One-dimensional Reaction-diffusion System Research Articles

Lyapunov-based nonlinear boundary control design with predefined convergence for a class of 1D linear reaction-diffusion equations

In this paper, we treat the problem of Lyapunov-based nonlinear boundary stabilization of a class of one-dimensional reaction-diffusion systems with any predefined convergence (asymptotic or non-asymptotic). As an application, we focus on the non-asymptotic notions (finite-time and fixed-time) for which we give some particular explicit control designs followed by some numerical simulations. The key idea of our approach is to use a “spatially weighted L2-norm” as a Lyapunov functional to design a nonlinear controller and to ensure stability with any desired convergence.

BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE

In this paper, through bifurcation analysis and numerical simulations, we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system. The hydra effect, which is a paradoxical result in both theoretical and applied ecology, refers to the phenomenon in which an increase in population mortality enhances its own population size. We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable. Meanwhile, numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steady-state solution. In addition, we introduce the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing-Hopf bifurcation with the parameters $ d_{2} $ and $ m_{C} $, respectively, as well as the normal form for the Turing-Hopf bifurcation. Based on the obtained normal form, we analyze the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point. Finally, the numerical simulations are carried out to corroborate the obtained theoretical results.

Temperature and Light Multiphysical Fields Modulate the Propagation of Chemical Waves on Multiple Length Scales in a System

Abstract The phenomenon of multiple length scales exists in plants during growth, which is influenced not only by light but also by other physical fields such as substance concentration and temperature, and the combined effects of various physical fields form a complex multi-scale phenomenon. However, the mechanism of the multi-physical interaction of temperature and light on the formation of multiple length scales in plants has not been investigated. In this paper, we coupled a gel with a Belousov-Zhabotinsky (BZ) chemical reaction system, constructed a one-dimensional open reaction diffusion system, and investigated the changes in the multiple length scale periodic structure of chemical waves in the one-dimensional gel system under the combined effect of multiple physical fields of light and temperature. It was found that when increasing the light intensity at T = 26 °C or decreasing the temperature at the light intensity I ≤ 50 µW·cm−2, the chemical wave multiple length scale period structure of the system changed as follows: P8 → P16 → P8 → P4 → P2 → P1, and the chemical wave period structure complexity showed a non-linear change of first increasing and then decreasing. When increasing the light intensity at T ≤ 24 °C or decreasing the temperature at the light intensity I > 50 µW·cm−2, the chemical wave period structure complexity of the system shows a linear decrease.

Spatiotemporal patterns emerging from a spatially localized time-delayed feedback scheme.

In attempts to manage spatiotemporal transient chaos in spatially extended systems, these systems are often subjected to protocols that perturb them as a whole and stabilize globally a new dynamic regime, as, for example, a uniform steady state. In this work we show that selectively perturbing only part of a system can generate space-time patterns that are not observed when controlling the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. Specifically, we use a spatially localized time-delayed feedback scheme on the one-dimensional Gray-Scott reaction-diffusion system in the transient chaotic regime and demonstrate, through the numerical integration of the resulting kinetic equations, the stabilization of spatially localized space-time patterns that can be perfectly periodic. The mechanism underlying the observed pattern generation is related to diffusion across the interfaces separating the perturbed and unperturbed regions.

Traveling wave solutions for a class of reaction-diffusion system

In this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

Multi-spike Patterns in the Gierer–Meinhardt System with a Nonzero Activator Boundary Flux

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

PurposeThis paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time.Design/methodology/approachIn a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight.FindingsThe proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems.Originality/valueThe originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

Adaptive moving mesh algorithm based on local reaction rate

An empirical mesh adaption algorithm is introduced for modeling one-dimensional reaction-diffusion systems with large moving gradients. Our new algorithm is based on the revelation, that in reaction-diffusion systems the high moving concentration gradients appear nearby to the region where the rate of reaction is maximal, thus the local reaction rate can be used to control the mesh adaption. We found, that the main advantage of such a method is its simplicity and easy implementation. As an example we study an acid-base diode, where large moving gradients appear. The mathematical model of the diode contains several parabolic PDEs, coupled with one elliptic PDE. An r-refinement technique is used and attached to the commercial finite element solver COMSOL. We investigated the time-dependent salt effects of the diode with our developed algorithm. Our mesh adaption method is advantageous for modeling of any reaction-diffusion systems with localized high concentration gradients.

Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

Investigation of a reaction-diffusion system, related to retinal patterning

In this paper we study a one-dimensional spatially-continuous reaction-diffusion system, related to patterning of Drosophila’s retina. We search for conditions of stability or diffusion-driven (Turing) instability of a homogeneous prepattern steady state.

Eckhaus selection: The mechanism of pattern persistence in a reaction-diffusion system.

