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**Stability and bifurcation in a harvested one-predator–two-prey model with delays.**
*(English)*
Zbl 1097.34051

This paper mainly concerns the stability and bifurcation of a delayed one-predator-two-prey model with harvesting of the predator at a constant rate. It is shown that time delay can cause a stable equilibrium to become unstable. By choosing the delay as a bifurcation parameter, Hopf bifurcation can occur as the delay crosses some critical values. The authors also investigate the direction and stability of the Hopf bifurcation by following the procedure of deriving a normal form given by T. Faria and L. T. Magalhães [J. Differ. Equations 122, No. 2, 181–200 (1995; Zbl 0836.34068)]. Finally, an example is given and numerical simulations are performed to justify the theoretical results.

Reviewer: Zhiming Guo (Guangzhou)

### MSC:

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

92D40 | Ecology |

92D25 | Population dynamics (general) |

### Citations:

Zbl 0836.34068
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\textit{Z. Liu} and \textit{R. Yuan}, Chaos Solitons Fractals 27, No. 5, 1395--1407 (2006; Zbl 1097.34051)

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### References:

[1] | Azar, C.; Holmberg, J.; Lindgren, K., Stability analysis of harvesting in a predator-prey model, J Theor Biol, 174, 13-19 (1995) |

[2] | Bush, A. W.; Cook, A. E., The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J Theor Biol, 63, 385-395 (1976) |

[3] | Chow, S.-N.; Hale, J., Methods of bifurcation theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0487.47039 |

[4] | Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J Math Anal Appl, 86, 592-627 (1982) · Zbl 0492.34064 |

[5] | Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J Math Anal Appl, 254, 433-463 (2001) · Zbl 0973.35034 |

[6] | Faria, T., On a planar system modelling a neuron network with memory, J Differen Equat, 168, 129-149 (2000) · Zbl 0961.92002 |

[7] | Faria, T.; Magalhães, L. T., Normal form for retarded functional differential equations and applications to Bogdanov-Takens singularity, J Differen Equat, 122, 201-224 (1995) · Zbl 0836.34069 |

[8] | Faria, T.; Magalhães, L. T., Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J Differen Equat, 122, 181-200 (1995) · Zbl 0836.34068 |

[9] | Hale, J.; Magalhães, L. T.; Oliva, W. M., Dynamics in infinite dimensions. Dynamics in infinite dimensions, Applied mathematical sciences, vol. 47 (2002), Springer: Springer New York · Zbl 1002.37002 |

[10] | Hale, J.; Verduyn Lunel, S., Introduction to functional differential equations (1993), Springer: Springer New York · Zbl 0787.34002 |

[11] | Ji, J., Stability and bifurcation in an electromechanical system with time delays, Mech Res Commun, 30, 217-225 (2003) · Zbl 1048.93502 |

[12] | Kuang, Y., Delay differential equations with applications in population dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002 |

[13] | Kumar, S.; Srivastava, S. K.; Chingakham, P., Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model, Appl Math Comput, 129, 107-118 (2002) · Zbl 1017.92041 |

[14] | Meng, X.; Wei, J., Stability and bifurcation of mutual system with time delay, Chaos, Solitons & Fractals, 21, 729-740 (2004) · Zbl 1048.34122 |

[15] | Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272 (1999) · Zbl 1066.34511 |

[16] | Yuan, S.; Han, M., Bifurcation analysis of a chemostat model with two distributed delays, Chaos, Solitons & Fractals, 20, 995-1004 (2004) · Zbl 1059.34059 |

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