Singular bifurcations in a modified Leslie-Gower model
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| Format: | Preprint |
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2024
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| author | García, Roberto Albarran Alvarez-Ramírez, Martha Jardón-Kojakhmetov, Hildeberto |
| author_facet | García, Roberto Albarran Alvarez-Ramírez, Martha Jardón-Kojakhmetov, Hildeberto |
| contents | We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular bifurcations in a modified Leslie-Gower model García, Roberto Albarran Alvarez-Ramírez, Martha Jardón-Kojakhmetov, Hildeberto Dynamical Systems Chaotic Dynamics 34E13, 34E15, 34E17, 37G15, 92D25 We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point. |
| title | Singular bifurcations in a modified Leslie-Gower model |
| topic | Dynamical Systems Chaotic Dynamics 34E13, 34E15, 34E17, 37G15, 92D25 |
| url | https://arxiv.org/abs/2411.18059 |