Dynamics of impulsive control in a predator–prey model with the fear effect and group defense

Longyang Song; Meng Zhang; Xiaofei Ge; Xiaojing Wang; Songbai Guo; Jinlong Lv; Longyang Song; Meng Zhang; Xiaofei Ge; Xiaojing Wang; Songbai Guo; Jinlong Lv

doi:10.3934/era.2026300

Electronic Research Archive

2026, Volume 34, Issue 9: 6905-6932. doi: 10.3934/era.2026300

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Dynamics of impulsive control in a predator–prey model with the fear effect and group defense

School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

Received: 09 June 2026 Revised: 20 July 2026 Accepted: 31 July 2026 Published: 12 August 2026

This paper investigates a predator–prey model incorporating both the fear effect and a Holling type Ⅳ functional response with group defense. In this model, the fear effect characterizes the suppression of prey reproduction caused by predator presence, whereas the Holling type Ⅳ functional response describes the enhanced group defense of prey at high densities. First, for the non-impulsive system, we analyze the positivity and boundedness of solutions, as well as the existence and local stability of equilibria. Second, when the equilibrium of the uncontrolled system fails to meet ecological management requirements, we introduce unilateral and bilateral state-feedback impulsive control strategies. The existence of order-one periodic solutions is established by the successor function method, and criteria for orbital asymptotic stability are derived using impulsive Floquet theory. Finally, numerical simulations validate the theoretical results, which provide a theoretical basis for designing ecologically balanced impulsive management strategies that suppress pest outbreaks while maintaining predator and prey populations within acceptable ranges.
- fear effect,
- group defense,
- Holling type Ⅳ functional response,
- state-feedback impulsive control,
- order-one periodic solutions,
- orbital asymptotic stability
Citation: Longyang Song, Meng Zhang, Xiaofei Ge, Xiaojing Wang, Songbai Guo, Jinlong Lv. Dynamics of impulsive control in a predator–prey model with the fear effect and group defense[J]. Electronic Research Archive, 2026, 34(9): 6905-6932. doi: 10.3934/era.2026300

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Abstract

This paper investigates a predator–prey model incorporating both the fear effect and a Holling type Ⅳ functional response with group defense. In this model, the fear effect characterizes the suppression of prey reproduction caused by predator presence, whereas the Holling type Ⅳ functional response describes the enhanced group defense of prey at high densities. First, for the non-impulsive system, we analyze the positivity and boundedness of solutions, as well as the existence and local stability of equilibria. Second, when the equilibrium of the uncontrolled system fails to meet ecological management requirements, we introduce unilateral and bilateral state-feedback impulsive control strategies. The existence of order-one periodic solutions is established by the successor function method, and criteria for orbital asymptotic stability are derived using impulsive Floquet theory. Finally, numerical simulations validate the theoretical results, which provide a theoretical basis for designing ecologically balanced impulsive management strategies that suppress pest outbreaks while maintaining predator and prey populations within acceptable ranges.

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