David Iron

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

Abstract The stability properties of an N-spike equilibrium solution to a simplified form of the Gierer–Meinhardt activator–inhibitor model in a one-dimensional domain is studied asymptotically in the limit of small activator diffusivity e. The equilibrium solution consists of a sequence of spikes of equal height. The two classes of eigenvalues that must be considered are the O(1) eigenvalues and the O(e2) eigenvalues, which are referred to as the large and small eigenvalues, respectively. The spike pattern is stable when the parameters in the Gierer–Meinhardt model are such that both sets of eigenvalues lie in the left half-plane. For a certain range of these parameters and for N≥2 and e→0, it is shown the O(1) eigenvalues are in the left half-plane only when D D N ∗ , where D N ∗ is another critical value of D, which satisfies D N ∗ N . Thus, when N≥2 and e≪1, the spike pattern is stable only when D N ∗ . An explicit formula for D N ∗ is given. For the special case N=1, it is shown that a one-spike equilibrium solution is stable when D D1(e). An asymptotic formula for D1(e) is given. Finally, the dynamics of a one-spike solution is studied by deriving a differential equation for the trajectory of the center of the spike.

THE DYNAMICS OF MULTISPIKE SOLUTIONS TO THE ONE-DIMENSIONAL GIERER-MEINHARDT MODEL

The dynamical behavior of spike-type solutions to a simplified form of the Gierer--Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically and numerically in the limit of small activator diffusivity

A metastable spike solution for a nonlocal reaction-diffusion model

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Wingless Directly Represses DPP Morphogen Expression via an Armadillo/TCF/Brinker Complex

. In the limit

A threshold area ratio of organic to conventional agriculture causes recurrent pathogen outbreaks in organic agriculture

\varepsilon \to 0

THE STABILITY AND DYNAMICS OF HOT-SPOT SOLUTIONS TO TWO ONE-DIMENSIONAL MICROWAVE HEATING MODELS

, a quasi-equilibrium solution for the activator concentration that has n localized peaks, or spikes, is constructed asymptotically using the method of matched asymptotic expansions. For an initial condition of this form, a differential-algebraicsystem of equations describing the evolution of the spike locations is derived. The equilibrium solutions for this system are discussed. The spikes are shown to evolve on a slow time scale

Spike pinning for the Gierer—Meinhardt model

\tau=\varepsilon^2 t

Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in {\mathbb {R}}^2

towards a stable equilibrium, provided that the inhibitor diffusivity D is below some threshold and that a certain stability criterion on the quasi-equilibrium solution is satisfied throughout the slow dynamics. If this stability condition is not satisfied initially or else is no l...

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos.

An asymptotic reduction of the Gierer--Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity leads to a singularly perturbed nonlocal reaction diffusion equation for the activator concentration. In the limit of small activator diffusivity, a one-spike solution to this nonlocal model is constructed. The spectrum of the eigenvalue problem associated with the linearization of the nonlocal model around such an isolated spike solution is studied in both a one-dimensional and a multidimensional context. It is shown that the principal eigenvalues in the spectrum are exponentially small in the limit of small activator diffusivity. The nonlocal term in the eigenvalue problem is essential for ensuring the existence of such exponentially small principal eigenvalues. These eigenvalues are responsible for the occurrence of an exponentially slow, or metastable, spike-layer motion for the time-dependent problem. Explicit metastable spike dynamics are derived by using a projection method, which...

Optimal on-line sampling of parallel reactions: general concept and a specific spectroscopic example

Background Spatially restricted morphogen expression drives many patterning and regeneration processes, but how is the pattern of morphogen expression established and maintained? Patterning of Drosophila leg imaginal discs requires expression of the DPP morphogen dorsally and the wingless (WG) morphogen ventrally. We have shown that these mutually exclusive patterns of expression are controlled by a self-organizing system of feedback loops that involve WG and DPP, but whether the feedback is direct or indirect is not known. Methods/Findings By analyzing expression patterns of regulatory DNA driving reporter genes in different genetic backgrounds, we identify a key component of this system by showing that WG directly represses transcription of the dpp gene in the ventral leg disc. Repression of dpp requires a tri-partite complex of the WG mediators armadillo (ARM) and dTCF, and the co-repressor Brinker, (BRK), wherein ARM•dTCF and BRK bind to independent sites within the dpp locus. Conclusions/Significance Many examples of dTCF repression in the absence of WNT signaling have been described, but few examples of signal-driven repression requiring both ARM and dTCF binding have been reported. Thus, our findings represent a new mode of WG mediated repression and demonstrate that direct regulation between morphogen signaling pathways can contribute to a robust self-organizing system capable of dynamically maintaining territories of morphogen expression.