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Archive of Issues

Russia Vladimir
Year
2018
Volume
28
Issue
2
Pages
213-221
 Section Mathematics Title Properties of average time profit in stochastic models of harvesting a renewable resource Author(-s) Rodina L.I.a Affiliations Vladimir State Universitya Abstract We consider models of harvesting a renewable resource given by differential equations with impulse action, which depend on random parameters. In the absence of harvesting the population development is described by the differential equation $\dot x =g (x),$ which has the asymptotic stable solution $\varphi (t) \equiv K,$ $K> 0.$ We assume that the lengths of the intervals $\theta_k =\tau_k-\tau _ {k-1}$ between the moments of impulses $\tau_k$ are random variables and the sizes of impulse action depend on random parameters $v_k,$ $k=1,2, \ldots.$ It is possible to exert influence on the process of gathering in such a way as to stop preparation in the case where its share becomes big enough to keep some part of a resource for increasing the size of the next gathering. We construct the control $\bar u = (u_1, \dots, u_k, \dots),$ which limits the share of an extracted resource at each instant of time $\tau_k$ so that the quantity of the remaining resource, starting with some instant $\tau _ {k_0}$, is no less than a given value $x> 0.$ We obtain estimates of average time profit from extraction of a resource and present conditions under which it has a positive limit (with probability one). It is shown that in the case of an insufficient restriction on the extraction of a resource the value of average time profit can be zero for all or almost all values of random parameters. Thus, we describe a way of long-term extraction of a resource for the gathering mode in which some part of population necessary for its further restoration constantly remains and there is a limit of average time profit with probability one. Keywords stochastic models of harvesting, renewable resource, average time profit UDC 517.935 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm180207 Received 10 April 2018 Language Russian Citation Rodina L.I. Properties of average time profit in stochastic models of harvesting a renewable resource, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 213-221. References Reed W.J. A stochastic model for the economic management of a renewable animal resource, Mathematical Biosciences, 1974, vol. 22, pp. 313-337. DOI: 10.1016/0025-5564(74)90097-2 Glait A. Optimal harvesting in continuous time with stochastic growth, Mathematical Biosciences, 1978, vol. 41, pp. 111-123. DOI: 10.1016/0025-5564(78)90069-X Reed W.J. Optimal escapement levels in stochastic and deterministic harvesting models, Journal of Environmental Economics and Management, 1979, vol. 6, pp. 350-363. DOI: 10.1016/0095-0696(79)90014-7 Clark C., Kirkwood G. 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