Reaction-diffusion Systems Research Articles

Quantum reaction-limited reaction–diffusion dynamics of noninteracting Bose gases

We investigate quantum reaction–diffusion (RD) systems in one-dimension with bosonic particles that coherently hop in a lattice, and when brought in range react dissipatively. Such reactions involve binary annihilation ( A+A→∅ ) and coagulation ( A+A→A ) of particles at distance d. We consider the reaction-limited regime, where dissipative reactions take place at a rate that is small compared to that of coherent hopping. In classical RD systems, this regime is correctly captured by the mean-field approximation. In quantum RD systems, for noninteracting fermionic systems, the reaction-limited regime recently attracted considerable attention because it has been shown to give universal power law decay beyond mean field for the density of particles as a function of time. Here, we address the question whether such universal behavior is present also in the case of the noninteracting Bose gas. We show that beyond mean-field density decay for bosons is possible only for reactions that allow for destructive interference of different decay channels. Furthermore, we study an absorbing-state phase transition induced by the competition between branching A→A+A , decay A→∅ and coagulation A+A→A . We find a stationary phase-diagram, where a first and a second-order transition line meet at a bicritical point which is described by tricritical directed percolation. These results show that quantum statistics significantly impact on both the stationary and the dynamical universal behavior of quantum RD systems.

Asymptotic behaviour of a conservative reaction-diffusion system associated with a Markovian process algebra model

This paper demonstrates a framework using lower and upper solutions approximation to investigate the asymptotic behaviour of conservative reaction-diffusion systems associated with Markovian process algebra models. In a case study, we proved the uniform convergence of the solution to its constant equilibrium as time tends to infinity, which is further illustrated by numerical experiments.

Finite-time synchronization of delayed semi-Markov reaction–diffusion systems: An asynchronous boundary control scheme

This paper tries to study the problem of finite-time synchronization for delayed semi-Markov reaction–diffusion systems. Based on the spatial and parametric characteristics of the considered systems, a new asynchronous boundary control scheme is proposed to ensure the finite-time synchronization of the drive and response systems. In the asynchronous boundary control scheme, only an actuator should be placed at the spatial boundary, which is more easier to implement and economical than the other non-boundary control strategies. Besides, the system parameters and controller follow two asynchronous semi-Markov chains for jumping, which is more practical than obeying one semi-Markov chain. Moreover, for the considered systems, we proposes a new lemma of finite-time stability, and by employing the inequality methods and variable substitution, we derive the criterion of finite-time synchronization and a correlative corollary. Finally, a numerical example and an application example on secure communication are carried out to support the developed approach.

Spatiotemporal control over self-assembly of supramolecular hydrogels through reaction–diffusion

Supramolecular self-assembly is ubiquitous in living system and is usually controlled to proceed in time and space through sophisticated reaction–diffusion processes, underpinning various vital cellular functions. In this contribution, we demonstrate how spatiotemporal self-assembly of supramolecular hydrogels can be realized through a simple reaction–diffusion-mediated transient transduction of pH signal. In the reaction–diffusion system, a relatively faster diffusion of acid followed by delayed enzymatic production and diffusion of base from the opposite site enables a transient transduction of pH signal in the substrate. By coupling such reaction–diffusion system with pH-sensitive gelators, dynamic supramolecular hydrogels with tunable lifetimes are formed at defined locations. The hydrogel fibers show interesting dynamic growing behaviors under the regulation of transient pH signal, reminiscent of their biological counterpart. We further demonstrate a proof-of-concept application of the developed methodology for dynamic information encoding in a soft substrate. We envision that this work may provide a potent approach to enable transient transduction of various chemical signals for the construction of new colloidal materials with the capability to evolve their structures and functionalities in time and space.

