Stationary multiple spots for reaction-diffusion systems

DOI10.1007/s00285-007-0146-yzbMath1141.92007OpenAlexW2080580900WikidataQ37023855 ScholiaQ37023855MaRDI QIDQ938180

Matthias Winter, Wei, Juncheng

Publication date: 18 August 2008

Published in: Journal of Mathematical Biology (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00285-007-0146-y

zbMATH Keywords

pattern formationmultiple spots

Mathematics Subject Classification ID

Related Items (45)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Asymptotic methods for reaction-diffusion systems: past and present
• Reaction and diffusion on growing domains: scenarios for robust pattern formation
• Point-condensation for a reaction-diffusion system
• Long time behavior of solutions of bistable nonlinear diffusion equations
• Unravelling the Turing bifurcation using spatially varying diffusion coefficients
• Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model
• A moving grid finite element method applied to a model biological pattern generator
• Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case
• Mathematical biology. Vol. 2: Spatial models and biomedical applications.
• Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case
• Mode-doubling and tripling in reaction-diffusion patterns on growing domains: a piecewise linear model
• Stability analysis of Turing patterns generated by the Schnakenberg model
• The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime
• Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach
• Stability and instability in the Gray-Scott model: The case of equal diffusivities
• A skeleton structure of self-replicating dynamics
• On stability and Hopf bifurcations for chemotaxis systems
• Existence and stability of multiple-spot solutions for the Gray-Scott model in \(\mathbb{R}^2\)
• Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model
• A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains
• Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus
• Static spike autosolitons in the Gray-Scott model
• Large stable pulse solutions in reaction-diffusion equations
• A stability index analysis of 1-D patterns of the Gray-Scott model
• Stability of the Static Spike Autosolitons in the Gray--Scott Model
• Asymmetric Spotty Patterns for the Gray-Scott Model in R2
• The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime
• Scattering of traveling spots in dissipative systems
• Stability of Singularly Perturbed Solutions to Systems of Reaction-Diffusion Equations
• Pattern formation in the one-dimensional Gray - Scott model
• Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems
• Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system
• Transition to chemical turbulence
• Exact Homoclinic and Heteroclinic Solutions of the Gray--Scott Model for Autocatalysis
• The chemical basis of morphogenesis
• Absolute instabilities of standing pulses
• On the Two-Dimensional Gierer--Meinhardt System with Strong Coupling
• On single interior spike solutions of the Gierer–Meinhardt system: uniqueness and spectrum estimates
• The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model
• Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model
• The Slow Dynamics of Two-Spike Solutions for the Gray--Scott and Gierer--Meinhardt Systems: Competition and Oscillatory Instabilities
• Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability
• Spatio-temporal chaos for the Gray-Scott model
• \(2^n\)-splitting or edge-splitting? A manner of splitting in dissipative systems
• The motion of weakly interacting pulses in reaction-diffusion systems

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