Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems
نویسندگان
چکیده
A new class of point-interaction problem characterizing the time evolution of spatially localized spots for reaction-diffusion (RD) systems on the surface of the sphere is introduced and studied. This problem consists of a differential algebraic system (DAE) of ODE’s for the locations of a collection of spots on the sphere, and is derived from an asymptotic analysis in the large diffusivity ratio limit of certain singularly perturbed two-component RD systems. In [27], this DAE system was derived for the Brusselator and Schnakenberg RD systems, and herein we extend this previous analysis to the Gray-Scott RD model. Results and open problems pertaining to the determination of equilibria of this DAE system, and its relation to elliptic Fekete point sets, are highlighted. The potential of deriving similar DAE systems for more complicated modeling scenarios is discussed.
منابع مشابه
The dynamics of localized spot patterns for reaction-diffusion systems on the sphere
Abstract. In the singularly perturbed limit corresponding to an asymptotically large diffusion ratio between two components, many reaction-diffusion (RD) systems will admit quasi-equilibrium spot patterns, where the concentration of one component will be localized at a discrete set of points in the domain. In this paper, we derive and study the differential algebraic equation (DAE) that charact...
متن کاملTopics in the Stability of Localized Patterns for some Reaction-Diffusion Systems
In the first part of this thesis, we study the existence and stability of multi-spot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reaction-diffusion model. The method of matched asymptotic expansions, tailored to problems with logarithmic gauge functions, is used to construct both symmetric and asymmetric spot patterns. There are three distinct typ...
متن کاملThe Stability of Localized Spot Patterns for the Brusselator on the Sphere
In the singularly perturbed limit of an asymptotically small diffusivity ratio ε, the existence and stability of localized quasi-equilibrium multi-spot patterns is analyzed for the Brusselator reaction-diffusion 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 probl...
متن کاملAnomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in 2-D
For three specific singularly perturbed two-component reaction diffusion systems in a bounded 2-D domain admitting localized multi-spot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In th...
متن کاملThe dynamics of localized spot patterns for reaction-diffusion
In the singularly perturbed limit corresponding to a large diffusivity ratio between two 9 components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, 10 producing a solution that concentrates at a discrete set of points in the domain. In this paper, we derive 11 and study the differential algebraic equation (DAE) that characterizes the slow dynamics for ...
متن کامل