The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System
Bibliographic record
Abstract
The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to a general class of reaction-diffusion (RD) system with and without sub-diffusion is analyzed. For both the case of regular and sub-diffusion, the method of matched asymptotic expansions is used to derive an ODE characterizing the spike locations in the absence of any 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fast instabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem (NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibrium solution. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined by the roots of some simple transcendental equations. From a rigorous analysis of these transcendental equations, explicit sufficient conditions are given to predict the occurrence of either Hopf bifurcations or competition instabilities of the two-spike quasi-equilibrium solution. The theory is illustrated for several specific choices of the reaction kinetics.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".