Dynamics of a nonlinear state-dependent feedback control ecological model with fear effect

Zhanhao Zhang; Yuan Tian; Zhanhao Zhang; Yuan Tian

doi:10.3934/math.20241181

AIMS Mathematics

2024, Volume 9, Issue 9: 24271-24296. doi: 10.3934/math.20241181

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Research article

Dynamics of a nonlinear state-dependent feedback control ecological model with fear effect

Zhanhao Zhang ¹,
Yuan Tian ^{1,2
,
,}

1.
School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China
2.
Hubei Key Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, 445000, China

Received: 17 June 2024 Revised: 05 August 2024 Accepted: 05 August 2024 Published: 16 August 2024
MSC : 34D23, 37N25, 93C27

Integrated pest management is a pest control strategy that combines biological and chemical methods to reduce environmental pollution and protect biodiversity. Recent research indicated that the fear caused by predators had a significant effect on the growth, development, and reproductive processes of prey. Therefore, we have proposed a pest-natrual enemy system, which is a nonlinear state-dependent feedback control model that incorporated the fear effect in the predator-prey relationship. We discussed impulsive sets and phase sets of the model and derived an expression for the Poincaré map. Furthermore, we analyzed the existence and stability of order-$ 1 $ periodic solutions and explored the existence of order-$ k $ $ (k\ge2) $ periodic solutions. Finally, numerical simulations were conducted to validate our theoretical results and reveal their biological implications.
- integrated pest management,
- Poincaré map,
- periodic solution,
- existence and stability,
- fear effect
Citation: Zhanhao Zhang, Yuan Tian. Dynamics of a nonlinear state-dependent feedback control ecological model with fear effect[J]. AIMS Mathematics, 2024, 9(9): 24271-24296. doi: 10.3934/math.20241181

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Abstract

Integrated pest management is a pest control strategy that combines biological and chemical methods to reduce environmental pollution and protect biodiversity. Recent research indicated that the fear caused by predators had a significant effect on the growth, development, and reproductive processes of prey. Therefore, we have proposed a pest-natrual enemy system, which is a nonlinear state-dependent feedback control model that incorporated the fear effect in the predator-prey relationship. We discussed impulsive sets and phase sets of the model and derived an expression for the Poincaré map. Furthermore, we analyzed the existence and stability of order-$ 1 $ periodic solutions and explored the existence of order-$ k $ $ (k\ge2) $ periodic solutions. Finally, numerical simulations were conducted to validate our theoretical results and reveal their biological implications.

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