In this work, we show theoretically and numerically that a one-dimensional reaction-diffusion system, near the Turing bifurcation, produces different number of stripes when, in addition to random noise, the Fourier mode of a prepattern used to initialize the system changes. We also show that the Fourier modes that persist are inside the Eckhaus stability regions, while those outside this region follow a wave number selection process not predicted by the linear analysis. To test our results, we use the Brusselator reaction-diffusion system obtaining an excellent agreement between the weakly nonlinear predictions of the real Ginzburg-Landau equations and the numerical solutions near the bifurcation. Although the persistence of patterns is not relevant as a simple generating mechanism of self-organization, it is crucial to understand the formation of patterns that occurs in multiple stages. In this work, we discuss the relevance of our results on the robustness and diversity of solutions in multiple-steps mechanisms of biological pattern formation and auto-organization in growing domains.

Boundary Observers for a Reaction–Diffusion System Under Time-Delayed and Sampled-Data Measurements

We construct finite-dimensional observers for a one-dimensional reaction–diffusion system with boundary measurements subject to time-delays and data sampling. The system has a finite number of unstable modes approximated by a Luenberger-type observer. The remaining modes vanish exponentially. For a given reaction coefficient, we show how many modes one should use to achieve a desired rate of convergence. The finite-dimensional part is analyzed using appropriate Lyapunov–Krasovskii functionals that lead to linear matrix inequalitie (LMI)-based convergence conditions feasible for small enough time-delay and sampling period. The LMIs can be used to find appropriate injection gains.

Exponential time differencing for stiff systems with nondiagonal linear part

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

Stabilization of a spatially uniform steady state in two systems exhibiting Turing patterns.

This paper deals with the stabilization of a spatially uniform steady state in two coupled one-dimensional reaction-diffusion systems with Turing instability. This stabilization corresponds to amplitude death that occurs in a coupled system with Turing instability. Stability analysis of the steady state shows that stabilization does not occur if the two reaction-diffusion systems are identical. We derive a sufficient condition for the steady state to be stable for any length of system and any boundary conditions. Our analytical results are supported with numerical examples.

Nonadiabatic combustion waves in a two-step competitive exothermic-reaction model