A reaction–diffusion vector-borne disease model with incubation period in almost periodic environments

To study the multiple effects of seasonal fluctuations, spatial heterogeneity and extrinsic incubation period on vector-borne disease transmission, a nonlocal almost periodic reaction–diffusion model of vector-borne diseases is proposed and studied. We first give a characterization of the upper Lyapunov exponent λ∗ for a class of linear almost periodic reaction–diffusion systems with time delay, and provide a numerical scheme to compute it. Then we show that λ∗ is a threshold value determining the uniform persistence and extinction of our model. Specifically, the disease will die out when λ∗<0, while the disease is uniformly persistent when λ∗>0. Some numerical simulations finally are presented to verify our theoretical results, and to investigate the effects of diffusion rates, extrinsic incubation period and spatial heterogeneity on disease transmission.

Correcting model misspecification in physics-informed neural networks (PINNs)

Data-driven discovery of governing equations in computational science has emerged as a new paradigm for obtaining accurate physical models and as a possible alternative to theoretical derivations. The recently developed physics-informed neural networks (PINNs) have also been employed to learn governing equations given data across diverse scientific disciplines, e.g., in biology and fluid dynamics. Despite the effectiveness of PINNs for discovering governing equations, the physical models encoded in PINNs may be misspecified in complex systems as some of the physical processes may not be fully understood, leading to the poor accuracy of PINN predictions. In this work, we present a general approach to correct the misspecified physical models in PINNs for discovering governing equations, given some sparse and/or noisy data. Specifically, we first encode the assumed physical models, which may be misspecified in PINNs, and then employ other deep neural networks (DNNs) to model the discrepancy between the imperfect models and the observational data. Due to the expressivity of DNNs, the proposed method is capable of reducing the computational errors caused by the model misspecification and thus enables the applications of PINNs in complex systems where the physical processes are not exactly known. Furthermore, we utilize the Bayesian physics-informed neural networks (B-PINNs) and/or ensemble PINNs to quantify uncertainties arising from noisy and/or gappy data in the discovered governing equations. A series of numerical examples including reaction-diffusion systems and non-Newtonian channel and cavity flows demonstrate that the added DNNs are capable of correcting the model misspecification in PINNs and thus reduce the discrepancy between the physical models encoded in PINNs and the observational data. In addition, the B-PINNs and ensemble PINNs can provide reasonable uncertainty bounds in the discovered physical models, which makes the predictions more reliable. We also demonstrate that we can seamlessly combine the present approach with the symbolic regression to obtain the explicit governing equations upon the training of PINNs. We envision that the proposed approach will extend the applications of PINNs for discovering governing equations in problems where the physico-chemical or biological processes are not well understood.

Intermittent boundary control for fixed-time stability of reaction–diffusion systems

This work is a pilot effort to investigate the fixed-time stability (FxTS) of reaction–diffusion systems under aperiodically intermittent boundary control (AIBC). The average control rate and a new Lyapunov function are proposed to overcome the challenges of handling the FxTS of reaction–diffusion systems with AIBC. Moreover, the proposed method is applicable to the study of finite-time stability and exponential stability of reaction–diffusion systems under AIBC. Based on the Wirtinger’s inequality and the Lyapunov method, a FxTS criterion for a reaction–diffusion system with AIBC is given. Finally, two examples are discussed along with the simulated results to verify the effectiveness of the proposed method.

Numerical analysis of light-controlled drug delivery systems

In this paper, we solve a non-linear reaction–diffusion system with Dirichlet–Neumann mixed boundary conditions using a finite difference method (FDM) in space and the implicit midpoint method in time. This type of system appears, e.g., in the mathematical modeling of light-controlled drug delivery. One of the key results of this paper is the proof that the method has superconvergence second-order in space in a discrete H1-norm and optimal second-order convergence in time in a discrete L2-norm. Our result relies on the direct analysis of a suitable error equation, avoiding the classic construction of consistency plus stability implies convergence. One advantage of such an analysis technique is the establishment of the method’s non-linear stability in an elegant way. Numerical examples support the theoretical convergence result.

