Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Tzou, Justin C.; Bayliss, A.; Matkowsky, B. J.; Volpert, Vitaly

doi:10.1017/s0956792511000179

Cited by 19 publications

(34 citation statements)

References 33 publications

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“…For the situation where only one of the two solution components is localized, the spots are said to exhibit semi-strong interactions. In this semi-strong interaction limit, and in a 1-D spatial domain, there have been many studies of the dynamics of localized patterns for specific reaction-diffusion systems; this includes the Gierer-Meinhardt (GM) model [11,13,34], the Gray-Scott (GS) model [7,9,10,34], the Schnakenberg model [31], a three-component RD system modeling gas-discharge [37], the Brusselator model [36], a model for hot-spots of urban crime [35], and a general class of RD models [24]. In these studies, a wealth of different analytical techniques have been used, including the method of matched asymptotic expansions, Lyapanov-Schmidt reductions, geometric singular perturbation theory, and the rigorous renormalization approach of [11].…”

Section: Connections and Differences With Other Workmentioning

confidence: 99%

The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

Trinh

Ward

2016

Nonlinearity

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In the singularly perturbed limit corresponding to a large diffusivity ratio between two components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, producing a solution that concentrates at a discrete set of points in the domain. In this paper, we derive and study the differential algebraic equation (DAE) that characterizes the slow dynamics for such spot patterns for the Brusselator RD model on the surface of a sphere. Asymptotic and numerical solutions are presented for the system governing the spot strengths, and we describe the complex bifurcation structure and demonstrate the occurrence of imperfection sensitivity due to higher order effects. Localized spot patterns can undergo a fast time instability and we derive the conditions for this phenomena, which depend on the spatial configuration of the spots and the parameters in the system. In the absence of these instabilities, our numerical solutions of the DAE system for N = 2 to N = 8 spots suggest a large basin of attraction to a small set of possible steady-state configurations. We discuss the connections between our results and the study of point vortices on the sphere, as well as the problem of determining a set of elliptic Fekete points, which correspond to globally minimizing the discrete logarithmic energy for N points on the sphere.

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Section: Connections and Differences With Other Workmentioning

confidence: 99%

The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

Trinh

Ward

2016

Nonlinearity

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“…In contrast to small amplitude patterns, in the singularly perturbed limit of a large diffusivity ratio O(ε −2 ) 1, many two-component RD systems in 1-D admit spike-type solutions. In this direction, there is a rather extensive analytical theory on the existence, linear stability and slow dynamics of spike-type solutions for many such RD systems in 1-D (see [5], [6], [14], [15], [24] [25], [26], and the references therein). To establish parameter regimes where spike-layer steady-states are linearly stable, one must analyze the spectrum of the operator associated with a linearization around the spike-layer solution.…”

Section: Introductionmentioning

confidence: 99%

Spike density distribution for the Gierer–Meinhardt model with precursor

Kolokolnikov

Xie

2020

Physica D: Nonlinear Phenomena

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“…[7]), and more recently, the Brusselator model (cf. [35]), [36]) and a reaction-diffusion model of urban crime (cf. [15]).…”

Section: Introductionmentioning

confidence: 99%

The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System

Nec

Ward

2013

Math. Model. Nat. Phenom.

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The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to a general class of reaction-diffusion (RD) system with and without sub-diffusion is analyzed. For both the case of regular and sub-diffusion, the method of matched asymptotic expansions is used to derive an ODE characterizing the spike locations in the absence of any O(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fast instabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem (NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibrium solution. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined by the roots of some simple transcendental equations. From a rigorous analysis of these transcendental equations, explicit sufficient conditions are given to predict the occurrence of either Hopf bifurcations or competition instabilities of the two-spike quasi-equilibrium solution. The theory is illustrated for several specific choices of the reaction kinetics.scale. This asymptotic limit, where only one of the two components of the RD system is localized, is called the semi-strong spike interaction limit (cf. [5]).Reaction-diffusion systems of the general class (1.1) arise in various applications. In particular, Turing [33] proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell. Through the use of a linearized analysis, he showed how stable spatially complex patterns can develop from small perturbations of spatially homogeneous initial data for a coupled system of reaction-diffusion equations. Later, Gierer and Meinhardt [8] (see the more recent book [17]), have demonstrated numerically the existence of stable spatially inhomogeneous equilibrium solutions for activator-inhibitor systems of the type (1.1) with u b = 0 where the inhibitor u diffuses more slowly than the activator v. A class of GM models is g(u) = u −q , f (u) = u −s , u b = 0, where qr/(p − 1) − (s + 1) > 0 (see [12]). Another application of (1.1) is to the study of chemical patterns, involving two reactants in a gel reactor where the reactor is maintained in contact with a reservoir of one of the two chemical species. This leads to the source or feed term u b in (1.1). In this class of substrate-depletion models, a ubiquitous model is the Gray-Scott (GS) model, which was introduced for continuously stirred systems in [10]. It is given by setting u b = 1, g(u) = Au, p = r = 2, and f = −u in (1.1), where A > 0 is a bifurcation parameter.We will also consider the sub-diffusive counterpart of (1.1), formulated aswhere the anomaly exponent γ is on the range 0 < γ < 1. The regular diffusion RD system (1.1) is obtained by setting γ = 1. In (1.2), the sub-diffusive operator as applied to a function h(t) is defined by

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Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Cited by 19 publications

References 33 publications

The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

Spike density distribution for the Gierer–Meinhardt model with precursor

The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System

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