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**Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay.**
*(English)*
Zbl 1245.92060

The dynamical behaviour of a bioeconomic model system of differential algebraic equations is analysed. The system describes a prey-predator fishery with two zones (free fishing and protected). It is shown that a singularity-induced bifurcation occurs when a variation of economic interest of harvesting is considered, thus the interior equilibrium changes from being stable to being unstable. A sufficient condition is obtained for the interior equilibrium to be stable when a state feedback controller is introduced. Further, when the time delay is incorporated as a parameter, Hopf bifurcation also occurs. Then these results are verified by some numerical simulations.

Reviewer: Zhanyuan Hou (London)

### MSC:

92D40 | Ecology |

91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |

34K18 | Bifurcation theory of functional-differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

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\textit{K. Chakraborty} et al., Nonlinear Anal., Hybrid Syst. 5, No. 4, 613--625 (2011; Zbl 1245.92060)

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