Dynamics and interactions of spikes on smoothly curved boundaries for reaction–diffusion systems in 2D

Ichiro, Shin; Ishimoto, Toshio

doi:10.1007/s13160-012-0088-7

Cited by 5 publications

(7 citation statements)

References 18 publications

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“…Of course, inhomogeneous solutions of nonlinear RDS may exhibit a variety of different behaviours, depending on the parameters and initial data of the system. In many cases (such as in the limit of large diffusivity ratios), it has been shown that spike solutions can approach, and be pinned to, boundaries, particularly points of extremal curvature in multiple spatial dimensions (Iron and Ward 2000a , b ; Kolokolnikov and Ward 2004 ; Miyamoto 2005 ; Ei and Ishimoto 2013 ). In other cases, initial data consisting only of internal spots can be shown to remain in a configuration of internal spots due to boundary repulsion (Kolokolnikov et al.…”

Section: Introductionmentioning

confidence: 99%

Isolating Patterns in Open Reaction–Diffusion Systems

et al. 2021

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Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

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Section: Introductionmentioning

confidence: 99%

Isolating Patterns in Open Reaction–Diffusion Systems

et al. 2021

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“…This can be understood from the linear theory as the unstable eigenmodes will achieve maximal or minimal values at the boundary (by the Maximum Principle). Of course, inhomogeneous solutions of nonlinear RDS may exhibit different behaviour, though in many cases it has been shown that spike solutions will approach and be pinned to boundaries, and particularly points of extremal curvature in multiple spatial dimensions [28,48,17]. Numerically, as we will show in Section 3, Neumann conditions will generically allow patterns to form up to the boundary for a variety of systems and parameters.…”

Section: Introductionmentioning

confidence: 94%

Isolating Patterns in Open Reaction-Diffusion Systems

Krause¹,

Klika²,

Maini³

et al. 2020

Preprint

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Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to or along these boundaries. Motivated by boundaries of patterning fields related to emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization, and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions, and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain, and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions, and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

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“…The existence and stability of a spike cluster made up of two boundary spikes has been established in [5].…”

Section: W Ao Et Al / J Math Pures Appl ••• (••••) •••-•••mentioning

confidence: 99%

“…The repulsive nature of spikes has been shown in [4]. Existence and stability of a spike cluster made up of two boundary spikes has been established in [3].…”

Section: Introductionmentioning

confidence: 99%