A variational PDNet network using a learning reaction–diffusion equation

Due to their high performance in modeling and forecasting a large amount of real-world complex phenomena, deep convolutional neural networks have received a great deal of attention over the past ten years. It has been extensively utilized in several recent vision problems, such as image denoising and deblurring. By incorporating recent developments in learning-based nonlinear diffusion models and highlighting the connection between partial differential equations (PDEs) and networks, especially residual networks, we propose a flexible learning reaction–diffusion system with spatially computed convolution kernels. This trained PDE is employed to prevent the low-level features from deteriorating, which is needed for tasks involving denoising. In fact, we introduce an improved network architecture for a denoising network entitled VPDNet through an optimal control problem. More precisely, the network architecture is governed by an explicit Euler discretization of a reaction–diffusion system. We give the main algorithm to identify the kernel parameters using the Alternating Direction Method of Multipliers (ADMM) combined with a Primal–Dual algorithm. Finally, numerical simulations are provided with appropriate comparisons to demonstrate the efficiency of the elaborated network.

Self-replicating segregation patterns in horizontally vibrated binary mixture of granules

Fluidized granular mixtures of various particle sizes exhibit intriguing patterns as different species segregate and condense. However, understanding the segregation dynamics is hindered by the inability to directly observe the time evolution of the internal structure. We discover self-replicating bands within a quasi-2D container subjected to horizontal agitation, resulting in steady surface waves. Through direct observation of surface flow and evolving internal structures, we reveal the crucial role of coupling among segregation, surface flow, and hysteresis in granular fluidity. We develop Bonhoeffer-van der Pol type equations grounded in experimental observations, reproducing complex band dynamics, such as replication, oscillation, and breathing. It suggests the similarity between pattern formation in granular segregation and that in reaction–diffusion systems.

How to regulate pattern formations for malware propagation in cyber-physical systems.

Malware propagation can be fatal to cyber-physical systems. How to detect and prevent the spatiotemporal evolution of malware is the major challenge we are facing now. This paper is concerned with the control of Turing patterns arising in a malware propagation model depicted by partial differential equations for the first time. From the control theoretic perspective, the goal is not only to predict the formation and evolution of patterns but also to design the spatiotemporal state feedback scheme to modulate the switch of patterns between different modes. The Turing instability conditions are obtained for the controlled malware propagation model with cross-diffusion. Then, the multi-scale analysis is carried out to explore the amplitude equations near the threshold of Turing bifurcation. The selection and stability of pattern formations are determined based on the established amplitude equations. It is proved that the reaction-diffusion propagation model has three types of patterns: hexagonal pattern, striped pattern, and mixed pattern, and selecting the appropriate control parameters can make the pattern transform among the three patterns. The results of the analysis are numerically verified and provide valuable insights into dynamics and control of patterns embedded in reaction-diffusion systems.

Global asymptotic stability of endemic equilibria and stability of traveling waves for a diffusive SIR epidemic model with logistic growth

AbstractThis paper deals with the asymptotic behavior of solutions for the diffusive epidemic model with logistic growth. In the first part, we consider the initial boundary value problem on the bounded domain and derive the stabilization of the solutions of the reaction–diffusion system to a constant equilibrium. In the second part, we consider the initial value problem on $${\mathbb {R}},$$ R , and derive the stability of forced waves under certain perturbations of a class of initial data.

Dynamics of an EIS spatially heterogeneous rabies model

This paper is devoted to the study of population dynamics of an EIS epidemic reaction-diffusion model in spatially heterogeneous environments, where the rates of disease-induced mortality and disease transmission are space dependent. For such a class of systems, we provide clear pictures on the global dynamics in terms of the basic reproduction number R0, which improves the local dynamics proposed in Wang and Zhao (2012) [20]. When R0≤1, the disease-free steady state is globally asymptotically stable. Different from earlier works for many epidemic models, we obtain the positive steady states by using bifurcation theory for R0>1, which can be easily generalized to the case where the reaction-diffusion system involves partial diffusion coefficients of zero.

