Turing pattern formation in coupled reaction-diffusion system with distributed delays

Turing pattern formation in coupled two-layer system with distributed delayed is investigated. Numerical simulations prove that, when the coupling is weak, it can apparently accelerate the formation process and enhance the spatial amplitude of the pattern. When it is strong, it will prolong the formation process or even inhibit the pattern and turn the whole system into bulk oscillatory state by its influence on the transient oscillatory state. If the coupling covers only part of the system, Turing pattern can be prominently oriented according to the shape of the coupling area at tiny coupling strength. However, if the coupling is too strong, the Turing pattern may also be destroyed. This means that in coupled systems, the delay effect in the cross-layer signal transfer may significantly influence the spatial character and/or the evolution dynamics in Turing pattern formation, even to destroy the pattern. This work is of practical significance in the study of Turing pattern in biosystems, where bilayer membranes or multilayer tissues are often found.     

Mobility-induced instability and pattern formation in a reaction-diffusion system

Ions undergoing a reaction-diffusion process are susceptible to electric field. We show that a constant external field may induce a kind of instability on the state stabilized by diffusion in a reaction-diffusion system giving rise to formation of pattern even when the diffusion coefficients of the reactants are equal. The origin of the pattern is due to the difference in mobilities of the two species and is thus markedly different from that of deformed Turing pattern in presence of the field. While this differential flow instability had been shown earlier to result in traveling waves, we realize in the context of stationary pattern formation in a typical reaction-diffusion-advective system. Our analysis is based on a numerical simulation of a generic model on a two-dimensional domain.     

Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion

Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.     

Amplitude equations for breathing spiral waves in a forced reaction-diffusion system

Based on a multiple scale analysis of a forced reaction-diffusion system leading to amplitude equations, we explain the existence of spiral wave and its photo-induced spatiotemporal behavior in chlorine dioxide-iodine-malonic acid system. When the photo-illumination intensity is modulated, breathing of spiral is observed in which the period of breathing is identical to the period of forcing. We have also derived the condition for breakup and suppression of spiral wave by periodic illumination. The numerical simulations agree well with our analytical treatment.     

Instability and pattern formation in reaction-diffusion systems: a higher order analysis

We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.     

The Gray-Scott model under the influence of noise: reentrant spatiotemporal intermittency in a reaction-diffusion system

We investigate the influence of noise on the spatiotemporal behavior of the Gray-Scott model, a prototype for a simple reaction-diffusion system. In the parameter regime studied it is characterized deterministically by a stable fixed point. As the noise increases a regular periodic pattern is replaced first by an irregularly oscillating periodic pattern and then by spatiotemporal intermittency. With further increasing noise strength the spatiotemporal intermittency is first replaced by a low amplitude noisy regime followed by spatiotemporal intermittency (STI) embedded into a noisy background. At sufficiently high noise intensity high amplitude noise prevails. We point out that the transition from spatiotemporal intermittency to low amplitude noise can be traced back to the fact that the spatially homogeneous state is a global attractor. As the noise strength grows further the "noisy" fixed point starts to communicate with STI leading to noise-induced spatiotemporal intermittency as an excitable state. At high enough noise strength high amplitude noise is left over wiping out all details of the underlying deterministic dynamical system.     

Turbulent states in a reaction-diffusion system with period-doubling bifurcations

We investigate the spatially extended Hastings–Powell model in one and two dimensions with constant diffusion coefficients and nonflux boundary conditions. Nowave zones, spirals and chaos are found. An absolute instability of the spirals produces a transition to chaos. A constant number of defects, linearly increasing with the bifurcation parameter of the system is found, i.e. there do not exist defect-creation or defect-destruction events. Defects behave as hard disks, with translational degrees of freedom, which result from a cooperative interaction between pairs of defects.     

Extended Born-Oppenheimer equation for a three-state system

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.     

Locally implicit solution of a reaction-diffusion system with stiff kinetics

The Tyson-Fife reaction-diffusion equations are solved numerically using a locally implicit approach. Since the variables evolve at very different time scales, the resulting system of equations is stiff. The reaction term is responsible for the stiffness and the time step is increased by using an implicit method. The diffusion operator is evaluated explicitly and the system of implicit nonlinear equations is decoupled. The method is particularly useful for parameter values in which the equations are very stiff, such as the values obtained directly from the experimental reaction rate constants. Previous efforts modified the parameters on the equations to avoid stiffness. The equations then become a simplified model of excitable media and, for those cases, the locally implicit method gives a faster although less accurate solution. Nevertheless, since the modified equations no longer represent a particular chemical system an accurate solution is not as important. The algorithm is applied to observe the transition from simple motion to compound motion of a spiral tip.     

Chemical oscillations and Turing patterns in a generalized two-variable model of chemical self-replication

Chemical self-replication of oligonucleotides and helical peptides show the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzimatic sink. Although the nonlinearity is necessary for complex dynamics, the nature of the sink is the essential feature in the mechanism that allows temporal and spatial patterns. We obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, autocatalator, and templator models as specific cases.     

Master equation simulations of bistable and excitable dynamics in a model of a thermochemical system

The effect of fluctuations on the dynamics of a model of a bistable thermochemical system is studied by means of the master equation. The system has three stationary states and exhibits two types of bistability: the coexistence of two stable focuses and the coexistence of a stable focus with a stable limit cycle separated by a saddle point. Stochastic effects are important when the system is close to the bifurcation, in which the stable limit cycle disappears through a homoclinic orbit. In this case the distribution of the first passage time from the stable limit cycle to the stable focus has a multipeak form. The dependence of this distribution on the number of particles is presented. Near the homoclinic orbit bifurcation, the system also exhibits excitability due to a particular shape of the basin of attraction of the stable focus.     

