Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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 it