Model Reaction-diffusion System Research Articles

Reaction diffusion systems and extensions of quantum stochastic processes

Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued Itô calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth–death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.

Parametric instability-induced synchronization in chemical oscillations and spatiotemporal patterns.

We consider a model reaction-diffusion system with two coupled layers in which one of the components in a layer is parametrically driven by a periodic force. On perturbation of a homogeneous stable steady state, the system exhibits parametric instability inducing synchronization in temporal oscillation at half the forcing frequency in absence of diffusion and spatiotemporal patterns in presence of diffusion, when strength of parametric forcing and the strength of coupling are kept above their critical thresholds. We have formulated a general scheme to derive analytically the critical thresholds and dispersion relation to locate the unstable spatial modes lying between the tilted Arnold tongue in the amplitude-frequency plot. Full numerical simulations on Gierer-Meinhardt activator-inhibitor model corroborate our theoretical analysis on parametric instability-induced antiphase synchronization in chemical oscillation and spatiotemporal pattern formation, between the two layers.

Statistical approach for parameter identification by Turing patterns

Prevailing theories in biological pattern formation, such as in morphogenesis or multicellular structures development, have been based on purely chemical processes, with the Turing models as the prime example. Recent studies have challenged the approach, by underlining the role of mechanical forces. A quantitative discrimination of competing theories is difficult, however, due to the elusive character of the processes: different mechanisms may result in similar patterns, while patterns obtained with a fixed model and fixed parameter values, but with small random perturbations of initial values, will significantly differ in shape, while being of the “same” type. In this sense each model parameter value corresponds to a family of patterns, rather than a fixed solution. For this situation we create a likelihood that allows a statistically sound way to distinguish the model parameters that correspond to given patterns. The method allows us to identify model parameters of reaction-diffusion systems by using Turing patterns only, i.e., the steady-state solutions of the respective equations without the use of transient data or initial values. The method is tested with three classical models of pattern formation: the FitzHugh-Nagumo model, Gierer-Meinhardt system and Brusselator reaction-diffusion system. We quantify the accuracy achieved by different amounts of training data by Bayesian sampling methods. We demonstrate how a large enough ensemble of patterns leads to detection of very small but systematic structural changes, practically impossible to distinguish with the naked eye.

Reactive multi-particle collision dynamics with reactive boundary conditions

In the present study, an off-lattice particle-based method called the reactive multi-particle collision (RMPC) dynamics is extended to model reaction-diffusion systems with reactive boundary conditions in which the a priori diffusion coefficient of the particles needs to be maintained throughout the simulation. To this end, the authors have made use of the so-called bath particles whose purpose is only to ensure proper diffusion of the main particles in the system. In order to model partial adsorption by a reactive boundary in the RMPC, the probability of a particle being adsorbed, once it hits the boundary, is calculated by drawing an analogy between the RMPC and Brownian Dynamics. The main advantages of the RMPC compared to other molecular based methods are less computational cost as well as conservation of mass, energy and momentum in the collision and free streaming steps. The proposed approach is tested on three reaction-diffusion systems and very good agreement with the solutions to their corresponding partial differential equations is observed.

A Lattice Boltzmann Model for Oscillating Reaction–Diffusion

AbstractA computational algorithm based on the lattice Boltzmann method (LBM) is proposed to model reaction–diffusion systems. In this paper, we focus on how nonlinear chemical oscillators like Belousov–Zhabotinsky (BZ) and the chlorite–iodide–malonic acid (CIMA) reactions can be modeled by LBM and provide with new insight into the nature and applications of oscillating reactions. We use Gaussian pulse initial concentrations of sulfuric acid in different places of a bidimensional reactor and nondiffusive boundary walls. We clearly show how these systems evolve to a chaotic attractor and produce specific pattern images that are portrayed in the reactions trajectory to the corresponding chaotic attractor and can be used in robotic control.

Regular and Discontinuous Solutions in a Reaction-Diffusion Model for Hair Follicle Spacing

Solutions of a model reaction-diffusion system inspired by a model for hair follicle initiation in mice are constructed and analysed for the case of a one-dimensional domain. It is shown that all regular spatially heterogeneous solutions of the problem are unstable. Numerical tests show that the only asymptotically stable weak solutions are those with large jump discontinuities.

Nonlinear effects on Turing patterns: Time oscillations and chaos

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction-diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns.

A model reaction diffusion system under spatial perturbation: theoretical and numerical investigation

We consider a model (Lengyel-Epstein) reaction-diffusion system under spatial parametric modulation and demonstrate the effect of resonance shift of the Hopf-Turing boundary. A systematic perturbative and numerical analysis shows that this shift may induce spatial inhomogeneity on an homogeneous stable state resulting in pattern formation.

Transverse instability of line defects of period-2 spiral waves.

