Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition
Bibliographic record
Abstract
Abstract In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε>0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).
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How this classification was reachedexpand
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 | 0.000 | 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 | 0.000 | 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)
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 teacher head, 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".