Global Hopf bifurcation and connected components in a delayed predator-prey model
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911151366864896 |
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| author | Khateeb, Wael El Fan, Guihong Shan, Chunhua Wang, Hao |
| author_facet | Khateeb, Wael El Fan, Guihong Shan, Chunhua Wang, Hao |
| contents | We study the dynamics of a delayed predator-prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies, and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is to show that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller's space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ODE model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Hopf bifurcation and connected components in a delayed predator-prey model Khateeb, Wael El Fan, Guihong Shan, Chunhua Wang, Hao Dynamical Systems Classical Analysis and ODEs 34K18, 34K20, 34K60, 92D25, 37N25, 37G15, 92D40 We study the dynamics of a delayed predator-prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies, and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is to show that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller's space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ODE model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics. |
| title | Global Hopf bifurcation and connected components in a delayed predator-prey model |
| topic | Dynamical Systems Classical Analysis and ODEs 34K18, 34K20, 34K60, 92D25, 37N25, 37G15, 92D40 |
| url | https://arxiv.org/abs/2509.10243 |