Competition of Spatial and Temporal Instabilities underTime Delay near Codimension-Two Turing-Hopf Bifurcations

Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion equation. The time delay changes remarkably the oscillation frequency, the intrinsic wave vector, and the intensities of both Turing and Hopf modes.
The application of appropriate time delay can control the competition between the Turing and Hopf modes. Analysis shows that individual or both feedbacks can realizethe control of the transformation between the Turing and Hopf patterns.Two-dimensional numerical simulations validate the analytical results.     

Spatial instabilities in reaction random walks with direction-independent kinetics

We study spatial instabilities in reacting and diffusing systems, where diffusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly, albeit with heavy damping. We present evolution equations for reaction random walks whose kinetics do not depend on the particles' direction of motion. The homogeneous steady state of such systems can undergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing instability in reaction-diffusion systems. The other type occurs in the ballistic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are derived and applied to two model systems. We also analyze the stability properties of one-variable systems and find that small wavelength perturbations decay in an oscillatory manner.     

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.     

Pattern formation in an N+Q component reaction-diffusion system

A general N+Q component reaction-diffusion system is analyzed with regard to pattern forming instabilities (Turing bifurcations). The system consists of N mobile species and Q immobile species. The Q immobile species form in response to reactions between the N mobile species and an immobile substrate and allow the Turing instability to occur. These results are valid both for bifurcations from a spatially uniform state and for systems with an externally imposed gradient as in the experimental systems in which Turing patterns have been observed. It is shown that the critical wave number and the location of the instability in parameter space are independent of the substrate concentration. It is also found that the system necessarily undergoes a Hopf bifurcation as the total substrate concentration is decreased. Further, in the case that all the mobile species diffuse at identical rates we show that if the full system is at a point of Turing bifurcation then the N component mobile subsystem is at transition from an unstable focus to an unstable node, and the critical wave number is simply related to the degenerate positive eigenvalue of the mobile subsystem. A sequence of bifurcations that occur in the eigenspectra as the total substrate concentration is decreased to zero is also discussed.     

Turing bifurcation in a reaction-diffusion system with density-dependent dispersal

Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction-diffusion system, where dispersal is described by simple Fickian diffusion. While activator-inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer-Meinhardt model.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.     

Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel-Epstein model. Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.     

Stable squares and other oscillatory turing patterns in a reaction-diffusion model

We study the Brusselator reaction-diffusion model under conditions where the Hopf mode is supercritical and the Turing band is subcritical. Oscillating Turing patterns arise in the system when bulk oscillations lose their stability to spatial perturbations. Spatially uniform external periodic forcing can generate oscillating Turing patterns when both the Turing and Hopf modes are subcritical in the autonomous system. Most of the symmetric patterns show period doubling in both space and time. Patterns observed include squares, rhombi, stripes, and hexagons.     

Turing patterns beyond hexagons and stripes

The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.     

Pattern formation in reaction-diffusion system in crossed electric and magnetic fields

We consider a reaction-diffusion system in crossed electric and magnetic fields lying on the reaction plane. It is shown that a charge separation along the direction normal to the reaction plane resulting in a diffusional flux may cause a differential flow induced chemical instability and stationary pattern formation on a homogeneous steady state. This pattern is generically different from a Turing pattern modified by the crossed fields. The special role of magnetic field is emphasized. Our theoretical analysis is corroborated by numerical simulation on a reaction-diffusion system in three dimensions.     

Spontaneous symmetry breaking turing-type pattern formation in a confined Dictyostelium cell mass

We have discovered a new type of patterning which occurs in a two-dimensionally confined cell mass of the cellular slime mold Dictyostelium discoideum. Besides the longitudinal structure reported earlier, we observed a spontaneous symmetry breaking spot pattern whose wavelength shows similar strain dependency to that of the longitudinal pattern. We propose that these structures are due to a reaction-diffusion Turing instability similar to the one which has been exemplified by CIMA (chlorite-iodide-malonic acid) reaction. The present finding may exhibit the first biochemical Turing structure in a developmental system with a controllable boundary condition.     

