Competition instabilities of spike patterns for the 1D Gierer–Meinhardt and Schnakenberg models are subcritical
Bibliographic record
Abstract
Abstract Spatially localized 1D spike patterns occur for various two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. A competition instability of a steady-state spike pattern is a linear instability that locally preserves the sum of the heights of the spikes. This instability, which results from a zero-eigenvalue crossing of a nonlocal eigenvalue problem at a certain critical value of the inhibitor diffusivity, has been implicated from full PDE numerical simulations of various RD systems of triggering a nonlinear event leading to spike annihilation. As a result, this linear instability is believed to be a key mechanism for initiating a coarsening process of 1D spike patterns. As an extension of the linear theory, we develop and implement a weakly nonlinear theory to analyse competition instabilities associated with symmetric two-boundary spike equilibria on a finite 1D domain for the Gierer–Meinhardt and Schnakenberg RD models. Two symmetric boundary spikes interacting through a long-range bulk diffusion field is the simplest spatial configuration of interacting localized spikes that can undergo a competition instability. Within a neighborhood of the parameter value for the competition instability threshold, a multi-scale asymptotic expansion is used to derive an explicit amplitude equation for the heights of the boundary spikes. This amplitude equation confirms that the competition instability is subcritical and, moreover, it shows that the competition instability threshold corresponds to a symmetry-breaking bifurcation point where an unstable branch of asymmetric two-boundary spike equilibria emerges from the symmetric branch. Results from our weakly nonlinear analysis are confirmed from full numerical solutions of the steady-state problem using numerical bifurcation software.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".