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Russia Vladimir
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.
  1. 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
  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
  3. 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
  4. Clark C., Kirkwood G. On uncertain renewable resourse stocks: Optimal harvest policies and the value of stock surveys, Journal of Environmental Economics and Management, 1986, vol. 13, issue 3, pp. 235-244. DOI: 10.1016/0095-0696(86)90024-0
  5. Ryan D., Hanson F.B. Optimal harvesting of a logistic population with stochastic jumps, Journal of Mathematical Biology, 1986, vol. 24, pp. 259-277. DOI: 10.1007/BF00275637
  6. Reed W.J., Clarke H.R. Harvest decisions and assert valuation for biological resources exhibiting size-dependent stochastic growth, International Economic Review, 1990, vol. 31, pp. 147-169. DOI: 10.2307/2526634
  7. Weitzman M.L. Landing fees vs harvest quotas with uncertain fish stocks, Journal of Environmental Economics and Management, 2002, vol. 43, pp. 325-338. DOI: 10.1006/jeem.2000.1181
  8. Kapaun U., Quaas M.F. Does the optimal size of a fish stock increase with environmental uncertainties? Economics Working Paper, 2012, vol. 9, pp. 1-40. DOI: 10.1007/s10640-012-9606-y
  9. Hansen L.G., Jensen F. Regulating fisheries under uncertainty, Resource and Energy Economics, 2017, vol. 50, pp. 164-177. DOI: 10.1016/j.reseneeco.2017.08.001
  10. Jensen F., Frost H., Abildtrup J. Fisheries regulation: A survey of the literature on uncertainty, compliance behavior and asymmetric information, Marine Policy, 2017, vol. 21, pp. 167-178. DOI: 10.1016/j.marpol.2017.03.028
  11. Rodina L.I. Optimization of average time profit for probability model of the population subject to a craft, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2018, vol. 28, issue 1, pp. 48-58 (in Russian). DOI: 10.20537/vm180105
  12. Riznichenko G.Yu. Lektsii po matematicheskim modelyam v biologii. Chast' 1 (Lectures on mathematical models in biology. Part 1), Izhevsk: Regular and Chaotic Dynamics, 2002, 232 p.
  13. Rodina L.I. On the invariant sets of control systems with random coefficients, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, issue 4, pp. 109-121 (in Russian). DOI: 10.20537/vm140409
  14. Shiryaev A.N. Veroyatnost' (Probability), Moscow: Nauka, 1989, 580 p.
  15. Filippov A.F. Vvedenie v teoriyu differentsial'nykh uravnenii (Introduction in the theory of the differential equations), Moscow: URSS, 2004, 240 p.
  16. Rodina L.I., Tyuteev I.I. About asymptotical properties of solutions of difference equations with random parameters, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 1, pp. 79-86 (in Russian). DOI: 10.20537/vm160107
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