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Russia Dolgoprudnyi; Moscow; Vladimir
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.
  1. 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.
  2. Quaas M.F., Tahvonen O. Strategic harvesting of age-structured populations, Marine Resource Economics, 2019, vol. 34, no. 4, pp. 291-309.
  3. Egorova A.V., Rodina L.I. On optimal harvesting of renewable resource from the structured population, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 501-517 (in Russian).
  4. Belkhodja K., Moussaoui A., Alaoui M.A.A. Optimal harvesting and stability for a prey-predator model, Nonlinear Analysis: Real World Applications, 2018, vol. 39, pp. 321-336.
  5. Frisman Y.Y., Kulakov M.P., Revutskaya O.L., Zhdanova O.L., Neverova G.P. The key approaches and review of current researches on dynamics of structured and interacting populations, Computer Research and Modeling, 2019, vol. 11, no. 1, pp. 119-151 (in Russian).
  6. Belyakov A.O., Davydov A.A. Efficiency optimization for the cyclic use of a renewable resource, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 299, suppl. 1, pp. 14-21.
  7. Davydov A.A. Existence of optimal stationary states of exploited populations with diffusion, Proceedings of the Steklov Institute of Mathematics, 2020, vol. 310, issue 1, pp. 124-130.
  8. Davydov A.A., Melnik D.A. Optimal states of distributed exploited populations with periodic impulse selection, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 99-107 (in Russian).
  9. Riznichenko G.Yu., Rubin A.B. Matematicheskie modeli biologicheskikh produktsionnykh protsessov (Mathematical models of biological production processes), Moscow: Moscow State University, 1993.
  10. Reed W.J. A stochastic model for the economic management of a renewable animal resource, Mathematical Biosciences, 1974, vol. 22, pp. 313-337.
  11. Gleit A. Optimal harvesting in continuous time with stochastic growth, Mathematical Biosciences, 1978, vol. 41, issues 1-2, pp. 111-123.
  12. Liu L., Meng X. Optimal harvesting control and dynamics of two-species stochastic model with delays, Advances in Difference Equations, 2017, vol. 2017, issue 1, article number: 18.
  13. Tahvonen O., Quaas M.F., Voss R. Harvesting selectivity and stochastic recruitment in economic models of age-structured fisheries, Journal of Environmental Economics and Management, 2018, vol. 92, pp. 659-676.
  14. Rodina L.I. Optimization of average time profit for a probability model of the population subject to a craft, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 1, pp. 48-58 (in Russian).
  15. Masterkov Yu.V., Rodina L.I. Estimation of average time profit for stochastic structured population, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2020, vol. 56, pp. 41-49 (in Russian).
  16. Jensen F., Frost H., Abildtrup J. Fisheries regulation: A survey of the literature on uncertainty, compliance behavior and asymmetric information, Marine Policy, 2017, vol. 81, pp. 167-178.
  17. Hening A., Nguyen D.H. Coexistence and extinction for stochastic Kolmogorov systems, The Annals of Applied Probability, 2018, vol. 28, no. 3, pp. 1893-1942.
  18. Hening A., Tran K.Q., Phan T.T., Yin G. Harvesting of interacting stochastic populations, Journal of Mathematical Biology, 2019, vol. 79, issue 2, pp. 533-570.
  19. Abramova E.P., Perevalova T.V. Influence of random effects on the equilibrium modes in the population dynamics model, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2020, vol. 55, pp. 3-18 (in Russian).
  20. Sharkovskii A.N., Kolyada S.F., Sivak A.G., Fedorenko V.V. Dinamika odnomernykh otobrazhenii (Dynamics of one-dimensional mappings), Kiev: Naukova dumka, 1989.
  21. Svirezhev Yu.M., Logofet D.O. Ustoichivost' biologicheskikh soobshchestv (Stability of biological communities), Moscow: Nauka, 1978.
  22. Shiryaev A.N. Veroyatnost'-1 (Probability-1), Moscow: Moscow Center for Continuous Mathematical Education, 2011.
  23. Feller W. An introduction to probability theory and its applications. Vol. 2, Wiley, 1971.
  24. Bellman R. A Markovian decision process, Journal of Mathematics and Mechanics, 1957, vol. 6, no. 5, pp. 679-684.
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