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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… (More)

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… (More)

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)… (More)

to all those who have taught me how beautiful math is, and for all those who wish to learn Abstract. Patterns are ubiquitous in nature, and research in the past fifty years have greatly advanced our understanding of the mechanisms through which patterns can originate. The objective of this paper is to review the derivation of models for pattern formation,… (More)

The formation of singularities in models of many physical systems can be described using self-similar solutions. One particular example is the finite-time rupture of a thin film of viscous fluid which coats a solid substrate. Previous studies have suggested the existence of a discrete, countably infinite number of distinct solutions of the nonlinear… (More)

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