Waves In Reaction-diffusion Systems Research Articles

Convergence to traveling waves in reaction-diffusion systems with equal diffusivities

In this paper, we first derive a theorem on the convergence of solutions to traveling waves in reaction-diffusion systems with equal diffusivities. Then we apply this theorem to some specific examples of predator-prey models which were studied recently in the literature. This gives a stability result for these traveling waves in the corresponding predator-prey system under certain perturbations of initial data.

Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schrodinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an inherent loss of regularity which, due to the low-order damping in the LLE, cannot be regained using standard methods. Therefore, in contrast to the case of reaction-diffusion systems, the nonlinear iteration cannot be closed for the modulated perturbation. In this work, we present a new nonlinear iteration scheme incorporating tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

Index theory for traveling waves in reaction diffusion systems with skew gradient structure

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation.

Unpinning of scroll waves attached to multiple heterogeneities

Scroll waves in reaction diffusion systems can be pinned to several unexcitable heterogeneities. The stable pinned filaments have highly enhanced lifetimes. An external gradient can be applied to unpin the scroll waves sequentially from the anchoring sites. As the number of pinning sites are increased, the strength of the field required to unpin the filament also increases. The time required for unpinning changes as a function of the number of anchors, the geometry in which they are arranged, and their placement relative to the external field direction. Experiments with the Belousov-Zhabotinsky reaction are supported by numerical simulations on a simple reaction diffusion model.

Paradoxical Onset of Arrhythmic Waves from Depolarized Areas in Cardiac Tissue Due to Curvature-Dependent Instability.

The generation of abnormal excitations in pathological regions of the heart is a main trigger for lethal cardiac arrhythmias. Such abnormal excitations, also called ectopic activity, often arise from areas with local tissue heterogeneity or damage accompanied by localized depolarization. Finding the conditions that lead to ectopy is important to understand the basic biophysical principles underlying arrhythmia initiation and might further refine clinical procedures. In this study, we are the first to address the question of how geometry of the abnormal region affects the onset of ectopy using a combination of experimental, in silico, and theoretical approaches. We paradoxically find that, for any studied geometry of the depolarized region in optogenetically modified monolayers of cardiac cells, primary ectopic excitation originates at areas of maximal curvature of the boundary, where the stimulating electrotonic currents are minimal. It contradicts the standard critical nucleation theory applied to nonlinear waves in reaction-diffusion systems, where a higher stimulus is expected to produce excitation more easily. Our in silico studies reveal that the nonconventional ectopic activity is caused by an oscillatory instability at the boundary of the damaged region, the occurrence of which depends on the curvature of that boundary. The onset of this instability is confirmed using the Schrödinger equation methodology proposed by Rinzel and Keener [SIAM J. Appl. Math. 43, 907 (1983)]. Overall, we show distinctively novel insight into how the geometry of a heterogeneous cardiac region determines ectopic activity, which can be used in the future to predict the conditions that can trigger cardiac arrhythmias.

An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([ 6 ]), we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. To this end, this gives rise to the N-barrier maximum principle for a second-order elliptic equation involving two distinct unknown functions and a quadratic nonlinearity. An immediate consequence of the N-barrier maximum principle is an a priori estimate for the total populations of the two species. As an application of this maximum principle, we show under certain conditions the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new \begin{document}$(1, 0, 0)$\end{document} - \begin{document}$(u^{*}, v^{*}, 0)$\end{document} waves are given in terms of the tanh function, provided that the system's parameters satisfy certain conditions.

Propagation Behaviors of an Acid Wavefront Through a Microchannel Junction.

Waves in reaction-diffusion systems yield a wealth of dynamic self-assembling phenomena in nature. Recent studies have been devoted to utilizing these active waves in conjunction with microscale technology. To provide a compass for controlling reaction-diffusion waves in microspaces, we have investigated the propagation behavior of one specific variety of the reaction-diffusion wave: an acid wave that utilizes an autocatalytic proton-production reaction. Furthermore, the acid wave that we have investigated occurs in a microchannel with a junction connecting circular and straight regions. The obtained results were compared with a neutralization wave that involves only a neutralization reaction. The acid wave was ignited by the addition of the appropriate amount of H2SO4 into the circular region that was filled with a substrate solution, where proton-consuming and proton-producing reactions followed a rapid neutralization reaction. At this stage, the wave penetrated and propagated into the channel region. Comparison between the acid and the neutralization waves clarified that the acid wave required a minimum threshold of H2SO4 concentration in order to be ignited and that the propagation of the acid wave was temporarily delayed because of the presence of intermediate chemical reaction steps. Furthermore, the propagation dynamics was found to be tuned through the configuration of the microchannel. The importance of microchannel configuration, especially for systems with a junction connecting different shapes, is discussed in terms of Fick's law and in terms of the proton flux from the circular to the straight regions.

Stability of position control of traveling waves in reaction-diffusion systems.

We consider the stability of position control of traveling waves in reaction-diffusion systems as proposed in Löber and Engel [Phys. Rev. Lett. 112, 148305 (2014)]. Instead of analyzing the controlled reaction-diffusion system, stability is studied on the reduced level of the equation of motion for the position over time of perturbed traveling waves. We find an interval of perturbations of initial conditions for which position control is stable. This interval can be interpreted as a localized region where traveling waves are susceptible to perturbations. For stationary solutions of reaction-diffusion systems with reflection symmetry, this region does not exist. Analytical results are in qualitative agreement with numerical simulations of the controlled Schlögl model.

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.

