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October  2012, 17(7): 2545-2559. doi: 10.3934/dcdsb.2012.17.2545

Optimal harvesting and planting control in stochastic logistic population models

Hiroaki Morimoto 1,

1. 

Department of Economics, Matsuyama University, Matsuyama 790-8578, Japan

Received  May 2012 Revised  May 2012 Published  July 2012

We consider the optimal harvesting and planting control problem to maximize the expected total net benefits in the stochastic logistic population model. The variational inequality associated with this problem is given by the degenerate form of elliptic type with quadratic coefficients. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a solution to the variational inequality. The optimal harvesting and planting policy is characterized in terms of two thresholds for the variational inequality.
Keywords: variational inequality, stochastic logistic model, viscosity solution., planting, Optimal harvesting.
Mathematics Subject Classification: Primary: 90E20, 92A15; Secondary: 60J7.
Citation: Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545
References:
[1]

L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177. doi: 10.1007/s002850050124.  Google Scholar

[2]

F. H. Clarke, "Optimizationand Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts., A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[3]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[4]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993.  Google Scholar

[5]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[6]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.  Google Scholar

[7]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537. doi: 10.1002/cpa.3160370408.  Google Scholar

[8]

H. Morimoto, "Stochastic Control and Mathematical Modeling: Applications in Economics," Encyclopedia of Mathematics and its Applications, 131, Cambridge University Press, Cambridge, 2010.  Google Scholar

[9]

H. Morimoto, A singular control problem with discretionary stopping for geometric Brownian motions, SIAM J. Control Optim., 48 (2010), 3781-3804. doi: 10.1137/080734856.  Google Scholar

[10]

S. P. Sethi and M. I. Taksar, Optimal financing of a corporation subject to random returns, Math. Finance, 12 (2002), 155-172. doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[11]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM J. Control Optim., 22 (1984), 55-75. doi: 10.1137/0322005.  Google Scholar

show all references

References:
[1]

L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177. doi: 10.1007/s002850050124.  Google Scholar

[2]

F. H. Clarke, "Optimizationand Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts., A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[3]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[4]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993.  Google Scholar

[5]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[6]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.  Google Scholar

[7]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537. doi: 10.1002/cpa.3160370408.  Google Scholar

[8]

H. Morimoto, "Stochastic Control and Mathematical Modeling: Applications in Economics," Encyclopedia of Mathematics and its Applications, 131, Cambridge University Press, Cambridge, 2010.  Google Scholar

[9]

H. Morimoto, A singular control problem with discretionary stopping for geometric Brownian motions, SIAM J. Control Optim., 48 (2010), 3781-3804. doi: 10.1137/080734856.  Google Scholar

[10]

S. P. Sethi and M. I. Taksar, Optimal financing of a corporation subject to random returns, Math. Finance, 12 (2002), 155-172. doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[11]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM J. Control Optim., 22 (1984), 55-75. doi: 10.1137/0322005.  Google Scholar

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