Spiral waves that arise in period-2 oscillatory media extended in space generically bear "line defects" along which the local kinetics exhibits a period-1 oscillation. Locally, these defect structures can be viewed as a front separating two period-2 oscillatory domains oscillating 2pi out of phase. Here we show that their shape can become sinusoidal with a transverse instability as in bistable fronts. This instability eventually leads to a line-defect filled spatiotemporal chaotic state having erratic proliferations, annihilations, and regenerations of line defects. The same sequence of phenomena is observed in a model reaction-diffusion system as well as in an experimental system.

Symmetric, asymmetric, and antiphase Turing patterns in a model system with two identical coupled layers.

We study Turing pattern formation in a model reaction-diffusion system with two coupled identical layers. The coupling creates a pitchfork bifurcation, which unfolds the symmetric steady state via primary Turing instability, into a pair of distinct, unstable, asymmetric steady states (a-SS). The a-SS gain stability at a reverse Turing bifurcation. The multiple stabilities created by the coupling generate a corresponding multiplicity of structures, including symmetric, asymmetric, antiphase, and localized Turing patterns. Coexistence and competition of the different types of Turing patterns are studied. A one-dimensional localized structure exhibits striking curvature effects.

Diffraction problems for quasilinear reaction–diffusion systems

The aim of this paper is to investigate existence, uniqueness and qualitative properties of the weak and piecewise classical solutions of the diffraction problem for weakly coupled reaction–diffusion systems. Monotony and stability with respect to the time variable of the weak solutions is proved. As an application is considered a model reaction–diffusion system describing an enzyme sensor for concentration of phenol in the chemical industry.

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.

Singular limit of reactiondiffusion systems and modified motion by mean curvature

We consider two model reaction-diffusion systems of bistable type arising in the theory of phase transition; they appear in various physical contexts, such as thin magnetic films and diblock copolymers. We prove the convergence of the solution of these systems to the solution of free-boundary problems involving modified motion by mean curvature.

On the growth of nonequilibrium spatial correlations in a model reaction diffusion system

The mesoscopic description of a system with chemical reactions predicts that if the detailed balance condition is not satisfied then the nonequilibrium spatial correlations between concentrations of reactants may appear. The present work is concerned with the dynamics of their growth. We assume that the system is homogeneous at the beginning and study how fast the nonequilibrium correlations develop. The theory based on the master equation is compared with molecular dynamics simulations performed for a model system of “reacting” hard spheres. The qualitative characteristics of the growth of the spatial correlations, predicted by the theory based on a master equation, are confirmed in our simulations. However, this theory fails to give the correct quantitative description of this process.

Modulated and alternating waves in a reaction-diffusion model with wave instability

Various patterns of modulated waves are found beyond the onset of the wave instability in a model reaction-diffusion system with length between 1 and 3 times the basic wavelength. With periodic boundary conditions, we observe low-frequency modulation of the speed of the travelling waves. A fascinating pattern of waves that periodically change their direction of propagation along the ring is found when the ring length is between 1.05 and 1.35 times the basic wavelength. With zero-flux boundary conditions, a related pattern of modulated standing waves is found between the domains of the half-wavelength and the one-wavelength standing waves; the period of modulation contains two phases of non-stationary travelling waves, moving alternately left and right, separated by phases of short-lived asymmetric standing waves.

Control of self-replicating patterns in a model reaction-diffusion system.

Spatiotemporal patterns, e.g., spots, which replicate, grow, and die as solutions of nonlinear reaction-diffusion models are interesting because of their striking resemblance to the self-replicating phenomena observed in many important physical, chemical, and biological processes, e.g., micelles, living cells, and DNA. With this viewpoint we considered the characterization of a model self-replicating system from limited and inaccurate system information. We also addressed control goals that are relevant to many natural processes exhibiting this phenomenon, such as altering and arresting the growth of replication.

Class of cellular automata for reaction-diffusion systems.

We introduce a class of cellular automata (CA) to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving-average procedure to implement diffusion and a probabilistic table lookup for the reactive part. The applicability of the present CA is demonstrated using the Ginzburg-Landau equation.

NUMERICAL STUDY OF TRAVELLING WAVES IN A REACTION-DIFFUSION SYSTEM: RESPONSE TO A SPATIOTEMPORAL FORCING

We investigate numerically the effect of a spatiotemporal forcing on travelling waves produced in a model reaction-diffusion system operating under periodic boundary conditions. It is shown that, for suitable parameter values, this external forcing can stabilize the progressive wave regime and impose its directionality through wave inversion. When the field fails to reverse waves travelling in an opposite direction, chaotic-like spatiotemporal patterns may be observed.

Regular and chaotic chemical spatiotemporal patterns.

The first experimental observation of a bifurcation sequence of patterns in a reaction-diffusion system is reported. The experiment on a one-dimensional system, maintained away from equilibrium, reveals steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic spatiotemporal states. Results from a model reaction-diffusion system agree qualitatively with the experiment and provide insight into the physical mechanism that drives the observed behavior.