
Probabilistic Models
On estimation of an average time profit in probabilistic environmental and economic models
L. I. Rodina^{a}, I. I. Tyuteev^{b} ^{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 environmentaleconomical 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.
DOI:
https://doi.org/10.18255/1818101520183257267
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UDC:
517.935 Received: 12.03.2018
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 257267
\mathnet{http://mi.mathnet.ru/mais626}
\crossref{https://doi.org/10.18255/1818101520183257267}
\elib{http://elibrary.ru/item.asp?id=35144408}
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http://mi.mathnet.ru/eng/mais626 http://mi.mathnet.ru/eng/mais/v25/i3/p257
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