An Explicitly Solvable Nonlocal Eigenvalue Problem and the Stability of a Spike for a Sub-Diffusive Reaction-Diffusion System
Bibliographic record
Abstract
The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 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) | 0.002 | 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".