Asymmetric spike patterns for the one-dimensional Gierer–Meinhardt model: equilibria and stability
Bibliographic record
Abstract
Equilibrium solutions to the one-dimensional Gierer–Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity ε. For a pattern with k spikes, the construction yields k1 spikes that have a common small amplitude and k2 = k− k1 spikes that have a common large amplitude. A k- spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value Dm that depends upon k1, on k2, and on other parameters associated with the Gierer–Meinhardt model. It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch sk, for which k spikes have equal height. These asymmetric solutions provide connections between the branch sk and the other symmetric branches sj , for j = 1,…, k− 1. The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(ε2) eigenvalues is also analyzed. It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > De, where De depends on k1 and k2, on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence. Numerical values for De are obtained from numerical solutions of a matrix eigenvalue problem. Another matrix eigenvalue problem that determines the small eigenvalues is derived. For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".