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 Model. Anal. Inform. Sist., 2018, Volume 25, Number 3, Pages 257–267 (Mi mais626)

Probabilistic Models

On estimation of an average time profit in probabilistic environmental and economic models

L. I. Rodinaa, I. I. Tyuteevb

a Vladimir State University named after Alexander and Nikolay Stoletovs, 87 Gorky str., Vladimir, 600000 Russia
b Udmurt State University, 1 Universitetskaya str., Izhevsk, 426034, Russia

Abstract: We consider environmental-economical models of optimal harvesting, given by the differential equations with impulse action, which depend on random parameters. We assume, that lengths of intervals $\theta_k$ between the moments of impulses $\tau_k$ are random variables and the sizes of impulse influence depend on random parameters $v_k,$ $k=1,2, \ldots$ One example of such objects is an equation with impulses, modelling dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation $\dot x =g (x)$ and in time moments $\tau_k$ some random share of resource $v_k,$ $k=1,2, \ldots$ is taken from population. We can control gathering process so that to stop harvesting when its share will appear big enough to keep possible biggest the rest of a resource to increase the size of the following gathering. Let the equation $\dot x =g (x)$ have an asymptotic stable solution $\varphi (t) \equiv K$ and the interval $(K_1, K_2)$ is the attraction area of the given solution (here $0 \leqslant K_1 <K< K_2$). We construct the control $u = (u_1, …, u_k, …),$ limiting a share of harvesting resource at each moment of time $\tau_k$, so that the quantity of the remained resource, since some moment $\tau _ {k_0},$ would be not less than the given value $x\in (K_1, K).$ For any $x\in (K_1, K)$ the estimations of average time profit, valid with probability one, are received. It is shown, that there is a unique $x ^*\in (K_1, K),$ at which the lower estimation reaches the greatest value. Thus, we described the way of population control at which the value of average time profit can be lower estimated with probability 1 by the greatest number whenever possible.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00346_a This work was supported by Russian Foundation of Basic Researches, project No 16-01-00346-a.

DOI: https://doi.org/10.18255/1818-1015-2018-3-257-267

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UDC: 517.935

Citation: L. I. Rodina, I. I. Tyuteev, “On estimation of an average time profit in probabilistic environmental and economic models”, Model. Anal. Inform. Sist., 25:3 (2018), 257–267

Citation in format AMSBIB
\Bibitem{RodTyu18} \by L.~I.~Rodina, I.~I.~Tyuteev \paper On estimation of an average time profit in probabilistic environmental and economic models \jour Model. Anal. Inform. Sist. \yr 2018 \vol 25 \issue 3 \pages 257--267 \mathnet{http://mi.mathnet.ru/mais626} \crossref{https://doi.org/10.18255/1818-1015-2018-3-257-267} \elib{http://elibrary.ru/item.asp?id=35144408}