Gray-Scott System Research Articles

Highly efficient NURBS-based isogeometric analysis for coupled nonlinear diffusion–reaction equations with and without advection

Nonlinear diffusion–reaction systems model a multitude of physical phenomena. A common situation is biological development modeling where such systems have been widely used to study spatiotemporal phenomena in cell biology. Systems of coupled diffusion–reaction equations are usually subject to some complicated features directly related to their multiphysics nature. Moreover, the presence of advection is source of numerical instabilities, in general, and adds another challenge to these systems. In this study, we propose a NURBS-based isogeometric analysis (IgA) combined with a second-order Strang operator splitting to deal with the multiphysics nature of the problem. The advection part is treated in a semi-Lagrangian framework and the resulting diffusion–reaction equations are then solved using an efficient time-stepping algorithm based on operator splitting. The accuracy of the method is studied by means of a advection–diffusion–reaction system with analytical solution. To further examine the performance of the new method on geometries more general than rectangles (e.g., L-shaped domains and parts of annuli), the well-known Schnakenberg–Turing problem is considered with and without advection. Finally, a Gray–Scott system on a circular domain is also presented. The results obtained demonstrate the efficiency of our new algorithm to accurately reproduce the solution in the presence of complex patterns on more complicated geometries. Moreover, the new method clarifies the effect of geometry on Turing patterns.

An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray–Scott system

This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order α∈(1,2) in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form u=0 in R∖Ω. The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order α: from self-replication to standing waves and from travelling waves to self-replication.

Entropy Production in Reaction-Diffusion Systems Confined in Narrow Channels.

This work analyzes the effect of wall geometry when a reaction-diffusion system is confined to a narrow channel. In particular, we study the entropy production density in the reversible Gray-Scott system. Using an effective diffusion equation that considers modifications by the channel characteristics, we find that the entropy density changes its value but not its qualitative behavior, which helps explore the structure-formation space.

Bifurcation Analysis and Steady-State Patterns in Reaction–Diffusion Systems Augmented with Self- and Cross-Diffusion

In this article, a study of long-term behavior of reaction–diffusion systems augmented with self- and cross-diffusion is reported, using an augmented Gray–Scott system as a generic example. The methodology remains general, and is therefore applicable to some other systems. Simulations of the temporal model (nonlinear parabolic system) reveal the presence of steady states, often associated with energy dissipation. A Newton method based on a mixed finite element method is provided, in order to directly evaluate the steady states (nonlinear elliptic system) of the temporal system, and validated against its solutions. Linear stability analysis using Fourier analysis is carried out around homogeneous equilibrium, and using spectral analysis around nonhomogeneous ones. For the latter, the spectral problem is solved numerically. A multiparameter bifurcation is reported. Original steady-state patterns are unveiled, not observable with linear diffusion only. Two key observations are: a dependency of the pattern with the initial condition of the system, and a dependency on the geometry of the domain.

High-fidelity simulations for Turing pattern formation in multi-dimensional Gray–Scott reaction-diffusion system

In this study, the authors present high-fidelity numerical simulations for capturing the Turing pattern in a multi-dimensional Gray–Scott reaction-diffusion system. For this purpose, an explicit mixed modal discontinuous Galerkin (DG) scheme based on multi-dimensional structured meshes is employed. This numerical scheme deals with high-order derivatives in diffusion, and highly nonlinear functions in reaction terms. Spatial discretization is accomplished using hierarchical basis functions based on scaled Legendre polynomials. A new reaction term treatment is also detailed, showing an intrinsic property of the DG scheme and preventing incorrect solutions due to excessively nonlinear reaction terms. The system is reduced to a set of time-dependent ordinary differential equations, which are solved with an explicit third-order Total Variation Diminishing (TVD) Runge–Kutta method. The Saddle-Node, Hopf, and Turing bifurcations’ boundary conditions are also examined for the system, and subsequently, Turing space is identified. The developed numerical scheme is applied to various multidimensional Gray–Scott systems to assess its ability to capture Turing pattern formation. Several test problems, such as stationary and non-stationary waves, pulse-splitting waves, self-replicating waves, and different types of spots, are chosen from the literature to demonstrate the different Turing patterns.

Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

AbstractIn this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.

Physics-informed neural networks approach for 1D and 2D Gray-Scott systems

Nowadays, in the Scientific Machine Learning (SML) research field, the traditional machine learning (ML) tools and scientific computing approaches are fruitfully intersected for solving problems modelled by Partial Differential Equations (PDEs) in science and engineering applications. Challenging SML methodologies are the new computational paradigms named Physics-Informed Neural Networks (PINNs). PINN has revolutionized the classical adoption of ML in scientific computing, representing a novel class of promising algorithms where the learning process is constrained to satisfy known physical laws described by differential equations. In this paper, we propose a PINN-based computational study to deal with a non-linear partial differential equations system. In particular, using this approach, we solve the Gray-Scott model, a reaction–diffusion system that involves an irreversible chemical reaction between two reactants. In the unstable region of the model, we consider some a priori information related to dynamical behaviors, i. e. a supervised approach that relies on a finite difference method (FDM). Finally, simulation results show that PINNs can successfully provide an approximated Grey-Scott system solution, reproducing the characteristic Turing patterns for different parameter configurations.

