribute to a detailed physical chemical description of the nonlinear behavior, bringing new insights about this reaction and pointing out the possibility to design switchable surface electrodes by taking advantage of the bistable behavior.

Self-organization in precipitation reactions far from the equilibrium

Far from the thermodynamic equilibrium, many precipitation reactions create complex product structures with fascinating features caused by their unusual origins. Unlike the dissipative patterns in other self-organizing reactions, these features can be permanent, suggesting potential applications in materials science and engineering. We review four distinct classes of precipitation reactions, describe similarities and differences, and discuss related challenges for theoretical studies. These classes are hollow micro- and macrotubes in chemical gardens, polycrystalline silica carbonate aggregates (biomorphs), Liesegang bands, and propagating precipitation-dissolution fronts. In many cases, these systems show intricate structural hierarchies that span from the nanometer scale into the macroscopic world. We summarize recent experimental progress that often involves growth under tightly regulated conditions by means of wet stamping, holographic heating, and controlled electric, magnetic, or pH perturbations. In this research field, progress requires mechanistic insights that cannot be derived from experiments alone. We discuss how mesoscopic aspects of the product structures can be modeled by reaction-transport equations and suggest important targets for future studies that should also include materials features at the nanoscale.

Sodium chlorite – iodide – acetylacetone oscillation reaction investigated by UV-Vis spectrophotometry

A new sodium chlorite – iodide – acetylacetone chemical oscillatory reaction has been studied by the UV-Vis spectrophotometric method. The initial concentrations of acetylacetone, sodium chlorite, iodide, and sulfuric acid and the pH value have great influence on the oscillation observed at a wavelength of 570 nm for the starch–triiodide complex. There is a pre-oscillatory or induction stage and the amplitude and number of oscillations depend on the initial concentration of the reactants. Equations for the starch–triiodide complex reaction rate change with reaction time and the initial concentrations in the oscillation stage were obtained. The induction time decreases linearly with the initial concentration of acetylacetone or sodium chlorite but increases linearly with the initial concentration of sulfuric acid. The oscillation reaction can be accelerated by increasing the reaction temperature. The apparent activation energies at the induction stage and the oscillation stage were 61.02 and 61.36 kJ/mol, respectively, indicating that the two stages have similar reaction mechanisms. Generating the enol isomer by keto–enol tautomerism is an important step to constrain the time of the induction period.

Chemical morphogenesis: recent experimental advances in reaction-diffusion system design and control

In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniform solution of reacting chemicals to develop stable spatially periodic concentration patterns. It took nearly 40 years before the first two unquestionable experimental demonstrations of such reaction-diffusion patterns could be made in isothermal single phase reaction systems. The number of these examples stagnated for nearly 20 years. We recently proposed a design method that made their number increase to six in less than 3 years. In this report, we formally justify our original semi-empirical method and support the approach with numerical simulations based on a simple but realistic kinetic model. To retain a number of basic properties of real spatial reactors but keep calculations to a minimal complexity, we introduce a new way to collapse the confined spatial direction of these reactors. Contrary to similar reduced descriptions, we take into account the effect of the geometric size in the confinement direction and the influence of the differences in the diffusion coefficient on exchange rates of species with their feed environment. We experimentally support the method by the observation of stationary patterns in red-ox reactions not based on oxihalogen chemistry. Emphasis is also brought on how one of these new systems can process different initial conditions and memorize them in the form of localized patterns of different geometries.

Chlorine dioxide-iodide-methyl acetoacetate oscillation reaction investigated by UV-vis and online FTIR spectrophotometric method

In order to study the chemical oscillatory behavior and mechanism of a new chlorine dioxide-iodide ion-methyl acetoacetate reaction system, a series of experiments were done by using UV-Vis and online FTIR spectrophotometric method. The initial concentrations of methyl acetoacetate, chlorine dioxide, potassium iodide, and sulfuric acid and the pH value have great influence on the oscillation observed at wavelength of 289 nm. There is a preoscillatory or induction period, and the amplitude and the number of oscillations are associated with the initial concentration of reactants. The equations for the triiodide ion reaction rate changing with reaction time and the initial concentrations in the oscillation stage were obtained. Oscillation reaction can be accelerated by increasing temperature. The apparent activation energies in terms of the induction period and the oscillation period were 26.02 KJ/mol and 17.65 KJ/mol, respectively. The intermediates were detected by the online FTIR analysis. Based upon the experimental data in this work and in the literature, a plausible reaction mechanism was proposed for the oscillation reaction.