Mechanism of branching morphogenesis inspired by diatom silica formation

The silica-based cell walls of diatoms are prime examples of genetically controlled, species-specific mineral architectures. The physical principles underlying morphogenesis of their hierarchically structured silica patterns are not understood, yet such insight could indicate novel routes toward synthesizing functional inorganic materials. Recent advances in imaging nascent diatom silica allow rationalizing possible mechanisms of their pattern formation. Here, we combine theory and experiments on the model diatom Thalassiosira pseudonana to put forward a minimal model of branched rib patterns-a fundamental feature of the silica cell wall. We quantitatively recapitulate the time course of rib pattern morphogenesis by accounting for silica biochemistry with autocatalytic formation of diffusible silica precursors followed by conversion into solid silica. We propose that silica deposition releases an inhibitor that slows down up-stream precursor conversion, thereby implementing a self-replicating reaction-diffusion system different from a classical Turing mechanism. The proposed mechanism highlights the role of geometrical cues for guided self-organization, rationalizing the instructive role for the single initial pattern seed known as the primary silicification site. The mechanism of branching morphogenesis that we characterize here is possibly generic and may apply also in other biological systems.

A diffusible small-RNA-based Turing system dynamically coordinates organ polarity.

The formation of a flat and thin leaf presents a developmentally challenging problem, requiring intricate regulation of adaxial-abaxial (top-bottom) polarity. The patterning principles controlling the spatial arrangement of these domains during organ growth have remained unclear. Here we show that this regulation in Arabidopsis thaliana is achieved by an organ-autonomous Turing reaction-diffusion system centred on mobile small RNAs. The data illustrate how Turing dynamics transiently instructed by prepatterned information is sufficient to self-sustain properly oriented polarity in a dynamic, growing organ, presenting intriguing parallels to left-right patterning in the vertebrate embryo. Computational modelling demonstrates that this self-organizing system continuously adapts to coordinate the robust planar polarity of a flat leaf while affording flexibility to generate the tissue patterns of evolutionarily diverse organ shapes. Our findings identify a small-RNA-based Turing network as a dynamic regulator of organ polarity that accounts for leaf shape diversity at the level of the individual organ, plant or species.

Supervised and unsupervised learning of (1+1) -dimensional even-offspring branching annihilating random walks

Machine learning (ML) of phase transitions (PTs) has gradually become an effective approach that enables us to explore the nature of various PTs more promptly in equilibrium and nonequilibrium systems. Unlike equilibrium systems, non-equilibrium systems display more complicated and diverse features because of the extra dimension of time, which is not readily tractable, both theoretically and numerically. The combination of ML and most renowned nonequilibrium model, directed percolation (DP), led to some significant findings. In this study, ML is applied to (1+1) -d, even offspring branching annihilating random walks (BAW), whose universality class is not DP-like. The supervised learning of (1+1) -d BAW via convolutional neural networks (CNN) results in a more accurate prediction of the critical point than the Monte Carlo (MC) simulation for the same system sizes. The dynamic exponent z and spatial correlation length correlation exponent ν⊥ were also measured and found to be consistent with their respective theoretical values. Furthermore, the unsupervised learning of (1+1) -d BAW via an autoencoder (AE) gives rise to a transition point, which is the same as the critical point. The latent layer of AE, through a single neuron, can be regarded as the order parameter of the system being properly re-scaled. Therefore, we believe that ML has exciting application prospects in reaction-diffusion systems such as BAW and DP.

Front Selection in Reaction–Diffusion Systems via Diffusive Normal Forms

Front Selection in Reaction–Diffusion Systems via Diffusive Normal Forms

Dihedral rings of patterns emerging from a Turing bifurcation

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in various ring-like dihedral configurations, bifurcating from quiescence near a Turing instability in generic two-component reaction-diffusion systems. The methods used are constructive and provide accurate initial conditions for numerical continuation methods to path-follow these ring-like patterns in parameter space. Our analysis is complemented by numerical investigations that illustrate our findings.

Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction–Diffusion Systems

Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction–Diffusion Systems

Minimal reaction schemes for pattern formation.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.