The dynamics of boundary spikes for a nonlocal reaction-diffusion model
Bibliographic record
Abstract
An asymptotic reduction of the Gierer–Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity and small activator diffusivity ε leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit ε → 0, this nonlocal problem for the activator concentration has localized spike-type solutions. In this limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or three-dimensional domain. By deriving asymptotic differential equations for the spike motion, it is shown that the spike moves towards a local maximum of the curvature in two dimensions and a local maximum of the mean curvature in three dimensions. The motion of a spike on a flat segment of a two-dimensional domain is also analyzed, and this motion is found to be metastable. The critical feature that allows for the slow boundary spike motion is the presence of the nonlocal term in the underlying reaction-diffusion equation.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".