Optimal impulsive harvesting policy for single population.

*(English)*Zbl 1011.92052Summary: We establish the exploitation of an impulsive harvesting single autonomous population model by the logistic equation. By some special methods, we analyse the impulsive harvesting population equation and obtain existence, an explicit expression and global attractiveness of impulsive periodic solutions for constant yield harvest and proportional harvest. Then, we choose the maximum sustainable yield as the management objective, and investigate the optimal impulsive harvesting policies, respectively. The optimal harvest effort that maximizes the sustainable yield per unit time, and the corresponding optimal population levels are determined. At last, we point out that the continuous harvesting policy is superior to the impulsive harvesting policy; however, the latter is more beneficial in realistic operations.

##### MSC:

92D40 | Ecology |

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

49N90 | Applications of optimal control and differential games |

34C25 | Periodic solutions to ordinary differential equations |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

##### Keywords:

logistic equation; impulsive differential equation; impulsive periodic solution; global attractivity; optimal impulsive harvesting policy
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\textit{X. Zhang} et al., Nonlinear Anal., Real World Appl. 4, No. 4, 639--651 (2003; Zbl 1011.92052)

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##### References:

[1] | Alvarez, H.R.; Shepp, L.A., Optimal harvesting of stochastically fluctuating populations, J. math. bio., 37, 155-177, (1998) · Zbl 0940.92029 |

[2] | Angelova, J.; Dishliev, A., Optimization problems for one-impulsive models from population dynamics, Nonlinear anal., 39, 483-497, (2000) · Zbl 0942.34010 |

[3] | Clark, C.W., Mathematical bio-economics: the optimal management of renewable resources, (1990), Wiley New York |

[4] | Fan, M.; Wang, K., Optimal harvesting policy for single population with periodic coefficients, Math. biosci., 152, 165-177, (1998) · Zbl 0940.92030 |

[5] | Franco, D.; Nieto, J.J., First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear anal., 42, 163-173, (2000) · Zbl 0966.34025 |

[6] | Lakshimikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Singapore |

[7] | Lcung, 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 |

[8] | Luo, Z.; Shen, J., Stability and boundedness for impulsive functional differential equations with infinite delays, Nonlinear anal., 46, 475-493, (2001) · Zbl 0997.34066 |

[9] | G.V. Tsretkva, Construction of an optimal policy taking into account ecological constraints (Russian), Modelling of natural system and optimal control problems (Russian). (China), Vo “Naukce”, Novosibirsk, 1995, pp. 65-74. |

[10] | T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer, Inc., New York, 1975, pp. 210-223. · Zbl 0304.34051 |

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