Complex dynamics of non-smooth pest-natural enemy Gomportz models with a variable searching rate based on threshold control

Yuan Tian; Xinlu Tian; Xinrui Yan; Jie Zheng; Kaibiao Sun; Yuan Tian; Xinlu Tian; Xinrui Yan; Jie Zheng; Kaibiao Sun

doi:10.3934/era.2025002

Electronic Research Archive

2025, Volume 33, Issue 1: 26-49. doi: 10.3934/era.2025002

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Research article

Complex dynamics of non-smooth pest-natural enemy Gomportz models with a variable searching rate based on threshold control

1.
School of Science, Dalian Maritime University, Dalian 116026, China
2.
Liaoning Ocean and Fisheries Science Research Institute, Dalian 116023, China
3.
School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China

Received: 09 October 2024 Revised: 23 November 2024 Accepted: 10 December 2024 Published: 06 January 2025

The implementation of ecological pest-management strategies is an important trend in the global agricultural development, which makes integrated pest management become an important research field. In this study, to achieve a scientific and reasonable pest-management objective, three aspects of work were carried out. 1) Modeling and analysis: a pest-natural enemy Gomportz-type model with a variable searching rate was put forward, and two pest-management models were formulated. The dynamic characteristics of the continuous model were investigated, and the results indicated that the search speed of natural enemies had an effect on the coexistence equilibrium. 2) Control effect: the sliding mode dynamics of the Filippov system including the existence of pseudo-equilibrium was analyzed to illustrate the effect of the non-smooth control strategy on the system. A Poincaré map was constructed for the system with a threshold control, and the complex dynamics induced by the threshold control was investigated. 3) Verifications: computer simulations were presented step by step to illustrate and verify the correctness of the theoretical results. A comprehensive study of predation relationships as well as the effects of different management strategies on the system can serve as a valuable reference for advancing sustainable agricultural practices and pest control.
- Filippov system,
- order-$ k $ periodic trajectory,
- Poincaré map,
- orbital asymptotic stability
Citation: Yuan Tian, Xinlu Tian, Xinrui Yan, Jie Zheng, Kaibiao Sun. Complex dynamics of non-smooth pest-natural enemy Gomportz models with a variable searching rate based on threshold control[J]. Electronic Research Archive, 2025, 33(1): 26-49. doi: 10.3934/era.2025002

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Abstract

The implementation of ecological pest-management strategies is an important trend in the global agricultural development, which makes integrated pest management become an important research field. In this study, to achieve a scientific and reasonable pest-management objective, three aspects of work were carried out. 1) Modeling and analysis: a pest-natural enemy Gomportz-type model with a variable searching rate was put forward, and two pest-management models were formulated. The dynamic characteristics of the continuous model were investigated, and the results indicated that the search speed of natural enemies had an effect on the coexistence equilibrium. 2) Control effect: the sliding mode dynamics of the Filippov system including the existence of pseudo-equilibrium was analyzed to illustrate the effect of the non-smooth control strategy on the system. A Poincaré map was constructed for the system with a threshold control, and the complex dynamics induced by the threshold control was investigated. 3) Verifications: computer simulations were presented step by step to illustrate and verify the correctness of the theoretical results. A comprehensive study of predation relationships as well as the effects of different management strategies on the system can serve as a valuable reference for advancing sustainable agricultural practices and pest control.

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