We consider travelling front solutions of a one-dimensional reaction-diffusion system corresponding to two-stage competitively exothermic reactions. We suppose all reactions occurring during the combustion may be lumped together as two different paths. Both exothermic reactions compete for the same reactant. Properties of travelling wave fronts, particularly flame speed, are determined numerically by solving the governing partial differential equations. The flame speed is analysed for different values of the heat loss parameter. It is demonstrated that, as the heat loss coefficient increases, the flame speed decays gradually until the front ceases to exist due to insufficient energy being available to sustain the flame front. Earlier studies for the adiabatic case showed the existence of bi-stability (fast and slow waves co-exist for the same parameter values). We study how heat loss affects the size of the bi-stable region. Furthermore, we investigate how the extinction limit depends on the heat loss parameter as well as the parameter representing the ratio of the activation energy to the heat release of the second reaction. Numerical solutions show that there is no travelling front when these parameters are above threshold values. The dependence of flame speed on the temperature profile is also investigated. The bi-stability phenomenon is demonstrated by perturbing the temperature profile. References W. Choi, S. Hong, J. T. Abrahamson, J.-H. Han, C. Song, N. Nair, S. Baik and M. S. Strano. Chemically driven carbon-nanotube-guided thermopower waves. Nat. Mater. 9:423–429, 2010. doi:10.1038/nmat2714 A. G. Merzhanov. Combustion and explosion processes in physical chemistry and technology of inorganic materials. Russ. Chem. Rev. 72:289–310, 2003. doi:10.1070/rc2003v072n04abeh000766 J. K. Bechtold and C. K. Law. The structure of premixed methane-air flames with large activation energy. Combust. Flame 97:317–338, 1994. doi:10.1016/0010-2180(94)90024-8 K. Seshadri, N. Peters and F. A. Williams. Asymptotic analysis of stoichiometric and lean hydrogen-air flames. Combust. Flame 96:407–427, 1994. doi:10.1016/0010-2180(94)90108-2 A. L. Sanchez, A. Lepinette, M. Bollig, A. Linan and B. Lazaro. The reduced kinetic description of lean premixed combustion. Combust. Flame 123:436–464, 2000. doi:10.1016/S0010-2180(00)00177-2 A. L. Sanchez, G. Balakrishnan, A. Linan and F. A. Williams. Relationships between bifurcation and numerical analyses for ignition of hydrogen-air diffusion flames. Combust. Flame 105:569–590, 1996. doi:10.1016/0010-2180(95)00241-3 V. V. Gubernov, A. V. Kolobov, A. A. Polezhaev and H. S. Sidhu. Analysing the stability of premixed rich hydrogen-air flame with the use of two-step models. Combust. Flame 160:1060–1069, 2013. doi:10.1016/j.combustflame.2013.01.021 R. Ball, A. McIntosh and J. Brindley. Thermokinetic models for simultaneous reactions: a comparative study. Combust. Theor. Model. 3:447–468, 1999. doi:10.1088/1364-7830/3/3/302 N. A. Martirosyan, S. K. Dolukhanyan and A. G. Merzhanov. Nonuniqueness of stationary states in combustion of mixtures of zirconium and soot powders in hydrogen. Combust. Explo. Shock 19:569–571, 1983. doi:10.1007/BF00750423 H. S. Sidhu, I. N. Towers, V. V. Gubernov, A. V. Kolobov and A. V. Polezhaev. Investigation of flame propagation in a model with competing exothermic reactions. Chemeca 2013 , Brisbane, Australia, 29 September–2 October, 2013. http://www.conference.net.au/chemeca2013/papers/29350.pdf I. N. Towers, V. V. Gubernov, A. V. Kolobov, A. V. Polezhaev and H. S. Sidhu. Bistability of flame propagation in a model with competing exothermic reactions. Proc. R. Soc. Lond. A 469:20130315, 2013. doi:10.1098/rspa.2013.0315 L. K. Forbes and W. Derrick. A combustion wave of permanent form in a compressible gas. Anziam J. 43:35–58, 2001. doi:10.1017/S144618110001141X Z. Huang, H. S. Sidhu, I. N. Towers, Z. Jovanoski and V. V. Gubernov. Investigating flame fronts in competitive exothermic reactions. MODSIM2015 , 21st International congress on Modelling and simulation, Gold coast, Australia, 29 November–4 December, 2015. http://www.mssanz.org.au/modsim2015/A1/huang.pdf F. A. Williams. Combustion theory . Westview Press, 1985. https://westviewpress.com/?s=Combustion+theory FlexPDE\(^TM\), PDE Solutions Inc. http://www.pdesolutions.com W. E. Schiesser. The numerical method of lines: integration of partial differential equations . Academic Press, San Diego, 1991. http://store.elsevier.com/The-Numerical-Method-of-Lines/isbn-9780128015513/ A. C. McIntosh, R. O. Weber and G. N. Mercer. Non-adiabatic combustion waves for general Lewis numbers: wave speed and extinction conditions. Anziam J. 46:1–16, 2004. doi:10.1017/S1446181100013638 A. R. Hall and H. G. Wolfhard. Multiple reaction zones in low pressure flames with ethyl and methyl nitrate, methyl nitrite and nitromethane. Proc. Combust. Inst , 6:190–199, 1957. doi:10.1016/S0082-0784(57)80029-0

A Model for Selection of Eyespots on Butterfly Wings.

Unsolved ProblemThe development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not.Key Idea and ModelWe illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature.ResultWe therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.

Parametric pattern selection in a reaction-diffusion model.

We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.

Exact Solutions for Pattern Formation in a Reaction-Diffusion System

Abstract Reaction-diffusion equations are ubiquitous as models of pattern formation in chemistry and biology. In this paper, the invariant subspace method is used to investigate exact solutions and the corresponding patterns in a reaction-diffusion system. Several forms of the exact solutions of the one-dimensional reaction-diffusion system are presented, including complexions. Furthermore, we show several patterns the two-dimensional reaction-diffusion system, involving stripes, spots and transitions between them. Additionally, we discuss the relations between patterns and exact solutions.

Validity of the Cahn–Hilliard approximation for modulations of slightly unstable pattern in the real Ginzburg–Landau equation

In order to describe slow modulations in time and space of slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, the Cahn–Hilliard equation can be derived via multiple scaling analysis as a formal approximation equation. By proving estimates between the approximations obtained via this procedure and the exact solutions of the original system the validity of the Cahn–Hilliard equation as an approximation equation can be rigorously justified. This has been done for a class of one-dimensional reaction–diffusion systems in Düll (2007) [18]. In this paper, we provide a simpler, more elementary and shorter validity proof for the case of the real Ginzburg–Landau equation as an original pattern forming system by exploiting the S1-symmetry of the real Ginzburg–Landau equation.

Spatial correlations in nonequilibrium reaction-diffusion problems by the Gillespie algorithm

We present a study of the spatial correlation functions of a one-dimensional reaction-diffusion system in both equilibrium and out of equilibrium. For the numerical simulations we have employed the Gillespie algorithm dividing the system into cells to treat diffusion as a chemical process between adjacent cells. We find that the spatial correlations are spatially short ranged in equilibrium but become long ranged in nonequilibrium. These results are in good agreement with theoretical predictions from fluctuating hydrodynamics for a one-dimensional system and periodic boundary conditions.