Dynamic analysis of a predator-prey state-dependent impulsive model with fear effect in which action threshold depending on the prey density and its changing rate

Yazhi Wu; Guangyao Tang; Changcheng Xiang; Yazhi Wu; Guangyao Tang; Changcheng Xiang

doi:10.3934/mbe.2022615

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 13152-13171. doi: 10.3934/mbe.2022615

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Research article

Dynamic analysis of a predator-prey state-dependent impulsive model with fear effect in which action threshold depending on the prey density and its changing rate

School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China

Received: 17 July 2022 Revised: 21 August 2022 Accepted: 30 August 2022 Published: 08 September 2022

In ecology, the impact of predators goes beyond killing prey, the mere presence of predators reduces the ability of prey to reproduce. In this study, we extend the predator-prey model with fear effect by introducing the state-dependent control with a nonlinear action threshold depending on the combination of the density of prey and its changing rate. We initially defined the Poincaré map of the proposed model and studied its fundamental properties. Utilizing the properties of the Poincaré map, periodic solution of the model is further investigated, including the existence and stability of the order-1 periodic solution and the existence of the order-k ($ k \ge 2 $) periodic solutions. In addition, the influence of the fear effect on the system's dynamics is explored through numerical simulations. The action threshold used in this paper is more consistent with the actual growth of the population than in earlier linear threshold studies, and the results show that the control objectives are better achieved using the action threshold strategy. The analytical approach used in this study provided several novel methods for analyzing the complex dynamics that rely on state-dependent impulsive.
- state-dependent impulsive system,
- fear effect,
- nonlinear threshold,
- Poincaré map,
- periodic solution
Citation: Yazhi Wu, Guangyao Tang, Changcheng Xiang. Dynamic analysis of a predator-prey state-dependent impulsive model with fear effect in which action threshold depending on the prey density and its changing rate[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13152-13171. doi: 10.3934/mbe.2022615

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Abstract

In ecology, the impact of predators goes beyond killing prey, the mere presence of predators reduces the ability of prey to reproduce. In this study, we extend the predator-prey model with fear effect by introducing the state-dependent control with a nonlinear action threshold depending on the combination of the density of prey and its changing rate. We initially defined the Poincaré map of the proposed model and studied its fundamental properties. Utilizing the properties of the Poincaré map, periodic solution of the model is further investigated, including the existence and stability of the order-1 periodic solution and the existence of the order-k ($ k \ge 2 $) periodic solutions. In addition, the influence of the fear effect on the system's dynamics is explored through numerical simulations. The action threshold used in this paper is more consistent with the actual growth of the population than in earlier linear threshold studies, and the results show that the control objectives are better achieved using the action threshold strategy. The analytical approach used in this study provided several novel methods for analyzing the complex dynamics that rely on state-dependent impulsive.

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