Turing patterns with O(3) symmetry

Callahan, T. K.

doi:10.1016/s0167-2789(03)00286-0

Cited by 16 publications

(28 citation statements)

References 26 publications

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“…Such an approach was used to [91] to analyze the small eigenvalues associated with spike patterns for the 1-D Brusselator model. • Perform a numerical bifurcation study to examine how solution branches of spot equilibria on the sphere are related to the weakly nonlinear patterns analyzed in [11] (see also [59]) near a Turing bifurcation of the spatially homogeneous steady-state of the Brusselator. A possible homotopy parameter for this study is ǫ.…”

Section: A)mentioning

confidence: 99%

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Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Ward

2018

Nonlinearity

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Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, localized spot patterns, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime. In a 2-D spatial domain we survey some recent and new results for the existence, linear stability, and slow dynamics of localized spot patterns by using the Brusselator RD model as the prototypical example. In the context of linear diffusive systems with localized solution behavior, we will discuss some previous results for the determination of the mean first capture time for a Brownian particle in a 2-D domain with localized traps, and the determination of the persistence threshold of a species in a 2-D landscape with patchy food resources. Common features in the analysis of all of these spatially localized patterns are emphasized, including the key role of certain matrices involving various Green's functions, and the derivation and study of new classes of interacting particle systems and discrete variational problems arising from various asymptotic reductions. The mathematical tools include matched asymptotic analysis based on strong localized perturbation theory, spectral analysis, the analysis of nonlocal eigenvalue problems, and bifurcation theory. Some specific open problems are highlighted and, more broadly, we will discuss a few new research frontiers for the analysis of localized patterns in multi-dimensional domains.

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Section: A)mentioning

confidence: 99%

“…This prototypical RD system has been a well-studied model for analyzing various aspects of weakly nonlinear patterns in RD systems (cf. [11]). In a 2-D bounded domain Ω, the Brusselator has the form…”

Section: Introductionmentioning

confidence: 99%

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

Ward

2018

Nonlinearity

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“…Another open problem, but with a numerical focus, is to quantify the bifurcation structure that links the weakly nonlinear regime associated with perturbations of the spatially uniform equilibrium state (as studied in [7], [9], [22], [29], [42], and the references therein) to the regime, studied in this paper, of localized spot patterns. In particular, do localized spot patterns arise from subcritical bifurcations of the weakly nonlinear amplitude equations?…”

Section: Platonic Solidmentioning

confidence: 99%

“…This degeneracy in the eigenspace associated with spherical harmonics leads to a coupled system of ODEs for the normal form amplitude equations. These normal form ODEs have an intricate bifurcation structure, and so the pattern that emerges from an interaction of these modes is difficult to predict theoretically (see [7], [9], [22], [23], [29], and [42]). It is the goal of this paper to develop and then implement an alternative theoretical framework for analyzing RD patterns on the sphere.…”

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The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Rozada¹,

Ruuth²,

Ward³

2014

SIAM J. Appl. Dyn. Syst.

142

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In the singularly perturbed limit of an asymptotically small diffusivity ratio ε 2 , the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reactiondiffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of "fast" O(1) time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of ν = −1/ log ε. From a leading-order-in-ν analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the closest point method. [39] 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. By means of a linearized analysis, he showed how stable spatially inhomogeneous patterns can develop from small perturbations of spatially homogeneous initial data for a coupled system of reaction-diffusion (RD) equations. Introduction. Turing inMotivated by this initial work, a systematic and rigorous approach has been developed over the last few decades for the analysis of small amplitude patterns in RD systems. After linearizing the RD system around a spatially uniform state to identify bifurcation points for the emergence of spatially nonuniform patterns, a weakly nonlinear analysis based on a

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