Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting

Xin-You Meng; Fan-Li Meng; Xin-You Meng; Fan-Li Meng

doi:10.3934/math.2021336

AIMS Mathematics

2021, Volume 6, Issue 6: 5695-5719. doi: 10.3934/math.2021336

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Research article

Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting

Xin-You Meng ^1,2,
Fan-Li Meng ^{1
,
,}

1.
College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110819, China
2.
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received: 05 February 2021 Accepted: 18 March 2021 Published: 26 March 2021
MSC : 92D25, 34D23, 34H05

In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results.
- predator-prey,
- herd behavior,
- two delays,
- prey harvesting,
- Hopf bifurcation
Citation: Xin-You Meng, Fan-Li Meng. Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting[J]. AIMS Mathematics, 2021, 6(6): 5695-5719. doi: 10.3934/math.2021336

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Abstract

In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results.

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