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**Hopf bifurcation in a delayed differential-algebraic biological economic system.**
*(English)*
Zbl 1221.34227

This paper deals with a predator-prey differential system incorporating harvesting effort on the prey and a constant time delay which represents the reaction time of the preys. The model is completed with an algebraic equation which incorporates the economic effect of harvesting, according to the economic theory proposed by Gordon in 1954.

The main goal of the paper is to analyze the behaviour of the positive steady-state of this differential-algebraic system in terms of the delay considered as a varying parameter. Using the general theory of normal forms and the center manifold theory, the existence of a Hopf bifurcation is analyzed. The paper includes some numerical simulations to illustrate the analytical findings.

The main goal of the paper is to analyze the behaviour of the positive steady-state of this differential-algebraic system in terms of the delay considered as a varying parameter. Using the general theory of normal forms and the center manifold theory, the existence of a Hopf bifurcation is analyzed. The paper includes some numerical simulations to illustrate the analytical findings.

Reviewer: Eva Sanchez (Madrid)

### MSC:

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

34K18 | Bifurcation theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K21 | Stationary solutions of functional-differential equations |

34K19 | Invariant manifolds of functional-differential equations |

34K17 | Transformation and reduction of functional-differential equations and systems, normal forms |

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\textit{G. Zhang} et al., Nonlinear Anal., Real World Appl. 12, No. 3, 1708--1719 (2011; Zbl 1221.34227)

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### References:

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