Complex Dynamics of a Leslie–Gower Predator–Prey Model with Crowley–Martin Functional Response
Bibliographic record
Abstract
Predator–prey models, describing the predation interaction between predators and their prey, are fundamental frameworks in studying ecosystems. Their dynamics reflect the variation tendency and the asymptotic states of the populations, as well as the interaction mechanism between them. The complexity of dynamics further illustrates the diversity of their development. In this paper, by employing methods of qualitative and quantitative analyses, we investigate a Leslie–Gower predator–prey model with Crowley–Martin functional response. Qualitatively, first, instead of discussing the cubic equation satisfied by the component of the prey at a positive equilibrium, we reformulate the equation appropriately to obtain explicit conditions on the existence of positive equilibria. Then we study their local stability via linearization. When a positive equilibrium loses stability, we show that Hopf bifurcation can occur and calculate the corresponding Lyapunov number to determine the stability of bifurcated periodic orbits. Moreover, we elucidate the existence of periodic solutions by means of the Poincaré–Bendixon Theorem for annular regions under certain conditions. Based on the qualitative results, we conduct quantitative analysis by continuously changing the intrinsic growth rate of the predator while fixing the values of the other parameters in six sets. The rich numerical simulations demonstrate that the model can have two coexisting periodic solutions and experience a series of bifurcation phenomena including the saddle-node bifurcation of equilibria, supercritical and subcritical Hopf bifurcations, homoclinic bifurcation, and the saddle-node bifurcation of nonconstant periodic solutions. In particular, in one set, the model has three simple positive equilibria and six critical values determining its dynamics. There can be 11 different types of dynamical properties. These results illustrate the diversity of the asymptotic states of the model. Moreover, our qualitative and quantitative analyses suggest that the change in the intrinsic growth rate of the predator can lead to complexity in the predator–prey interaction process and asymptotic states in different situations.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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) | 0.002 | 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".