Pattern alternations induced by nonlocal interactions
Bibliographic record
Abstract
Pattern formation is a visual understanding of the dynamics of complex systems. Patterns arise in many ways, such as the segmentation of animals, bacterial colonies during growth, vegetation, chemical reactions, etc. In most cases, the long-range diffusion occurs, and the usual reaction-diffusion (RD) model can not capture such phenomena. The nonlocal RD model, on the other hand, can fill the gap. Analytical derivation of the amplitude equations (AE) for an RD system is a valuable tool to predict the pattern selections, in particular, the stationary Turing patterns when they occur. In this paper, we analyze the conditions for the Turing bifurcation for the nonlocal model and also derive the AE for the nonlocal RD model near the Turing bifurcation threshold to describe the reason behind the pattern selections. This derivation of the AE is not only limited to the nonlocal prey-predator model, as shown in our representative example but also can be applied to other nonlocal models near the Turing bifurcation threshold. The analytical prediction agrees with numerical simulation near the Turing bifurcation threshold. Moreover, the analytical and numerical results fit each other well even more remote from the Turing bifurcation threshold for the small values of the nonlocal parameter but not for the higher values.
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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 | 0.001 | 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".