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About the methods of renewable resourse extraction from the structured population

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The problem of optimal extraction of a resource from the structured population consisting of individual species or divided into age groups, is considered. Population dynamics, in the absence of exploitation, is given by a system of ordinary differential equations and at certain time moments, part of the population, is extracted. In particular, it can be assumed that we extract various types of fish, each of which has a certain value. Moreover, there exist predatorprey interactions or competition relationships for food and habitat between these species. We study the properties of the average time benefit which is equal to the limit of the average cost of the resource with an unlimited increase in times of withdrawals. Conditions are obtained under which the average time benefit goes to infinity and a method for constructing a control system to achieve this value is indicated. We show that for some models of interaction between two species, this method of extracting a resource can lead to the complete extinction of one of the species and unlimited growth to the other. Therefore, it seems appropriate to study the task of constructing a control to achieve a fixed final value of the average time benefit. The results obtained here are illustrated with examples of predator-prey models and models of competition of two species and can be applied to other various models of population dynamics.

Keywords

model of the population subject to harvesting, structured population, average time benefit

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DOI

10.20310/2686-9667-2022-27-137-16-26

UDC

517.935

Pages

16-26

References

[1] D.D. Bainov, “Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population”, Applied Mathematics and Computation, 39:1 (1990), 37-48. [2] A.I. Abakumov, “Populations optimal harvest (time continuous models)”, Matem. Mod., 5:11 (1993), 41–52 (In Russian). [3] G.P. Neverova, O.L. Zhdanova, E.Ya. Frisman, “Dynamics of predator-prey community with age structures and its changing due to harvesting”, Mathematical Biology and Bioinformatics, 15:1 (2020), 73–92 (In Russian). [4] A.I. Abakumov, Yu.G. Izrailsky, “The harvesting effect on a fish population”, Mathematical Biology and Bioinformatics, 11:2 (2016), 191–204 (In Russian). [5] G.P. Neverova, A.I. Abakumov, E.Ya. Frisman, “Dynamic modes of exploited limited population: results of modeling and numerical study”, Mathematical Biology and Bioinformatics, 11:1 (2016), 1–13 (In Russian). [6] A.O. Belyakov, A.A. Davydov, “Efficiency optimization for the cyclic use of a renewable resource”, Proc. Steklov Inst. Math. (Suppl.), 299:suppl. 1 (2017), 14–21. [7] A.A. Davydov, “Existence of optimal stationary states of exploited populations with diffusion”, Proc. Steklov Inst. Math., 310 (2020), 124–130. [8] A.V. Egorova, L.I. Rodina, “On optimal harvesting of renewable resourse from the structured population”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:4 (2019), 501–517 (In Russian). [9] A.V. Egorova, “Optimization of discounted income for a structured population exposed to harvesting”, Russian Universities Reports. Mathematics, 26:133 (2021), 15–25 (In Russian). [10] M.S. Woldeab, L.I. Rodina, “About the methods of biological resourse extraction, providing the maximum average time benefit”, Russian Mathematics, 1 (2022), 12–24 (In Russian). [11] O.A. Kuzenkov, E.A. Ryabova, Matematicheskoe Modelirovanie Processov Otbora, Nizhny Novgorod University Press, Nizhny Novgorod, 2007 (In Russian). [12] G.Yu. Riznichenko, Lectures on Mathematical Models in Biology. Part 1, Mathematical biology, biophysics, Publisher “Regular and Chaotic Dynamics”, Izhevsk, 2002 (In Russian), 560 pp.

Received

2021-12-03

Section of issue

Scientific articles

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