Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting

Xin-You Meng; Yu-Qian Wu; Xin-You Meng; Yu-Qian Wu

doi:10.3934/mbe.2019133

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2668-2696. doi: 10.3934/mbe.2019133

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Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting

Xin-You Meng ^,,
Yu-Qian Wu

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, P. R. China

Received: 26 January 2019 Accepted: 14 March 2019 Published: 26 March 2019

In this paper, a differential algebraic predator-prey model including two delays, Beddington-DeAngelis functional response and nonlinear predator harvesting is proposed. Without considering time delay, the existence of singularity induced bifurcation is analyzed by regarding economic interest as bifurcation parameter. In order to remove singularity induced bifurcation and stabilize the proposed system, state feedback controllers are designed in the case of zero and positive economic interest respectively. By the corresponding characteristic transcendental equation, the local stability of interior equilibrium and existence of Hopf bifurcation are discussed in the different case of two delays. By using normal form theory and center manifold theorem, properties of Hopf bifurcation are investigated. Numerical simulations are given to demonstrate our theoretical results.
- bioeconomic system,
- predator-prey model,
- nonlinear predator harvesting,
- Beddington-DeAngelis functional response,
- singularity induced bifurcation,
- Hopf bifurcation
Citation: Xin-You Meng, Yu-Qian Wu. Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133

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Abstract

In this paper, a differential algebraic predator-prey model including two delays, Beddington-DeAngelis functional response and nonlinear predator harvesting is proposed. Without considering time delay, the existence of singularity induced bifurcation is analyzed by regarding economic interest as bifurcation parameter. In order to remove singularity induced bifurcation and stabilize the proposed system, state feedback controllers are designed in the case of zero and positive economic interest respectively. By the corresponding characteristic transcendental equation, the local stability of interior equilibrium and existence of Hopf bifurcation are discussed in the different case of two delays. By using normal form theory and center manifold theorem, properties of Hopf bifurcation are investigated. Numerical simulations are given to demonstrate our theoretical results.

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