Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems

Jamieson-Lane, Alastair; Trinh, Philippe H.; Ward, Michael J.

doi:10.1007/978-3-319-30379-6_58

Cited by 4 publications

(5 citation statements)

References 27 publications

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“…. , 10, stable spot configurations of the DAE dynamics (2.14) are found to coincide with elliptic Fekete point distributions (see [38,89]), which correspond to minimizing the discrete logarithmic energy…”

Section: Localized Spot Patterns In 2d: Existence Linear Stability An...mentioning

confidence: 90%

“…. , 8 and for N = 9, 10 such equilibria have been classified in [89] and [38] as follows: N = 2: two antipodal spots; N = 3: three equally-spaced spots on an equator of the sphere; N = 4: four spots centered at the vertices of a regular tetrahedron; N = 5, 6, 7: an (N − 2) + 2 pattern consisting of a pair of antipodal spots, with the remaining N − 2 spots equally-spaced on the equatorial mid-plane between the two polar spots; N = 8: a 'twisted cuboidal' shape, consisting of two parallel rings of four equally-spaced spots, symmetrically placed above and below an equator. The spots are phase shifted by 45 • between each ring.…”

Section: Localized Spot Patterns In 2d: Existence Linear Stability An...mentioning

confidence: 99%

“…We now list several specific open problems related to the DAE dynamics for the Brusselator in a planar domain (2.8) and on the sphere (2.14). For the sphere, a few of these were also listed in [38].…”

Section: Localized Spot Patterns In 2d: Existence Linear Stability An...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: Localized Spot Patterns In 2d: Existence Linear Stability An...mentioning

confidence: 90%

Section: Localized Spot Patterns In 2d: Existence Linear Stability An...mentioning

confidence: 99%

See 1 more Smart Citation

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

Ward

2018

Nonlinearity

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“…However the long-time integration of the full PDE system (1.3) required to verify the DAE (4.10) provides a significant numerical challenge. This is true also for the DAE derived in [16] for the uncoupled Brusselator model (see also open problems in [6]).…”

Section: Two-spot Configurationsmentioning

confidence: 55%

The Stability and Dynamics of Localized Spot Patterns for a Bulk-Membrane Coupled Brusselator Model

Gomez¹

2019

Preprint

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We consider a bulk-membrane-coupled partial differential equation in which a single diffusion equation posed within the unit ball is coupled to a two-component reaction diffusion equation posed on the bounding unit sphere through a linear Robin boundary condition. Specifically, within the bulk we consider a process of linear diffusion with point-source generation for a bulk-bound activator. On the bounding surface we consider the classical two-component Brusselator model where the feed term is replaced by the restriction of the bulk-bound activator to the membrane. By considering the singularly perturbed limit of a small diffusivity ratio between the membrane-bound activator and inhibitor species, we use formal asymptotic expansions to construct strongly localized quasi-equilibrium spot solutions and study their linear stability. Our analysis reveals that bulk-membrane-coupling can restrict the existence of localized spot solutions through a recirculation mechanism. In addition we derive stability thresholds that illustrate the effect of coupling on both competition and splitting instabilities. Finally, we use higher-order matched asymptotic expansions to derive a system of differential algebraic equations that describe the slow motion of spots. The potential for new coupling induced dynamical behaviour is illustrated by considering examples of one-, two-, and three-spot solutions.

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“…Additionally, there are a number of results on pattern formation on curved surfaces in certain singular limits where spots become strongly localized (i.e., far from onset), and where the interaction of spots is described by DAEs for the spot locations. See, e.g., [JLTW16,TT19] and the references therein. For this, one major step in the analysis are accurate approximations of certain Green's functions on the curved surface, and an interesting (semi-analytical) result is that on tori spots slowly drift towards the inner radius.…”

Section: Pattern Formation On Curved Surfacesmentioning

confidence: 99%

Pattern formation with pde2path -- a tutorial

Uecker

2019

Preprint

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We explain some pde2path setups for pattern formation in 1D, 2D and 3D. A focus is on new pde2path functions for branch switching at steady bifurcation points of higher multiplicity, typically due to discrete symmetries, but we also review general concepts of pattern formation and their handling in pde2path, including localized patterns and homoclinic snaking, again in 1D, 2D and 3D, based on the demo sh (Swift-Hohenberg equation). Next, the demos schnakpat (a Schnakenberg reaction-diffusion system) and chemtax (a quasilinear RD system with crossdiffusion from chemotaxis) simplify and unify previous results in a simple and concise way, CH (Cahn-Hilliard) deals with mass constraints, hexex deals with (multiple) branch points of higher degeneracy in a scalar problem on a hexagonal domain, and shgc illustrates some global coupling. The demos acS, actor,schnakS and schnaktor (the Allen-Cahn and Schnakenberg models on spheres and tori) consider pattern formation on curved surfaces, and bruosc (Brusselator) explains how to augment autonomous systems by a time periodic forcing. Along the way we also comment on the choice of meshes, on time integration, and we give some examples of branch point continuation and Hopf point continuation to approximate stability boundaries.

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Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems

Cited by 4 publications

References 27 publications

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

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

The Stability and Dynamics of Localized Spot Patterns for a Bulk-Membrane Coupled Brusselator Model

Pattern formation with pde2path -- a tutorial

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