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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 2, Pages 213–221 (Mi vuu632)  

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

Properties of average time profit in stochastic models of harvesting a renewable resource

L. I. Rodina

Vladimir State University, ul. Gor'kogo, 87, Vladimir, 600000, Russia

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, …, u_k, …),$ 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00346_а


DOI: https://doi.org/10.20537/vm180207

Full text: PDF file (247 kB)
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Bibliographic databases:

UDC: 517.935
MSC: 34A60, 37N35, 49J15, 93B03
Received: 10.04.2018

Citation: L. I. Rodina, “Properties of average time profit in stochastic models of harvesting a renewable resource”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:2 (2018), 213–221

Citation in format AMSBIB
\Bibitem{Rod18}
\by L.~I.~Rodina
\paper Properties of average time profit in stochastic models of harvesting a renewable resource
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 2
\pages 213--221
\mathnet{http://mi.mathnet.ru/vuu632}
\crossref{https://doi.org/10.20537/vm180207}
\elib{http://elibrary.ru/item.asp?id=35258688}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Egorova, L. I. Rodina, “Ob optimalnoi dobyche vozobnovlyaemogo resursa iz strukturirovannoi populyatsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 501–517  mathnet  crossref
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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