Pattern formation in excitable media with concentration-dependent diffusivities

Square Turing patterns in reaction-diffusion systems with coupled layers

Square Turing patterns are usually unstable in reaction-diffusion systems and are rarely observed in corresponding experiments and simulations. We report here an example of spontaneous formation of square Turing patterns with the Lengyel-Epstein model of two coupled layers. The squares are found to be a result of the resonance between two supercritical Turing modes with an appropriate ratio. Besides, the spatiotemporal resonance of Turing modes resembles to the mode-locking phenomenon. Analysis of the general amplitude equations for square patterns reveals that the fixed point corresponding to square Turing patterns is stationary when the parameters adopt appropriate values.

Regulation of Turing patterns in a spatially extended chlorine-iodine-malonic-acid system with a local concentration-dependent diffusivity

A chlorine-iodine-malonic-acid Turing system involving a local concentration-dependent diffusivity (LCDD) has fundamental significance for physical, chemical, and biological systems with inhomogeneous medium. We investigated such a system by both numerical computation and mathematical analysis. Our research reveals that a variable local diffusivity has an evident effect on regulating the Turing patterns for different modes. An intrinsic square-root law is given by λ ∼ (c(1)+c(2)k)(1/2), which relates the pattern wavelength (λ) with the LCDD coefficient (k). This law indicates that the system pattern has the properties of an equivalent Turing pattern. The current study confirms that, for the Turing system with LCDD, the system pattern form retains the basic characteristics of a traditional Turing pattern in a wide range of LCDD coefficients.

Replication of noise-sustained autocatalytic chemical structures

Autocatalytic systems in a differential-flow reactor may undergo a differential-flow-induced chemical instability toward a convectively unstable regime, in which noise-sustained structures may appear. This is the case of a system with Gray-Scott kinetics in a packed-bed reactor, as reported in [B. von Haeften and G. Izus, Phys. Rev. E 67, 056207 (2003)]. In this work, two identical copies of such a system are coupled in master-slave configuration and submitted to independent spatiotemporal Gaussian white noise sources. Numerical simulation of two-dimensional reactors with uniform and Poiseuille flows reveals that the slave system replicates to a very high degree of precision and the convective patterns arising in the master one due to the presence of noise. The quality of this synchronization is assessed through several measures. A convective instability in the synchronization manifold is theoretically predicted and numerically confirmed.

Stochastic resonance on weakly paced scale-free networks

We study the impact of additive Gaussian noise and weak periodic forcing on the dynamics of a scale-free network of bistable overdamped oscillators. The periodic forcing is introduced to a single oscillator and therefore acts as a pacemaker trying to impose its rhythm on the whole ensemble. We show that an intermediate intensity of temporally and spatially uncorrelated noise is able to optimally assist the pacemaker in achieving this goal, thus providing evidence for stochastic resonance on weakly paced scale-free networks. Because of the inherent degree inhomogeneity of individual oscillators forming the scale-free network, the placement of the pacemaker within the network is thereby crucial. As two extremes, we consider separately the introduction of the pacemaker to the oscillator with the highest degree and to one of the oscillators having the lowest degree. In both cases the coupling strength plays a crucial role, since it determines to what extent the whole network will follow the pacemaker on the expense of a weaker correlation between the pacemaker and the units that are directly linked with the paced oscillator. Higher coupling strengths facilitate the global outreach of the pacemaker, but require higher noise intensities for the optimal response. In contrast, lower coupling strengths and comparatively low noise intensities localize the optimal response to immediate neighbors of the paced oscillator. If the pacemaker is introduced to the main hub, the transition between the locally and globally optimal responses is characterized by a double resonance that postulates the existence of an optimal coupling strength for the transmission of weak rhythmic activity across scale-free networks. We corroborate the importance of the inhomogeneous structure of scale-free networks by additionally considering regular networks of oscillators with different degrees of coupling.

Stochastic resonance on excitable small-world networks via a pacemaker

We show that the correlation between the frequency of subthreshold pacemaker activity and the response of an excitable array is resonantly dependent on the intensity of additive spatiotemporal noise. Thereby, the effect of the underlying network, defining the interactions among excitable units, largely depends on the coupling strength. Only for intermediate coupling strengths is the small world property able to enhance the stochastic resonance, whereas for smaller and larger couplings the impact of the transition from diffusive to random networks is less profound. Thus, the optimal interplay between a localized source of weak rhythmic activity and the response of the whole array demands a delicate balance between the strength of excitation transfer and the effectiveness of the network structure to support it.

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.

Reaction–diffusion models of development with state-dependent chemical diffusion coefficients

Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.