Earlier the authors proved the equivalence of a proposed formulation of a renewable ecoresource sustainable exploitation problem (based on representing the exploited ecosystem by a discrete dynamic system) and a certain mathematical program. In this paper we prove the concavity of a map describing the dependence of the state vector of the ecosystem on the control in the case where the step operator of the dynamic system is concave. In the particular case of a structured ecosystem described by Leslie’s binary model, conditions for the objective function are characterized under which there are optimal controls preserving all structural divisions of the system. In this case, we used the notion of local irreducibility, which generalizes the classical notion of map irreducibility.
Keywords: rational exploitation of ecosystems, optimal nondestructive controls, irreducible map, concave programming
Received June 11, 2020
Revised July 20, 2020
Accepted July 24, 2020
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-07-01243).
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
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
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Cite this article as: Vl.D. Mazurov, A.I. Smirnov. A criterion for the existence of nondestructive controls in the problem of optimal exploitation of an ecosystem with a binary structure, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 101–117; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 315, Suppl. 1, pp. S203–S218.
