Antipredator behavior of a nonsmooth ecological model with a state threshold control strategy

Shuai Chen; Wenjie Qin; Shuai Chen; Wenjie Qin

doi:10.3934/math.2024360

AIMS Mathematics

2024, Volume 9, Issue 3: 7426-7448. doi: 10.3934/math.2024360

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Antipredator behavior of a nonsmooth ecological model with a state threshold control strategy

Shuai Chen ,
Wenjie Qin ^,

Department of Mathematics, Yunnan Minzu University, Kunming 650500, China

Received: 11 December 2023 Revised: 05 February 2024 Accepted: 06 February 2024 Published: 21 February 2024
MSC : 34A34, 34A37

A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist.
- anti-predator behavior,
- state-dependent feedback control,
- nonlinear impulsive equautions,
- Poincaré map,
- order-k periodic solution
Citation: Shuai Chen, Wenjie Qin. Antipredator behavior of a nonsmooth ecological model with a state threshold control strategy[J]. AIMS Mathematics, 2024, 9(3): 7426-7448. doi: 10.3934/math.2024360

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Abstract

A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist.

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