Dash waves in a reaction-diffusion system

Patterns in reaction-diffusion systems generally consist of smooth traveling waves or of stationary, discontinuous Turing structures. Hybrid patterns that blend the properties of waves and Turing structures have not previously been observed. We report observation of dash waves, which consist of wave segments regularly separated by gaps, moving coherently in the Belousov-Zhabotinsky system dispersed in water-in-oil microemulsion. Dash waves emerge from the interaction between excitable and pseudo-Turing-unstable steady states. We are able to generate dash waves in simulations with simple models.     

Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes

Spatial resonances leading to superlattice hexagonal patterns, known as "black-eyes," and superposition patterns combining stripes and/or spots are studied in a reaction-diffusion model of two interacting Turing modes with different wavelengths. A three-phase oscillatory interlacing hexagonal lattice pattern is also found, and its appearance is attributed to resonance between a Turing mode and its subharmonic.     

Effect of a Local Source or Sink of Inhibitor on Turing Patterns

We investigate a reaction-diffusion model in which a Turing pattern develops and reproduces the formation of periodic segments behind a propagating chemical wave front. The chemical scheme involves two species known as activator and inhibitor. The model can be used to mimic the formation of prevertebrae during the early development of vertebrate embryo. Deterministic and stochastic analyses of the reaction-diffusion processes are performed for two typical sets of parameter values, far from and close to the Turing bifurcation. The effects of a local source or sink of inhibitor on the growing structure are studied and successfully compared with experiments performed on chick embryos. We show that fluctuations may lead to the formation of additional prevertebra.     

一类大气非线性动力热力学反应-扩散系统解的稳定性态

研究了一类大气非线性系统. 利用大气非线性热力动力学理论, 讨论了大气非线性强迫耗散系统的速度、温度和湿度所满足的非线性反应-扩散系统, 再利用微分方程Lyapunov稳定性理论, 得到了大气非线性反应-扩散系统在均匀定态解邻域内的微扰解. 最后由系统的控制参数的变化, 得到了大气非线性强迫耗散系统有关的物理量的有序-无序-有序状态的转化过程, 从而可以预报和预测相应的局部大气非线性强迫耗散系统的稳定性态.     

Controlling the transition between Turing and antispiral patterns by using time-delayed-feedback

The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model.We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing-Hopf codimension-two phase space.Numerical simulations show that the transition between the Turing patterns(hexagon,stripe,and honeycomb),the dual-mode antispiral,and the antispiral by applying appropriate feedback parameters.The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities.Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.     

Superlattice Turing structures in a photosensitive reaction-diffusion system

Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.     

Pattern formation in the thiourea-iodate-sulfite system: Spatial bistability, waves, and stationary patterns

We present a detailed study of the reaction-diffusion patterns observed in the thiourea-iodate-sulfite (TuIS) reaction, operated in open one-side-fed reactors. Besides spatial bistability and spatio-temporal oscillatory dynamics, this proton autoactivated reaction shows stationary patterns, as a result of two back-to-back Turing bifurcations, in the presence of a low-mobility proton binding agent (sodium polyacrylate). This is the third aqueous solution system to produce stationary patterns and the second to do this through a Turing bifurcation. The stationary pattern forming capacities of the reaction are explored through a systematic design method, which is applicable to other bistable and oscillatory reactions. The spatio-temporal dynamics of this reaction is compared with that of the previous ferrocyanide-iodate-sulfite mixed Landolt system.     

Complex patterns in reaction-diffusion systems: A tale of two front instabilities

Two front instabilities in a reaction-diffusion system are shown to lead to the formation of complex patterns. The first is an instability to transverse modulations that drives the formation of labyrinthine patterns. The second is a nonequilibrium Ising-Bloch (NIB) bifurcation that renders a stationary planar front unstable and gives rise to a pair of counterpropagating fronts. Near the NIB bifurcation the relation of the front velocity to curvature is highly nonlinear and transitions between counterpropagating fronts become feasible. Nonuniformly curved fronts may undergo local front transitions that nucleate spiral-vortex pairs. These nucleation events provide the ingredient needed to initiate spot splitting and spiral turbulence. Similar spatiotemporal processes have been observed recently in the ferrocyanide-iodate-sulfite reaction.