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**Bifurcations of a class of singular biological economic models.**
*(English)*
Zbl 1197.37129

Summary: This paper studies systematically a prey-predator singular biological economic model with time delay. It shows that this model exhibits two bifurcation phenomena when the economic profit is zero. One is transcritical bifurcation which changes the stability of the system, and the other is singular induced bifurcation which indicates that zero economic profit brings impulse, i.e., rapid expansion of the population in biological explanation. On the other hand, if the economic profit is positive, at a critical value of bifurcation parameter, the system undergoes a Hopf bifurcation, i.e., the increase of delay destabilizes the system and bifurcates into small amplitude periodic solution. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here. In addition, we study numerically that the system undergoes a saddle-node bifurcation when the bifurcation parameter goes through critical value of positive economic profit.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

### MSC:

37N25 | Dynamical systems in biology |

92D25 | Population dynamics (general) |

34C23 | Bifurcation theory for ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

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

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

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\textit{X. Zhang} et al., Chaos Solitons Fractals 40, No. 3, 1309--1318 (2009; Zbl 1197.37129)

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### References:

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