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Russian Universities Reports. Mathematics, 2021, Volume 26, Issue 133, Pages 15–25
(Mi vtamu212)
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This article is cited in 2 scientific papers (total in 2 papers)
Scientific articles
Optimization of discounted income for a structured population exposed to harvesting
A. V. Egorova Vladimir State University
Abstract:
A structured population the individuals of which are divided into $n$ age or typical groups $x_1,\ldots,x_n$ is considered.
We assume that at any time moment $k,$ $k=0,1,2\ldots$ the size of the population $x(k)$ is determined by
the normal autonomous system of difference equations $x(k+1)=F\bigl(x(k)\bigr)$,
where $F(x)={\rm col}\bigl(f_1(x),\ldots,f_n(x)\bigr)$ are given vector functions with real non-negative components $f_i(x),$ $i=1,\ldots,n.$
We investigate the case when it is possible to influence the population size by means of harvesting.
The model of the exploited population under discussion has the form
$$
x(k+1)=F\bigl((1-u(k))x(k)\bigr),$$
where $u(k)=\bigl(u_1(k),\dots,u_n(k)\bigr)\in[0,1]^n$ is a control vector, which can be varied to achieve the best result of harvesting the resource.
We assume that the cost of a conventional unit
of each of $n$ classes is constant and equals to $C_i\geqslant 0,$ $i=1,\ldots,n.$
To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form
$$
H_\alpha\bigl(\overline u,x(0)\bigr)={\sum\limits_{j=0}^{\infty}}\sum\limits_{i=1}^{n}C_ix_i(j)u_i(j)e^{-\alpha j},
$$ where $\alpha>0$ is the discount coefficient.
The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is
solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for $n =1$).
Keywords:
structured population; optimization problem for the average temporary gain; discounted income; optimal exploitation; mode of exploitation of the population.
Citation:
A. V. Egorova, “Optimization of discounted income for a structured population exposed to harvesting”, Russian Universities Reports. Mathematics, 26:133 (2021), 15–25
Linking options:
https://www.mathnet.ru/eng/vtamu212 https://www.mathnet.ru/eng/vtamu/v26/i133/p15
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Abstract page: | 241 | Full-text PDF : | 92 | References: | 19 |
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