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## Archive of Issues

Russia Dolgoprudnyi; Moscow; Vladimir
Year
2022
Volume
32
Issue
2
Pages
211-227
 Section Mathematics Title On how to exploit a population given by a difference equation with random parameters Author(-s) Rodin A.A.a, Rodina L.I.bc, Chernikova A.V.c Affiliations Moscow Institute of Physics and Technologya, National University of Science and Technology MISISb, Vladimir State Universityc Abstract We consider a model of an exploited homogeneous population given by a difference equation depending on random parameters. In the absence of exploitation, the development of the population is described by the equation $$X(k+1)=f\bigl(X(k)\bigr), \quad k=1,2,\ldots,$$ where $X(k)$ is the population size or the amount of bioresources at time $k,$ $f(x)$ is a real differentiable function defined on $I=[0,a]$ such that $f(I)\subseteq I$. At moments $k=1,2,\ldots$, a random fraction of the resource $\omega(k)\in\omega\subseteq[0,1]$ is extracted from the population. The harvesting process can be stopped when the share of the harvested resource exceeds a certain value of $u(k)\in[0,1)$ to keep as much of the population as possible. Then the share of the extracted resource will be equal to $\ell(k)=\min (\omega(k),u(k)).$ The average temporary benefit $H_*$ from the extraction of the resource is equal to the limit of the arithmetic mean from the amount of extracted resource $X(k)\ell(k)$ at moments $1,2,\ldots,k$ when $k\to\infty.$ We solve the problem of choosing the control of the harvesting process, in which the value of $H_*$ can be estimated from below with probability one, as large a number as possible. Estimates of the average time benefit depend on the properties of the function $f(x)$, determining the dynamics of the population; these estimates are obtained for three classes of equations with $f(x)$, having certain properties. The results of the work are illustrated, by numerical examples using dynamic programming based on, that the process of population exploitation is a Markov decision process. Keywords difference equations, equations with random parameters, optimal exploitation, average time profit UDC 517.929, 519.857.3 MSC 39A23, 49L20, 49N90, 90C40, 93C55 DOI 10.35634/vm220204 Received 25 August 2021 Language Russian Citation Rodin A.A., Rodina L.I., Chernikova A.V. On how to exploit a population given by a difference equation with random parameters, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 211-227. References Belyakov A.O., Davydov A.A., Veliov V.M. Optimal cyclic exploitation of renewable resources, Journal of Dynamical and Control Systems, 2015, vol. 21, issue 3, pp. 475-494. https://doi.org/10.1007/s10883-015-9271-x Quaas M.F., Tahvonen O. 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