Stable asymmetric spike equilibria for the Gierer–Meinhardt model with a precursor field
Bibliographic record
Abstract
Abstract Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer–Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric and two-spike patterns, corresponding to localized peaks in the activator of different heights. These stable, asymmetric patterns co-exist in the same parameter space as symmetric two-spike patterns. This is in contrast to a constant precursor case, for which asymmetric spike patterns are known to be unstable. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach, we analyse the spectrum of the linearization of the GM model around the two-spike steady state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis, a new class of vector-valued non-local eigenvalue problem is derived and analysed.
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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".