Optimization Problems for Models of Harvesting a Renewable Resource
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
OPTIMIZATION PROBLEMS FOR MODELS OF HARVESTING A RENEWABLE RESOURCE L. I. Rodina ∗ A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
A. H. Hammadi University of Al-Qadisiyah Al Diwaniyah, Iraq [email protected]
UDC 517.935
We consider models of exploited populations defined by equations with random parameters. For the deterministic population model we study the resource share control problem of maximizing profit. For the probabilistic model we propose a mining method providing an estimate of the average profit from below by the largest possible number. Bibliography: 7 titles.
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Introduction
A resource extraction method was described in [1, 2] in the case where some part of population, necessary for further recovery, is constantly preserved and there exists (with probability one) a positive limit of average time profit. In this paper, we assume that the value of later profit from resource extraction is reduced and, therefore, we consider the profit from infinitely many harvests with taking into account discount revenue (similar characteristics were studied in [3, 4]). We introduce the main definitions for the exploited population model defined by equations with random parameters. We assume that, in the absence of exploitation, the population development is described by the differential equation x˙ = g(x) and the resource is harvested at the times τk while some random share vk , k = 1, 2, . . ., is extracted from the population. We assume that the lengths of intervals θk = τk − τk−1 between the harvest times τk are random variables and the amount of extracted resource depends on random parameters vk ; here ∗
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Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 103-110. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0113
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θk ∈ Ω1 ⊆ [α1 , β1 ], where 0 < α1 β1 < ∞, vk ∈ Ω2 ⊆ [0, 1], k = 1, 2, . . . . Assume that it is possible to affect the harvesting process to stop a workpiece if its share turns out to be large enough (larger than some value uk ∈ [0, 1) at the time τk ). Then some resource share is preserved to increase the size of the next harvest. In this case, the extracted resource share has the form v k , v k < uk , k = (vk , uk ) = uk , v k uk . Thus, we consider an exploited population with dynamics described by the differential equation with impulse action x˙ = g(x), t = τk , (1.1) x(τk ) = (1 − k ) · x(τk − 0), k = 1, 2, . . . , where θk = τk − τk−1 ∈ Ω1 , (x, vk , uk ) ∈ [0, +∞) × Ω2 × [0, 1]. We assume that the solutions to Equation (1.1) are continuous from the right and the function g(x) is defined and continuously differentiable for all x ∈ [0, +∞). Below, we assume that the following condition is satisfied. Assumption 1.1. The equation x˙ = g(x) has the asymptotically steady solution ϕ(t) ≡ K, and (K1 , K2 ) is the domain of attraction (the asymptotical steadiness domain) of this solution
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