Controlling the position of traveling waves in reaction-diffusion systems.

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a prespecified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space (x) and time (t) dependent control function f(x,t) that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

Stability of planar waves in reaction-diffusion system

This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on ℝ n , where n ⩾ 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t → ∞. The convergence is uniform in ℝ n . Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Rotating spirals of curvature flows: a center manifold approach

Rigidly rotating spiral wave patterns of Archimedean shape are a prominent phenomenon in the spatiotemporal evolution of planar reaction diffusion systems. A first analysis by Wiener and Rosenblueth [27] was motivated by electrical waves in heart tissue. For experimental evidence of Archimedean spirals in Belousov– Zhabotinsky systems and other excitable chemical reaction media see, for example, [24–26,5,22,20,21]. See also the recent survey [17] and references therein. Fluid convection experiments, although modeled by the Navier–Stokes system rather than by reaction and diffusion, also exhibit dynamically intricate spiral wave patterns; see [19]. For a survey of mathematical issues related to spiral waves in reaction diffusion systems, see [10]. One mathematical approach, initiated already by Wiener, aims at a geometric description of the spatiotemporal dynamics of the sharp wave fronts that seem to emanate from a highly focused “core” region or “tip” and that take the form of an Archimedeam spiral, far away from the tip. This approach, where the wave front is represented simply by a planar curve z = z(s) ∈ R2, parametrized over arclength s ≥ 0, is sometimes called kinematic theory of spiral waves and has been developed on a mostly formal level. See, for example, [15,14,18,17] and references therein. Typically the dynamics of the curve z(s) is then described by a curvature driven flow

Spiral and Antispiral Waves in Reaction-Diffusion Systems**The project supported by National Natural Science Foundation of China under Grant No. 10174019, the Natural Science Foundation of Henan Province of China under Grant No. 0111050800, and the Natural Science Foundation of Education Commission of Henan Province of China under Grant Nos. 2001-89 and 2003140028

Spiral waves are ubiquitous phenomena in nonlinear chemical, physical, and biological systems. But antispiral waves are infrequent to date. The transition between spiral and antispiral waves has been rarely explored. We have analyzed the extended Brusselator model and the extended Oregonator model by linear stability analysis. We have demonstrated that it is possible and plausible to realize the transition between them by control of diffusion coefficient of inactivator from theoretical analysis and numerical simulations.

Antispiral waves in reaction-diffusion systems.

We report spontaneous antispiral wave formation in typical reaction-diffusion systems. Our findings qualitatively reproduce a series of phenomena recently observed in a Belousov-Zhabotinsky-type chemical reaction. We found that antispiral waves can occur only near the Hopf bifurcation, when the system is characterized by small amplitude oscillatory (as opposed to excitable) dynamics. For reaction-diffusion systems in the vicinity of the Hopf bifurcation, the specific conditions required for antispiral formation are established here through theoretical analyses and numerical simulations. Thus, this work provides a comprehensive description of the mechanisms underlying antispiral waves in reaction-diffusion systems.

Nonlinear waves in reaction-diffusion systems: The effect of transport memory

Motivated by the problem of determining stress distributions in granular materials, we study the effect of finite transport correlation times on the propagation of nonlinear wave fronts in reaction-diffusion systems. We obtain results such as the possibility of spatial oscillations in the wave-front shape for certain values of the system parameters and high enough wave-front speeds. We also generalize earlier known results concerning the minimum wave-front speed and shape-speed relationships stemming from the finiteness of the correlation times. Analytic investigations are made possible by a piecewise linear representation of the nonlinearity.

Target patterns and spiral waves in reaction-diffusion systems with limit cycle kinetics and weak diffusion

In this paper, we study periodic waves in reaction-diffusion systems with limit cycle kinetics and weak diffusion. The results are based on the analysis of nonlinear partial differential equation which is derived from singular perturbation techniques. We obtain the first order approximation expression of the dispersion curve for periodic wave trains and then show that in the generic case of limit cycle kinetics there is a one parameter family of target patterns and the rotating spiral waves of Archimedian type exist., Finally, we apply these results to the general λ−ω systems.

On the speed of amplitude transition waves in reaction-diffusion systems of λ−ω type

On the speed of amplitude transition waves in reaction-diffusion systems of λ−ω type

Rotating waves in reaction-diffusion systems with folded slow manifolds

A scaling approach to multiple time scale systems with folded slow manifolds is applied to reaction-diffusion systems. Plane periodic wave trains, rotating waves, and static patterns are obtained. The periodic (planar and rotating) waves are shown to arise as a finite amplitude bifurcation at zero frequency.

Geometrical Characteristics Associated with Stability and Bifurcations of Periodic Travelling Waves in Reaction-Diffusion Systems

A periodic travelling wave propagating steadily in a spatially homogeneous reaction-diffusion system corresponds one-to-one to a closed orbit in a certain finite dimensional dynamical system, and hence the travelling wave can be characterized qualitatively by using topological characteristics of the corresponding closed orbit. We introduce a sequence of characteristics $[ \delta ( \gamma _l )^k ],\, k = 1,2, \cdots $, of the closed orbit and show that they are closely associated with the stability and bifurcations of the travelling wave.In this paper we mainly consider the first characteristic of the closed orbit. The value of the characteristic can be determined if the form of the dispersion relation curve for the travelling wave (i.e. the propagation speed $c( l )$ as a function of the spatial period l) is known. We give three theorems to determine the value of the characteristic for three qualitatively different types of periodic travelling waves.As an example, we compute numerically the dispersion rel...