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**Periodic environments and periodic harvesting.**
*(English)*
Zbl 1067.92056

Nat. Resour. Model. 16, No. 3, 233-244 (2003); errata ibid. 16, No. 4, 505 (2003).

In this nicely written paper, the authors discuss the effect of harvesting in several well-known mathematical models of the growth of a population, namely, logistic, Gompertz, Nisbet-Gurney and spruce budworm. A general result on the existence of periodic solutions is formulated for an autonomous system \(y^{\prime}=f(t,y)\), and then specified for the differential equation (1) \(y^{\prime}=g(y)-H(t)\), which models population growth in the presence of harvesting \(H(t).\) In the remaining part of the paper, particular cases of (1) are studied.

It has been shown that under periodic harvesting there is either extinction of the population or an asymptotically stable periodic solution. Furthermore, the examples considered in the paper suggest that a seasonal harvesting may produce a larger average yield compared to a constant harvesting, and there could be situations in which moderate seasonal harvesting may be useful for suppressing the population to a lower density, the latter useful, for instance, to fight harmful bugs or insects.

It has been shown that under periodic harvesting there is either extinction of the population or an asymptotically stable periodic solution. Furthermore, the examples considered in the paper suggest that a seasonal harvesting may produce a larger average yield compared to a constant harvesting, and there could be situations in which moderate seasonal harvesting may be useful for suppressing the population to a lower density, the latter useful, for instance, to fight harmful bugs or insects.

Reviewer: Svitlana P. Rogovchenko (Famagusta)

### MSC:

92D40 | Ecology |

34C25 | Periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

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

### Keywords:

population dynamics; periodic environment; harvesting; periodic solutions; asymptotic stability
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\textit{F. Brauer} and \textit{D. A. Sánchez}, Nat. Resour. Model. 16, No. 3, 233--244 (2003; Zbl 1067.92056)

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### References:

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[2] | Brauer, Constant Rate Population Harvesting: Equilibrium and Stability, Theoret. Pop. Biol. 8 pp 12– (1975) · Zbl 0313.92012 |

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