Abstract
The paper contributes to the topic of optimal utilization of spatially distributed renewable resources. Namely, a problem of “sustainable” optimal cyclic exploitation of a renewable resource with logistic law of recovery is investigated. The resource is distributed on a circle and is collected by a single harvester moving along the circle. The recovering and harvesting rates are position dependent, and the latter depends also on the velocity of the harvester, which is considered as a control. The existence of an optimal solution is proved, as well as necessary optimality conditions for the velocity of the harvester. On this base, a numerical approach is proposed, and some qualitative properties of the optimal solutions are established. The results are illustrated by numerical examples, which reveal some economically meaningful features of the optimal harvesting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The justification of this change of variables, as well as the one that appears below, requires attention. First of all, within one round the point x(t) can be identified with the distance along the circle to the origin. Since we have \(v(x) \geq 1/\bar r\), x(t) is a strictly monotone absolutely continuous function satisfying \(\dot x(t) = 1/r(x(t))\), x(0) = 0. Notice that the superposition r(x(t)) is measurable thanks to the strict monotonicity of x(⋅). For the change of the variable t from s = x(t), where x(t) appears as an argument of a measurable function, one may use [14, Theorem I.4.43].
References
Alekseev V M, Tikhomirov V M, Fomin S V. Optimal Control. New York: Consultants Bureau; 1987.
Brock W, Xepapadeas A. Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. J Econ Dyn Control. 2008;32:2745–2787.
Brock W, Xepapadeas A. Pattern formation, spatial externalities and regulation in coupled economic–ecological systems. J Environ Econ Manag. 2010;59:149–164.
Behringer S, Upmann T. Optimal harvesting of a spatial renewable resource. J Econ Dyn Control. 2014;42:105–120.
Cohen Ph J, Foale S J. Sustaining small-scale fisheries with periodically harvested marine reserves. Mar Policy 2013;37:278–287.
Colonius F, Kliemann W. Infinite time optimal control and periodicity. Appl Math Optim. 1989;20(1):113–130.
Ekeland I, Temam R. Convex analysis and variational problems. Classics in Mathematics Society for Industrial and Applied Mathematics. Philadelphia; 1999.
Koopman B O. The theory of search. Part III. The optimal distribution of searching efforts. Oper Res. 1957;5(5):613–626.
Smith M D, Sanchirico J N, Wilen J E. The economics of spatial–dynamic processes: Applications to renewable resources. J Environ Econ Manag. 2009;57:104–121.
Repetto R, Gillis M. Public policies and misuse of forest resources: Cambridge University Press; 1990.
Rudin W. Functional analysis. NewYork: McGraw-Hill Inc; 1991.
Stone L D. Theory of optimal search. New York: Academic Press; 1975.
Undersander D J, Albert B, Porter P, Crossley A. Pastures for profit: a hands on guide to rotational grazing. University of Wisconsin Cooperative Extension Publication A3529, Madison. WL 1994:26.
Warga J. Optimal control of differential and functional equations: Academic Press; 1972.
Acknowledgments
This research financed by a joint project of the Austrian Science Foundation (FWF) and Russian Foundation for Basic Research (RFBR) under grant No I 476-N13 and 10-01-91004 AИΦ_a, respectively .
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Belyakov, A.O., Davydov, A.A. & Veliov, V.M. Optimal Cyclic Exploitation of Renewable Resources. J Dyn Control Syst 21, 475–494 (2015). https://doi.org/10.1007/s10883-015-9271-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-015-9271-x