A.I. Smirnov, Vl.D. Mazurov. A solution algorithm for a problem of optimal exploitation of a system with a binary structure ... P. 142-160

We consider a dynamic problem of an optimal sustainable exploitation of a renewable bioresource system that in equilibrium is equivalent to a mathematical programming problem. The latter, in the case of a system with a binary structure described by a nonlinear generalization of Leslie’s model, for a fixed value of some aggregated variable, turns into a linear program. A solution algorithm is proposed for the optimal sustainable exploitation problem. The algorithm employs the peculiarities of the constraint system of the problem dual to this linear program and reduces the original problem to a series of one-dimensional optimization problems.

Keywords: rational exploitation of ecosystems, optimal nondestructive controls, concave programming

Received May 18, 2021

Revised July 9, 2021

Accepted July 19, 2021

Aleksandr Ivanovich Smirnov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: asmi@imm.uran.ru

Vladimir Danilovich Mazurov, Dr. Phys.-Math. Sci. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: mazurov@imm.uran.ru

REFERENCES

1.   The state of world fisheries and aquaculture. Sustainability in action. Rome: Food and Agriculture Organization of the United Nations, 2020, 210 p. doi: 10.4060/ca9229en 

2.   Link J.S., Watson R.A. Global ecosystem overfishing: clear delineation within real limits to production. Science Advances, 2019, vol. 5, no. 6, art. no. eaav0474, 11 p. doi: 10.1126/sciadv.aav0474 

3.   The state of the world’s forests. Forests, biodiversity and people. Rome: Food and Agriculture Organization of the United Nations, 2020, 188 p. doi: 10.4060/ca8642en 

4.   Scott Mills L.S. Conservation of wildlife populations: Demography, genetics and management. 2nd Ed. NJ: Wiley-Blackwell, 2013, 342 p. ISBN: 9781118406670 .

5.   Fonseca C.R., Paterno G.B., Guadagnin D.L. et al. Conservation biology: four decades of problem- and solution-based research. Perspectives in Ecology and Conservation, 2021, vol. 19, no. 2, pp. 121–130. doi: 10.1016/j.pecon.2021.03.003 

6.   Roxburgh T., Ellis K., Johnson J.A., et al. Global futures: Assessing the global economic impacts of environmental change to support policy-making. Technical report. World Wildlife Fund, 2020, 102 p. Available on: https://www.wwf.org.uk/sites/default/files/2020-02/Global_Futures_Techni...

7.   Clark C.W. Mathematical bioeconomics: The mathematics of conservation. Ser. Pure and applied mathematics: A Wiley Series of Texts, Monographs and Tracts, 3rd ed., NY: Wiley Interscience, 2010, 368 p. ISBN: 978-0-470-37299-9 .

8.   Getz W.M., Haight R.G. Population harvesting: demographic models of fish, forest and animal resources. Princeton, NJ: Princeton University Press, 1989, 391 p. ISBN: 9780691085166 .

9.   Andersen K.H. Fish ecology, evolution and exploitation: A new theoretical synthesis. Monographs in Population Biology, vol. 62, Princeton, NJ: Princeton University Press, 2019, 257 p. doi: 10.23943/princeton/9780691192956.001.0001 

10.   Quaas M.F., Tahvonen O. Strategic harvesting of age-structured populations. Marine Resource Economics, 2019, vol. 34, no. 4, pp. 291–309. doi: 10.1086/705905 

11.   Tuljapurkar T., Coulson C., Steiner S. Structured population models: Introduction. Theoretical Population Biology, 2012, vol. 82, no. 4, pp. 241–243. doi: 10.1016/j.tpb.2012.10.007 

12.   Botsford L.W., White J.W., Hastings A. Population dynamics for conservation. Oxford: Oxford University Press, 2019, 352 p. doi: 10.1093/oso/9780198758365.003.0003 

13.   De Lara M., Doyen L. Sustainable management of natural resources: Mathematical models and methods. Berlin; Heidelberg: Springer-Verlag, 2008, 266 p. doi: 10.1007/978-3-540-79074-7 

14.   Smirnov A.I., Mazurov Vl.D. On existence of optimal non-destructive controls for ecosystem exploitation problem applied to a generalization of Leslie model. DEStech Transactions on Computer Science and Engineering (Suppl. Vol.): Proc. Internat. Conf. on Optimization and Applications (OPTIMA-2018), Yu. G. Evtushenko et al. (eds.), 2018, pp. 199–213. doi: 10.12783/dtcse/optim2018/27933 

15.   Smirnov A.I., Mazurov Vl.D. Generalization of controls bimodality property in the optimal exploitation problem for ecological population with binary structure. In: Proc. Internat. Conf. on Optimization and Applications (OPTIMA 2019): Optimization and Applications, M. Jacimovic et al. (eds), Ser. Communications in Computer and Information Science, vol. 1145, Cham: Springer, 2020, pp. 206–221. doi: 10.1007/978-3-030-38603-0_16 

16.   Mazurov V.D., Smirnov A.I. A criterion for the existence of nondestructive controls in the problem of optimal exploitation of an ecosystem with a binary structure. Trudy Inst. Mat. i Mekh. UrO RAN, 2020, vol. 26, no. 3, pp. 101–117 (in Russian). doi: 10.21538/0134-4889-2020-26-3-101-117 

Cite this article as: A.I. Smirnov, Vl.D. Mazurov. A solution algorithm for a problem of optimal exploitation of a system with a binary structure, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 142–160.