Experimental Study of Closed System in the Chlorine Dioxide-Iodide-Sulfuric Acid Reaction by UV-Vis Spectrophotometric Method

The mole ratio r(r = [I⁻]₀/[ClO₂]₀) has great influence on ClO₂-I⁻-H₂SO₄ closed reaction system. By changing the initiate concentration of potassium iodide, the curve of absorbance along with the reaction time was obtained at 350 nm and 297 nm for triiodide ion, and 460 nm for iodine. The changing point of the absorbance curve's shape locates at r = 6.00. For the reaction of ClO₂-I⁻ in the absence of H₂SO₄, the curve of absorbance along with the reaction time can be obtained at 350 nm for triiodide ion, 460 nm for iodine. The mole ratio r is equal to 1.00 is the changing point of the curve's shape no matter at which wavelength to determine the reaction. For the reaction of ClO₂-I⁻-H⁺ in different pH buffer solution, the curve of absorbance along with the reaction time was recorded at 460 nm for iodine. When r is greater than 1.00, the transition point of the curve's shape locates at pH 2.0, which is also the point of producing chlorite or chloride for chlorine dioxide at different pH. When r is less than 1.00, the transition point locates at pH 7.0.

Lattice Boltzmann study of pattern formation in reaction-diffusion systems

Pattern formation in reaction-diffusion systems is of great importance in surface micropatterning [Grzybowski et al., Soft Matter 1, 114 (2005)], self-organization of cellular micro-organisms [Schulz et al., Annu. Rev. Microbiol. 55, 105 (2001)], and in developmental biology [Barkai et al., FEBS Journal 276, 1196 (2009)]. In this work, we apply the lattice Boltzmann method to study pattern formation in reaction-diffusion systems. As a first methodological step, we consider the case of a single species undergoing transformation reaction and diffusion. In this case, we perform a third-order Chapman-Enskog multiscale expansion and study the dependence of the lattice Boltzmann truncation error on the diffusion coefficient and the reaction rate. These findings are in good agreement with numerical simulations. Furthermore, taking the Gray-Scott model as a prominent example, we provide evidence for the maturity of the lattice Boltzmann method in studying pattern formation in nonlinear reaction-diffusion systems. For this purpose, we perform linear stability analysis of the Gray-Scott model and determine the relevant parameter range for pattern formation. Lattice Boltzmann simulations allow us not only to test the validity of the linear stability phase diagram including Turing and Hopf instabilities, but also permit going beyond the linear stability regime, where large perturbations give rise to interesting dynamical behavior such as the so-called self-replicating spots. We also show that the length scale of the patterns may be tuned by rescaling all relevant diffusion coefficients in the system with the same factor while leaving all the reaction constants unchanged.

Pattern formation in the ferrocyanide-iodate-sulfite reaction: the control of space scale separation

We revisit the conditions for the development of reaction-diffusion patterns in the ferrocyanide-iodate-sulfite bistable and oscillatory reaction. This hydrogen ion autoactivated reaction is the only example known to produce sustained stationary lamellar patterns and a wealth of other spatio-temporal phenomena including self-replication and localized oscillatory domain of spots, due to repulsive front interactions and to a parity-breaking front bifurcation (nonequilibrium Ising-Bloch bifurcation). We show experimentally that the space scale separation necessary for the observation of stationary patterns is mediated by the presence of low mobility weak acid functional groups. The presence of such groups was overlooked in the original observations made with hydrolyzable polyacrylamide gels. This missing information made the original observations difficult to reproduce and frustrated further experimental exploitation of the fantastic potentialities of this system. Using one-side-fed spatial reactors filled with agarose gel, we can reproduce all the previous pattern observations, in particular the stationary labyrinthine patterns, by introducing, above a critical concentration, well controlled amounts of polyacrylate chains in the gel network. We use two different geometries of spatial reactors (annular and disk shapes) to provide complementary information on the actual three-dimensional character of spatial patterns. We also reinvestigate the role of other feed parameters and show that the system exhibits both a domain of spatial bistability and of large-amplitude pH oscillations associated in a typical cross-shape diagram. The experimental method presented here can be adapted to produce patterns in the large number of oscillatory and bistable reactions, since the iodate-sulfite-ferrocynide reaction is a prototype of these systems.

