Existence and stability of multiple-spot solutions for the Gray-Scott model in \(\mathbb{R}^2\)

DOI10.1016/S0167-2789(02)00743-1zbMath1014.37036OpenAlexW2002181677MaRDI QIDQ1858117

Matthias Winter, Wei, Juncheng

Publication date: 11 February 2003

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s0167-2789(02)00743-1

zbMATH Keywords

stabilityGreen functionthreshold behaviornonlocal eigenvalue problemsmultiple interior spot solutions

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Stability in context of PDEs (35B35) Reaction-diffusion equations (35K57) Stability problems for finite-dimensional Hamiltonian and Lagrangian systems (37J25)

Related Items

Spatially Periodic Multipulse Patterns in a Generalized Klausmeier--Gray--Scott Model ⋮ Oscillatory translational instabilities of spot patterns in the Schnakenberg system on general 2D domains ⋮ Long-time behavior of solution for a reactor model ⋮ Existence and stability of a spike in the central component for a consumer chain model ⋮ Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion ⋮ Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots ⋮ Lattice Boltzmann model for the bimolecular autocatalytic reaction-diffusion equation ⋮ Existence and stability of a spike in the central component for a consumer chain model in a two-dimensional domain ⋮ Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems ⋮ Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain ⋮ The stability and slow dynamics of spot patterns in the 2D Brusselator model: the effect of open systems and heterogeneities ⋮ Robust IMEX schemes for solving two-dimensional reaction-diffusion models ⋮ Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator ⋮ Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions ⋮ Stationary patterns in a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis ⋮ On accurately estimating stability thresholds for periodic spot patterns of reaction-diffusion systems in² ⋮ Dynamics of two-compartment Gray-Scott equations ⋮ The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model ⋮ Asymmetric Spotty Patterns for the Gray-Scott Model in R2 ⋮ Zigzag and Breakup Instabilities of Stripes and Rings in the Two‐Dimensional Gray–Scott Model ⋮ The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime ⋮ Logarithmic expansions and the stability of periodic patterns of localized spots for reaction-diffusion systems in \(\mathbb R^2\) ⋮ Stationary patterns caused by cross-diffusion for a three-species prey-predator model ⋮ Stationary multiple spots for reaction-diffusion systems ⋮ Information geometric analysis of phase transitions in complex patterns: the case of the Gray-Scott reaction–diffusion model ⋮ Qualitative analysis for a biological depletion model ⋮ MULTIPLE BIFURCATION ANALYSIS AND SPATIOTEMPORAL PATTERNS IN A 1-D GIERER–MEINHARDT MODEL OF MORPHOGENESIS ⋮ Robustness of global attractors for reversible Gray-Scott systems ⋮ Refined stability thresholds for localized spot patterns for the Brusselator model in ⋮ Stable and unstable periodic spiky solutions for the Gray-Scott system and the Schnakenberg system ⋮ On a reversible Gray-Scott type system from energetic variational approach and its irreversible limit ⋮ Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems ⋮ Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems ⋮ A ring of spikes in a Schnakenberg model ⋮ Instability of multi-spot patterns in shadow systems of reaction-diffusion equations ⋮ The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Cites Work

• Unnamed Item
• Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case
• Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case
• 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
• 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
• 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
• Exact Homoclinic and Heteroclinic Solutions of the Gray--Scott Model for Autocatalysis
• On the Two-Dimensional Gierer--Meinhardt System with Strong Coupling
• On single interior spike solutions of the Gierer–Meinhardt system: uniqueness and spectrum estimates
• Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system
• Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability