Resonant-pattern formation induced by additive noise in periodically forced reaction-diffusion systems

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction-diffusion systems

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.

Numerical and renormalization group analysis of the phase diagram of a stochastic cubic autocatalytic reaction-diffusion system

The renormalization group is a set of tools that can be used to incorporate the effect of fluctuations in a dynamical system as a rescaling of the system's parameters. Here, we apply the renormalization group to a pattern-forming stochastic cubic autocatalytic reaction-diffusion model and compare its predictions with numerical simulations. Our results demonstrate a good agreement within the range of validity of the theory and show that external noise can be used as a control parameter in such systems.

Turing conditions for pattern forming systems on evolving manifolds

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.

Pattern transitions in spatial epidemics: Mechanisms and emergent properties

Infectious diseases are a threat to human health and a hindrance to societal development. Consequently, the spread of diseases in both time and space has been widely studied, revealing the different types of spatial patterns. Transitions between patterns are an emergent property in spatial epidemics that can serve as a potential trend indicator of disease spread. Despite the usefulness of such an indicator, attempts to systematize the topic of pattern transitions have been few and far between. We present a mini-review on pattern transitions in spatial epidemics, describing the types of transitions and their underlying mechanisms. We show that pattern transitions relate to the complexity of spatial epidemics by, for example, being accompanied with phenomena such as coherence resonance and cyclic evolution. The results presented herein provide valuable insights into disease prevention and control, and may even be applicable outside epidemiology, including other branches of medical science, ecology, quantitative finance, and elsewhere.

Competition promotes the persistence of populations in ecosystems

Competition is one of the most common form in ecological systems, which plays important roles in population dynamics. However, the influences of competition on persistence of populations remain unclear when space effect is included. In this paper, we investigated a predator-prey model with competition and spatial diffusion. Based on pattern formations and time series of populations, we found that competitions induce the persistence of populations, which denies competitive exclusion principle. Moreover, we testify the robustness of these effects. Our results also suggest that space may lead to the emergence of new phenomenon in ecosystems.

Parametric spatiotemporal oscillation in reaction-diffusion systems

We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.

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.

Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response

In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H(0) hexagon patterns and stripe patterns, H(π) hexagon patterns, and H(0) hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H(π) hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H(0) hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.

Noise-controlled pattern formation and threshold shift for electroconvection in the conduction and dielectric regimes

We report noise-controlled electrohydrodynamic pattern formations and threshold shifts in nematic liquid crystals. In the electrohydrodynamic system superposed with noise, experimental results obtained in the dielectric regime are compared to those in the conventional conduction regime. The noise intensity dependences of thresholds and pattern evolution processes are remarkably different for each regime. The pattern formation mechanisms in the presence of noise for both regimes are discussed based on the results. Moreover, it is found that the thresholds and characteristic wavelengths of dissipative structures can be effectively controlled by external multiplicative noises with appropriate intensity and correlation times.

Spontaneously periodic wave generation in coupled excitable media

We studied a modified reaction-diffusion model theoretically by coupling two ideal excitable media systems. In the simulated homogeneous system, we observed the propagation of reaction-diffusion wave trains that required no external force after the initial stimulation. We investigated the dependence of the system's oscillation patterns on model parameters, and we discussed the influence of the different dynamic constants of the individual coupled systems on the dynamics of the coupled systems. Some complex two-dimensional patterns generated by our model are shown. We also found similar phenomena in the models for catalytic CO oxidation on Pt(110), and for cardiac tissue.

Pattern Formation Induced by Internal Microscopic Fluctuations

We report spatiotemporal patterns induced by microscopic fluctuations in the Gray-Scott model. In the framework of stochastic kinetics, the macroscopic effect of internal noise of the system was investigated by simulating the reaction-diffusion master equation using Gillespie's algorithm. Pattern formation at the level of stochastic description is presented in comparison with that given by deterministic equations. Complex spatiotemporal patterns, including spiral waves, Turing patterns, self-replicating spots and others, which are not captured or correctly predicted by the deterministic reaction-diffusion equations, are induced by internal reaction fluctuations. Furthermore the intrinsic noise selects and controls the pattern formation with different intensities of fluctuation.