Sustained Spatiotemporal Patterns in the Bromate−Sulfite Reaction

The acid autoactivated bromate-sulfite reaction exhibits spatial bistability, travelling acid-base fronts, and spatiotemporal oscillations when operated in an unstirred one-side-fed spatial reactor. We show that a skeleton kinetic model, recently proposed by Szántó and Rábai, in which we take into account the charge and the actual diffusivity of solvated ions, provides theoretical results in good agreement with experimental observations. The differences with previous observations made with the homologous iodate-sulfite reaction are discussed. Despite the analogies in the phase diagram of these two systems, it is concluded that the relevant kinetic mechanism of the iodate-sulfite system cannot be just a homologous transcription of that presently working well for the bromate-sulfite system, even in excess sulfite conditions.

Spatiotemporal dynamics of the Landolt reaction in an open spatial reactor with conical geometry

In a previous study, the iodate-sulfite proton autoactivated reaction (Landolt reaction) was shown to exhibit spatial bistability and spatiotemporal oscillations when operated in an open spatial reactor with fixed "thickness", i.e., feed boundary to core distance. Here, we show that the spatial reactors with conical geometry enable one to rapidly probe the sensitivity of the above phenomena over a large range of the "thickness" parameter. This often-neglected parameter in chemical pattern studies plays an important role on the selection and stability of states. We reveal that the quenching capacity of slow diffusing polyacrylate ions on the spatiotemporal oscillations depends on this "thickness". The presented results should be useful for further research on reaction diffusion patterns and chemomechanical structures.

Spatial bistability: a source of complex dynamics. From spatiotemporal reaction-diffusion patterns to chemomechanical structures

We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate "spatial bistability" when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatiotemporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatiotemporal shape patterns, referred to as chemomechanical structures.

Spatial Bistability in a pH Autocatalytic System: From Long to Short Range Activation

The acid-auto-activated chlorite-tetrathionate reaction is studied in a one-side-fed spatial reactor. It was previously shown that in these conditions the unstirred reaction-diffusion system can generate oscillatory and excitable states even though under well-stirred nonequilibrium conditions only steady-state bistability is observed. Numerical simulations suggest that these temporal reaction-diffusion instabilities result from long-range activation by rapidly diffusing protons. We study here experimentally and numerically the effect of introducing into this reaction-diffusion system macromolecular carboxylate species that reduce the effective diffusivity of protons. Consistent with the original assumption, the introduction of such slow mobility proton-binding species quenches both oscillatory and excitability dynamics. Within the bistability domain the direction of the propagation of an interface between the two steady states depends on control parameter value. We elaborate on the fact that beyond a low critical concentration of macromolecular carboxylate species, the stability limit of the "thermodynamic" branch of spatial steady state does not depend on this concentration. Despite the relative simplicity of the kinetic model used in the numerical simulations, the results are in quasi-quantitative agreement with the experimental observations.

Turing Patterns, Spatial Bistability, and Front Interactions in the [ClO2, I2, I-, CH2(COOH)2] Reaction

Recent experiments by Szalai and De Kepper performed in open spatial reactors have shown that the rich variety of dynamic properties of the chlorine dioxide-iodide-chlorite-iodine-malonic acid family of reactions is far from being exhausted: stable inhomogeneous patterns due to front interactions and transient labyrinthine structures are now added to the spatial bistability and Turing patterns as possible spatial behavior. The two latter phenomena, already observed in the chlorine dioxide-iodide (CDI) and the chlorine dioxide-iodide-malonic acid (CDIMA) reactions, respectively, were kept as limiting cases in the new setup. In this paper, we numerically analyze an extension of the most detailed available model of the CDI system (Lengyel et al.) including a reaction between I2 and MA that comes from the presence of the latter into the flow. The resulting nine-variable model is simulated in one and two dimensions, taking into account the proper constraints of the boundary-fed system. The nonequilibrium phase diagram closely follows the results of the experiments of ref 1. In particular, the model reproduces observations on spatial bistability, stationary front interactions, and Turing patterns. In addition, it predicts a new region of spatial bistability.

Regular Liesegang patterns and precipitation waves in an open system

We investigate the regular and moving Liesegang pattern formation phenomena in an open system. First, simulations have been performed at fixed coupling between the reactive medium and the reservoir, later this control parameter was varied during the simulations resulting in various phenomena. We predicted and monitored for the first time various--dynamically changing--precipitation structures and a spatial hysteresis phenomenon, which is beyond the scope of the Turing instability. The dynamics of the reaction is well detectable using specific quantities: the total amount of precipitate and its center of gravity.