On the study of the limit cycles for a class of population models with time-varying factors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913572515217408 |
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| author | Tian, Renhao Huang, Jianfeng Zhao, Yulin |
| author_facet | Tian, Renhao Huang, Jianfeng Zhao, Yulin |
| contents | In this paper, we study a class of population models with time-varying factors, represented by one-dimensional piecewise smooth autonomous differential equations.
We provide several derivative formulas in "discrete" form for the Poincaré map of such equations, and establish a criterion for the existence of limit cycles.
These two tools, together with the known ones, are then combined in a preliminary procedure that can provide a simple and unified way to analyze the equations.
As an application, we prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao in 2016.
We also apply our results to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles, is three.
Finally, a population suppression model for mosquitos considered by Yu and Li in 2020 and Zheng et al. in 2021 is studied using our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the study of the limit cycles for a class of population models with time-varying factors Tian, Renhao Huang, Jianfeng Zhao, Yulin Classical Analysis and ODEs Dynamical Systems Populations and Evolution In this paper, we study a class of population models with time-varying factors, represented by one-dimensional piecewise smooth autonomous differential equations. We provide several derivative formulas in "discrete" form for the Poincaré map of such equations, and establish a criterion for the existence of limit cycles. These two tools, together with the known ones, are then combined in a preliminary procedure that can provide a simple and unified way to analyze the equations. As an application, we prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao in 2016. We also apply our results to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles, is three. Finally, a population suppression model for mosquitos considered by Yu and Li in 2020 and Zheng et al. in 2021 is studied using our approach. |
| title | On the study of the limit cycles for a class of population models with time-varying factors |
| topic | Classical Analysis and ODEs Dynamical Systems Populations and Evolution |
| url | https://arxiv.org/abs/2410.21848 |