Turing structures and stability for the 1-D Lengyel–Epstein system

This paper continues the analysis on the Lengyel–Epstein reaction- diffusion system of the chlorite-iodide-malonic acid-starch (CIMA) reaction for the rich Turing structures. The steady state structures, especially the double bifurcation one, and their stability and multiplicity are studied by the use of Lyapunov–Schmidt reduction technique and singularity theory. Numerical simulations are presented to support our theoretical studies. The results show that the richer stationary Turing patterns heavily rely both on the size of the reactor and on the effective diffusion rate in the CIMA reaction.     

Turing patterns created by cross-diffusion for a Holling II and Leslie-Gower type three species food chain model

In this paper, we develop a theoretical framework for a research into spatial patterns in a three-species Holling II and Leslie-Gower type food chain model with cross-diffusion, the results of which show that the cross-diffusion induces the spatial patterns. When biological pattern formation has been concerned with the method of reaction-diffusion theory, in most of the previous works, as a precondition, the assumption of the existence of nonhomogeneous steady state is presented essentially. We give a rigorous proof to the assumption that the model has at least a nonhomogeneous stationary solution by the Leray-Schauder degree theory. Moreover, the numerical simulations for spatial pattern is also carried out, we propose a method to estimate the wavenumber of the spatial patterns.     

Spatiotemporal chaos arising from standing waves in a reaction-diffusion system with cross-diffusion

We show that quasi-standing wave patterns appear in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction when a cross-diffusion term is added, no wave instability is required in this case. These standing waves have a frequency that is half the frequency of bulk oscillations displayed in the absence of diffusive coupling. The standing wave patterns show a dependence on the systems size. Regular standing waves can be observed for small systems, when the system size is an integer multiple of half the wavelength. For intermediate sizes, irregular patterns are observed. For large sizes, the system shows an irregular state of spatiotemporal chaos, where standing waves drift, merge, and split, and also phase slips may occur.     

Complex wave patterns in an effective reaction-diffusion model for chemical reactions in microemulsions

An effective medium theory is employed to derive a simple qualitative model of a pattern forming chemical reaction in a microemulsion. This spatially heterogeneous system is composed of water nanodroplets randomly distributed in oil. While some steps of the reaction are performed only inside the droplets, the transport through the extended medium occurs by diffusion of intermediate chemical reactants as well as by collisions of the droplets. We start to model the system with heterogeneous reaction-diffusion equations and then derive an equivalent effective spatially homogeneous reaction-diffusion model by using earlier results on homogenization in heterogeneous reaction-diffusion systems [S.Alonso, M.Ba?r, and R.Kapral, J. Chem. Phys. 134, 214102 (2009)]. We study the linear stability of the spatially homogeneous state in the resulting effective model and obtain a phase diagram of pattern formation, that is qualitatively similar to earlier experimental results for the Belousov-Zhabotinsky reaction in an aerosol OT (AOT)-water-in-oil microemulsion [V.K.Vanag and I.R.Epstein, Phys. Rev. Lett. 87, 228301 (2001)]. Moreover, we reproduce many patterns that have been observed in experiments with the Belousov-Zhabotinsky reaction in an AOT oil-in-water microemulsion by direct numerical simulations.     

Transition from traveling to standing waves as a function of frequency in a reaction-diffusion system

The addition of polyethylene glycol to the Belousov-Zhabotinsky reaction increases the frequency of oscillations, which in an extended system causes a transition from traveling to standing waves. A further increase in frequency causes another transition to bulk oscillations. The standing waves are composed of two domains, which oscillate out of phase with a small delay between them, the delay being smaller as the frequency of oscillations is increased.     

Integral equation theory for atactic polystyrene melt with a coarse-grained model

In this work, an integral equation approach to investigate the atactic polystyrene (aPS) melt based on polymer reference interaction site model (PRISM) theory is proposed. The intramolecular structure factors, required as input to PRISM theory, are obtained from the semiflexible chain model. With a novel coarse-graining procedure and the explicit-atom molecular-dynamics (MD) simulations for aPS, the parameters needed for the coarse-grained model are obtained by using an automatic simplex optimization. These parameters can be used to describe the structure and thermodynamic properties of the complex aPS melt and good agreement is obtained between the theory and MD simulations. The proposed integral equation approach provides a basis for describing the structure and properties of PS nanocomposites where the application of molecular simulation is difficult.     

Isometric graphing and multidimensional scaling for reaction-diffusion modeling on regular and fractal surfaces with spatiotemporal pattern recognition

Heterogeneous surface reactions exhibiting complex spatiotemporal dynamics and patterns can be studied as processes involving reaction-diffusion mechanisms. In many realistic situations, the surface has fractal characteristics. This situation is studied by isometric graphing and multidimensional scaling (IGMDS) of fractal surfaces for extracting geodesic distances (i.e., shortest scaled distances that obtain edges of neighboring surface nodes and their interconnections) and the results obtained used to model effects of surface diffusion with nonlinear reactions. Further analysis of evolved spatiotemporal patterns may be carried out by IGMDS because high-dimensional snapshot data can be efficiently projected to a transformed subspace with reduced dimensions. Validation of the IGMDS methodology is carried out by comparing results with reduction capabilities of conventional principal component analysis for simple situations of reaction and diffusion on surfaces. The usefulness of the IGMDS methodology is shown for analysis of complex patterns formed on both regular and fractal surfaces, and using generic nonlinear reaction-diffusion systems following FitzHugh Nagumo and cubic reaction kinetics. The studies of these systems with nonlinear kinetics and noise show that effects of surface disorder due to fractality can become very relevant. The relevance is shown by studying properties of dynamical invariants in IGMDS component space, viz., the Lyapunov exponents and the KS entropy for interesting situations of spiral formation and turbulent patterns.