Multi-scale dynamics of predator-prey systems with Holling-IV functional response

Kexin Zhang; Caihui Yu; Hongbin Wang; Xianghong Li; Kexin Zhang; Caihui Yu; Hongbin Wang; Xianghong Li

doi:10.3934/math.2024174

AIMS Mathematics

2024, Volume 9, Issue 2: 3559-3575. doi: 10.3934/math.2024174

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Research article

Multi-scale dynamics of predator-prey systems with Holling-IV functional response

1.
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shi Jiazhuang 050043, China
2.
School of Management, Shijiazhuang Tiedao University, Shi Jiazhuang 050043, China

Received: 07 November 2023 Revised: 21 December 2023 Accepted: 29 December 2023 Published: 09 January 2024
MSC : 34K18, 34K25, 34K27, 34D23, 34H05, 92B05, 92D25

In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.
- predator-prey system,
- bursting oscillations,
- bifurcation,
- hysteresis,
- slow-fast analysis method
Citation: Kexin Zhang, Caihui Yu, Hongbin Wang, Xianghong Li. Multi-scale dynamics of predator-prey systems with Holling-IV functional response[J]. AIMS Mathematics, 2024, 9(2): 3559-3575. doi: 10.3934/math.2024174

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Abstract

In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.

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