Hopf bifurcation and hybrid control in a delayed predator-prey model

Jiao-Guo Wang; Xin-You Meng; Jiao-Guo Wang; Xin-You Meng

doi:10.3934/math.20251124

AIMS Mathematics

2025, Volume 10, Issue 11: 25380-25405. doi: 10.3934/math.20251124

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Research article

Hopf bifurcation and hybrid control in a delayed predator-prey model

Jiao-Guo Wang ^1,2,
Xin-You Meng ^{1,3
,
,}

1.
School of Automation and Electrical Engineering, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
2.
College of General Education, Lanzhou University of Information Science and Technology, Lanzhou, Gansu 730300, China
3.
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received: 10 August 2025 Revised: 21 October 2025 Accepted: 22 October 2025 Published: 05 November 2025
MSC : 34H20, 34K18, 37G15, 92B05

In this paper, we considered a predator-prey model with self-feedback delay to investigate its Hopf bifurcation and control. First, non-negativity, boundedness, and the existence and uniqueness of solutions were discussed. Next, the conditions of the existence of the unique positive equilibrium were analyzed. Then, by using delay as the bifurcation parameter, the local stability of the positive equilibrium, and the existence and characteristics of Hopf bifurcation were investigated. Furthermore, by designing a hybrid controller integrating linear time-delayed state feedback and parametric regulation, the local asymptotic stability at the positive equilibrium was achieved within the target delay interval. Finally, numerical simulations verified the theoretical results. From the perspectives of unknown and known control gains, the stability switching mechanism and the maximum stable delay interval were explored. The results demonstrated that the proposed hybrid controller maintains system stability over a wider delay range and eliminates population extinction risks, providing a theoretical basis for stability regulation in ecosystem management.
- predator-prey,
- delay,
- Hopf bifurcation,
- hybrid control
Citation: Jiao-Guo Wang, Xin-You Meng. Hopf bifurcation and hybrid control in a delayed predator-prey model[J]. AIMS Mathematics, 2025, 10(11): 25380-25405. doi: 10.3934/math.20251124

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Abstract

In this paper, we considered a predator-prey model with self-feedback delay to investigate its Hopf bifurcation and control. First, non-negativity, boundedness, and the existence and uniqueness of solutions were discussed. Next, the conditions of the existence of the unique positive equilibrium were analyzed. Then, by using delay as the bifurcation parameter, the local stability of the positive equilibrium, and the existence and characteristics of Hopf bifurcation were investigated. Furthermore, by designing a hybrid controller integrating linear time-delayed state feedback and parametric regulation, the local asymptotic stability at the positive equilibrium was achieved within the target delay interval. Finally, numerical simulations verified the theoretical results. From the perspectives of unknown and known control gains, the stability switching mechanism and the maximum stable delay interval were explored. The results demonstrated that the proposed hybrid controller maintains system stability over a wider delay range and eliminates population extinction risks, providing a theoretical basis for stability regulation in ecosystem management.

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