The stability of spike solutions to the one-dimensional Gierer—Meinhardt model

By D. Iron, Michael Jeffrey Ward, Jun-cheng Wei. 2001

248 citations recorded by the source corpus.

Subfields: Cognitive Neuroscience, Computer Networks and Communications, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/570240b5a532d24d95ac4d7708ae7f96b9882525

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Citations in the corpus

Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion (2004, methodology): “These include the Gierer–Meinhardt model [11, 18 ,38,41], the Sel’kov model [7,35], the Gray–Scott model [39], the Ginzburg–Landau equation [29], the Brusselator model [1], the Noyes–Field model for Belousov–Zhabotinskii reaction [32], the chemotactic diffusion model [22,37], the Lotka–Volterra comp”
The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model (2002, methodology): “[17]) in the form written in Appendix E of [6].”
Strategy and stationary pattern in a three-species predator–prey model (2004, methodology): “These include the Gierer–Meinhardt model [14,18,39–41], the Sel’kov model [11,36], the Noyes–Field model for Belousov–Zhabotinskii reaction [31], the chemotactic diffusion model [21,38], the competition model [6,10,22–24], the predator–prey model [7,8,12,13,17,19,20,29,30,37], as well as models of s”
Chapter 6 Existence and stability of spikes for the Gierer-Meinhardt system (2008, background): “From Case 2 and Case 3 of Theorem 5.9, we see that for multiple spikes ( K> 1)l argeτ may increase stability, provided that η0 >K .T his is anew phenomenon in R2. It is known that in R1, large τ implies linear instability for multiple spikes [8, 34 ,59,60].”
Positive steady states of the Holling–Tanner prey–predator model with diffusion (2005, methodology): “These include the Gierer–Meinhardt model [ 11 , 26], the Sel’kov model [5, 25], the Brusselator model [2], the chemotactic diffusion model [24], the Lotka–Volterra competition model [14,15], the Lotka–Volterra predator–prey model [6,7,12,19,20], as well as models of semiconductors, plasmas, chemical”
An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations (2015, unknown): “The essential difference between the present stability analysis and the existing literature on the stability of pulses in GS/GM-type models [3, 4, 18, 22, 23] is made explicit by the terms ν1 dF1 dU (u s ∗(|εξ|))”
Asymmetric Spotty Patterns for the Gray–Scott Model in R2 (2003, methodology): “K−peaked solutions was completely characterized for τ small in [ 15 ] by us-”
The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime (2005, background): “29b) The spectrum of B can be calculated explicitly as was done in Appendix D of [18].”
Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. (2005, background): “, a generalization of the ‘classical’ Gierer-Meinhardt equation originally proposed by Gierer and Meinhardt as a model for biological pattern formation or ‘morphogenesis’ (see [15, 18, 25, 29, 30] and the references therein).”