An Organizing Center of Codimension Four in a Predator-Prey Model with Generalist Predator: From Tristability and Quadristability to Transients in a Nonlinear Environmental Change
Bibliographic record
Abstract
Abstract. In this paper, we take the Rosenzweig–MacArthur (RM) model with generalist predator as an example in a constant or changing environment. When the environment is fixed, we provide a more easily verifiable classification, in terms of the coefficients of the system with nilpotent linear part and general higher terms, to determine the types and codimension of nilpotent singularities in a general planar system. Second, by using the existing classification and some algebraic methods, we show that the highest codimension of a nilpotent focus is 4 and the sample RM model with generalist predator can exhibit nilpotent focus bifurcation of codimension 4. Our results indicate that generalist predation can cause not only richer bifurcations and dynamics (such as multitype tristability and quadristability, a figure-eight loop) but also the possible extirpation of prey. When the environment is changing, we study the impact of the rate [Formula: see text] and intensity [Formula: see text] of a nonlinear environmental change on dynamics. The key observations on the asymptotic and transient dynamics include (i) transient tracking on unstable steady states or oscillations, and transient-related regime shifts; (ii) slow and fast regime shifts; (iii) regulation of transient dynamics by the environmental change parameters [Formula: see text] and [Formula: see text]; (iv) slow negative or fast positive environmental change can delay or even avoid population extirpation.
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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 | 0.001 |
| 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) | 0.001 | 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".