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**Optimal harvesting policy for single population with periodic coefficients.**
*(English)*
Zbl 0940.92030

Summary: We examine the exploitation of a single population modeled by a time-dependent logistic equation with periodic coefficients. First, it is shown that the time-dependent periodic logistic equation has a unique positive periodic solution, which is globally asymptotically stable for positive solutions, and we obtain its explicit representation. Further, we choose the maximum annual-sustainable yield as the management objective, and investigate the optimal harvesting policies for constant harvest and periodic harvest.

The optimal harvest effort that maximizes the annual-sustainable yield, the corresponding optimal population level, the corresponding harvesting time-spectrum, and the maximum annual-sustainable yield are determined, and their explicit expressions are obtained in terms of the intrinsic growth rate and the carrying capacity of the considered population. Our interesting and brief results generalize the classical results of C. W. Clark [Mathematical bioeconomics. The optimal management of renewable resources. (1976; Zbl 0364.90002)] for a population described by the autonomous logistic equation in renewable resources management.

The optimal harvest effort that maximizes the annual-sustainable yield, the corresponding optimal population level, the corresponding harvesting time-spectrum, and the maximum annual-sustainable yield are determined, and their explicit expressions are obtained in terms of the intrinsic growth rate and the carrying capacity of the considered population. Our interesting and brief results generalize the classical results of C. W. Clark [Mathematical bioeconomics. The optimal management of renewable resources. (1976; Zbl 0364.90002)] for a population described by the autonomous logistic equation in renewable resources management.

### MSC:

92D40 | Ecology |

49N20 | Periodic optimal control problems |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

### Keywords:

Euler-Lagrange equation; havesting time-spectrum; time-dependent logistic equation; periodic coefficients; optimal harvesting policies### Citations:

Zbl 0364.90002
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\textit{M. Fan} and \textit{K. Wang}, Math. Biosci. 152, No. 2, 165--177 (1998; Zbl 0940.92030)

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### References:

[4] | Leung, A. W., Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim., 31, 2, 219 (1995) · Zbl 0820.49011 |

[6] | Bhatta charya, D. K.; Begum, S., Bionomic equilibrium of two-species System I, Math. Biosci., 135, 2, 111 (1996) · Zbl 0856.92018 |

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