Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model
Bibliographic record
Abstract
In the limit of small activator-diffusivity $\varepsilon$ , a formal asymptotic analysis is used to derive a differential equation for the motion of a one-spike solution to a simplified form of the Gierer–Meinhardt activator-inhibitor model in a two-dimensional domain. The analysis, which is valid for any finite value of the inhibitor diffusivity $D$ with $D\,{\gg}\,\varepsilon^2$ , is delicate in that two disparate scales $\varepsilon$ and ${-1/\ln\varepsilon}$ must be treated. This spike motion is found to depend on the regular part of a reduced-wave Green's function and its gradient. Limiting cases of the dynamics are analyzed. For $D$ small with $\varepsilon^2 \,{\ll}\, D \,{\ll}\, 1$ , the spike motion is metastable. For $D\,{\gg}\, 1$ , the motion now depends on the gradient of a modified Green's function for the Laplacian. The effect of the shape of the domain and of the value of $D$ on the possible equilibrium positions of a one-spike solution is also analyzed. For $D\,{\ll}\,1$ , stable spike-layer locations correspond asymptotically to the centres of the largest radii disks that can be inserted into the domain. Thus, for a dumbbell-shaped domain when $D\,{\ll}\,1$ , there are two stable equilibrium positions near the centres of the lobes of the dumbbell. In contrast, for the range $D\,{\gg}\,1$ , a complex function method is used to derive an explicit formula for the gradient of the modified Green's function. For a specific dumbbell-shaped domain, this formula is used to show that there is only one equilibrium spike-layer location when $D\,{\gg}\,1$ , and it is located in the neck of the dumbbell. Numerical results for other non-convex domains computed from a boundary integral method lead to a similar conclusion regarding the uniqueness of the equilibrium spike location when $D\,{\gg}\,1$ . This leads to the conjecture that, when $D\,{\gg}\, 1$ , there is only one equilibrium spike-layer location for any convex or non-convex simply connected domain. Finally, the asymptotic results for the spike dynamics are compared with corresponding full numerical results computed using a moving finite element method.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| 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".