Spike density distribution for the Gierer–Meinhardt model with precursor

Kolokolnikov, Théodore; Xie, Shouyi

doi:10.1016/j.physd.2019.132247

Cited by 8 publications

(13 citation statements)

References 38 publications

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“…In particular, we will derive the effective spike density and spike heights (see also comparison to analytical results in Figure 4). This extends in part the results in [14] from one dimension to two dimensions.…”

supporting

confidence: 83%

“…We then solve (14a) subject to the constraint (14b) to obtain the lattice spacing u(r) as well as the lattice radius R. Figure 3 illustrates a very good agreement between the continuum limit (14) and the full simulation of (13).…”

mentioning

confidence: 85%

“…Here, a and h represent activator and inhibitor concentrations, respectively, and we make the standard assumption that the activator diffuses much faster than the activator, that is ε 2 1. The function µ(x) is called a precursor [7], and spike clusters form near the minimum of µ(x) [7,9,18,22,14]. A typical choice for µ(x) that we will use for illustration is µ(x) = 1 + α |x| 2 , α > 0.…”

mentioning

confidence: 99%

“…As in 1D, let's examine how the density changes with a. By scaling r = βr and lettingâ = aβ 2 ,R = R/β,N = N/β 2 , the equations (14) remain invariant after dropping the hats. In other words, if we double N, we can half a and retain the same spike density but on a domain that has twice the area (whose radius is √ 2 larger).…”

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confidence: 99%

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Hexagonal spike clusters for some PDE's in 2D

Kolokolnikov¹,

Wei²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of R 2. These clusters consist of N spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE 0 = ∆u − u + u 2 + ε|x| 2 with a small "confinement well" ε|x| 2. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.

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confidence: 83%

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confidence: 85%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hexagonal spike clusters for some PDE's in 2D

Kolokolnikov¹,

Wei²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…For the one-dimensional Gierer-Meinhardt system with precursors, the existence and stability of N -peak solutions and the stable spike clusters were studied in [15] and [18], respectively. Moreover, the spike density was studied in [11]. The Gray-Scott model is a closely related model (see e.g [14,18] and the references therein.)…”

Section: Theorem 12 ([7 Theorem 12]mentioning

confidence: 99%

Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

Ishii¹

2020

Communications on Pure &Amp; Applied Analysis

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In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:        ut − ε 2 uxx = dε − u + g(x)u 2 v, x ∈ (−1, 1), t > 0, εvt − Dvxx = 1 2 − c ε g(x)u 2 v, x ∈ (−1, 1), t > 0, ux(±1) = vx(±1) = 0. where d, c, D > 0 are given constants, ε > 0 is sufficiently small, and g(x) is a given positive function. Let N ≥ 1 be an arbitrary natural number. We assume that g(x) is a periodic and symmetric function, namely g(x) = g(−x) and g(x) = g(x + 2N −1). We study the stability of N-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity g(x) above on their stability. For the standard Schnakenberg model, namely the case of g(x) = 1, with d = 0, the stability of N-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of N-peak solution. In particular, we investigate how N-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case g(x) = 1.

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Concentration phenomena on Y-shaped metric graph for the Gierer–Meinhardt model with heterogeneity

Ishii

2021

Nonlinear Analysis

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Spike density distribution for the Gierer–Meinhardt model with precursor

Cited by 8 publications

References 38 publications

Hexagonal spike clusters for some PDE's in 2D

Hexagonal spike clusters for some PDE's in 2D

Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

Concentration phenomena on Y-shaped metric graph for the Gierer–Meinhardt model with heterogeneity

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