On pattern formation in reaction–diffusion systems containing self- and cross-diffusion

In this article we propose a unified framework in order to study reaction–diffusion systems containing self- and cross-diffusion using a free energy approach. This framework naturally leads to the formulation of an energy law, and to a numerical method respecting a discrete version of the latter. It constitutes an alternative method and complements the standard linear stability analysis, as it allows for the numerical study of nonlinear patterns, while monitoring the energy evolution, even in complex geometries. As an application, we propose and study a modified Gray–Scott system augmented with self- and cross-diffusion terms. Numerical simulations unveil original patterns, clearly distinct from those obtained with linear diffusion only.

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

<p style='text-indent:20px;'>In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.</p>

Dynamics of multi-pulse splitting process in one-dimensional Gray-Scott system with fractional order operator

In this paper, the dynamic process of pulse-splitting patterns is reported in fractional medium. In the classical Gray-Scott system, the integer-order derivative is replaced with the known Atangana-Baleanu fractional order derivative in the sense of Caputo. mathematical analysis such as the existence of stationary solutions for pulse-splitting process, existence and uniqueness of solutions for the fractional system are presented. The beauty of the work is further demonstrated by presenting numerical results for different values of γ in one dimensional space. We deduced from the numerical experiments that pulse-splitting patterns in both integer and noninteger order scenarios are almost the same.

Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise

<p style='text-indent:20px;'>We first show that the stochastic two-compartment Gray-Scott system generates a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic two-compartment Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.

Stable and Unstable Periodic Spiky Solutions for the Gray–Scott System and the Schnakenberg System

The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the Hopf bifurcation is supercritical for the Schnakenberg system, and hence the bifurcating periodic solutions are linearly stable. For the Gray–Scott system, there is a critical feeding rate, when the feeding rate of the system is greater than this critical feeding rate, the Hopf bifurcation is supercritical which implies the bifurcating periodic solutions are linearly stable, while when the feeding rate is smaller than this critical feeding rate, the Hopf bifurcation is subcritical, implying the bifurcating periodic solutions are linearly unstable.

Generative complexity of Gray–Scott model

In the Gray–Scott reaction–diffusion system one reactant is constantly fed in the system, another reactant is reproduced by consuming the supplied reactant and also converted to an inert product. The rate of feeding one reactant in the system and the rate of removing another reactant from the system determine configurations of concentration profiles: stripes, spots, waves. We calculate the generative complexity—a morphological complexity of concentration profiles grown from a point-wise perturbation of the medium—of the Gray–Scott system for a range of the feeding and removal rates. The morphological complexity is evaluated using Shannon entropy, Simpson diversity, approximation of Lempel–Ziv complexity, and expressivity (Shannon entropy divided by space-filling). We analyse behaviour of the systems with highest values of the generative morphological complexity and show that the Gray–Scott systems expressing highest levels of the complexity are composed of the wave-fragments (similar to wave-fragments in sub-excitable media) and travelling localisations (similar to quasi-dissipative solitons and gliders in Conway’s Game of Life).

The travelling wave of Gray-Scott systems – existence, multiplicity and stability

ABSTRACTThis article studies existence and stability of travelling wave of unstirred Gray-Scott system in biological pattern formation which models an isothermal chemical reaction involving two chemical species, a reactant A and an auto-catalyst B, and a linear decay , where C is an inert product. Our result shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different travelling pulses than related models in the literature. In particular, the existence of multiple travelling waves which have distinctive number of local maxima is proved. Furthermore, the stability of travelling wave of the reaction–diffusion system of isothermal diffusion system , is also studied.

REGULARITY OF PULLBACK ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC COUPLED REACTION-DIFFUSION SYSTEMS

We consider the dynamical behavior of the typical non-autonomous autocatalytic stochastic coupled reaction-diffusion systems on the entire space Rn. Some new uniform asymptotic estimates are implemented to investigate the existence of pullback attractors in the Sobolev space H1(Rn)3 for the three-component reversible Gray-Scott system.

Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method

In this paper, we have applied modified cubic B-spline based differential quadrature method to get numerical solutions of one dimensional reaction-diffusion systems such as linear reaction-diffusion system, Brusselator system, Isothermal system and Gray-Scott system. The models represented by these systems have important applications in different areas of science and engineering. The most striking and interesting part of the work is the solution patterns obtained for Gray Scott model, reminiscent of which are often seen in nature. We have used cubic B-spline functions for space discretization to get a system of ordinary differential equations. This system of ODE’s is solved by highly stable SSP-RK43 method to get solution at the knots. The computed results are very accurate and shown to be better than those available in the literature. Method is easy and simple to apply and gives solutions with less computational efforts.

Upper semicontinuity of random attractors for stochastic three-component reversible Gray–Scott system

We consider the upper semicontinuity of the global random attractors for the stochastic three-component reversible Gray–Scott system on unbounded domains when the intensity of the noise converges to zero.

Random attractors and robustness for stochastic reversible reaction-diffusion systems

For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.

Uniform attractor of non-autonomous three-component reversible Gray–Scott system

We discuss the long time behavior of solutions for the non-autonomous three-component reversible Gray–Scott system with Dirichlet boundary condition on a bounded domain of space dimension n⩽3. When the external force is translation compact in appropriate space, we obtain the existence and the structure of the uniform attractor for the processes associated to the system by constructing skew product flow on the extended phase space. Also, we establish that there exists a unique bounded asymptotically stable solution to the system when the external forces are properly small.