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

MATHEMATICS

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

L. I. Rodina

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_à

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

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Bibliographic databases:

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

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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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
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