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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 1, Pages 48–58 (Mi vuu619)

MATHEMATICS

Optimization of average time profit for a probability model of the population subject to a craft

L. I. Rodina

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

Abstract: We consider the model of population subject to a craft, in which sizes of the trade preparations are random variables. In the absence of operation the population development is described by the logistic equation $\dot x = (a-bx) x,$ where coefficients $a$ and $b$ are indicators of growth of population and intraspecific competition respectively, and in time moments $\tau_k=kd$ some random share of a resource $\omega_k,$ $k=1,2, \ldots,$ is taken from population. We assume that there is a possibility to exert influence on the process of resource gathering so that to stop preparation in the case when its share becomes big enough (more than some value $u_k\in (0,1)$ in the moment $\tau_k$) in order to keep the biggest possible rest of a resource and to increase the size of next gathering. We investigate the problem of an optimum way to control population $\bar u = (u_1, …, u_k, …)$ at which the extracted resource is constantly renewed and the value of average time profit can be lower estimated by the greatest number whenever possible. It is shown that at insufficient restriction of a share of the extracted resource the value of average time profit can be equaled to zero for all or almost all values of random parameters. We also consider the following problem: let a value $u\in (0,1)$ be given, by which we limit a random share of a resource $\omega_k,$ extracted from population in time moments $\tau_k,$ $k=1,2, \ldots .$ It is required to find minimum time between neighboring withdrawals, necessary for resource renewal, in order to make it possible to do extractions until the share of the taken resource does not reach the value $u.$

Keywords: model of the population subject to a craft, average time profit, optimal exploitation.

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

DOI: https://doi.org/10.20537/vm180105

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Bibliographic databases:

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

Citation: L. I. Rodina, “Optimization of average time profit for a probability model of the population subject to a craft”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:1 (2018), 48–58

Citation in format AMSBIB
\Bibitem{Rod18} \by L.~I.~Rodina \paper Optimization of average time profit for a probability model of the population subject to a craft \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2018 \vol 28 \issue 1 \pages 48--58 \mathnet{http://mi.mathnet.ru/vuu619} \crossref{https://doi.org/10.20537/vm180105} \elib{http://elibrary.ru/item.asp?id=32697215} 

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This publication is cited in the following articles:
1. L. I. Rodina, I. I. Tyuteev, “Ob otsenke srednei vremennoi vygody v veroyatnostnykh ekologo-ekonomicheskikh modelyakh”, Model. i analiz inform. sistem, 25:3 (2018), 257–267
2. L. I. Rodina, “Svoistva srednei vremennoi vygody v stokhasticheskikh modelyakh sbora vozobnovlyaemogo resursa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:2 (2018), 213–221
3. 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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