Geodesic
Optimization of average time profit for a probability model of the population subject to a craft
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 1, pp. 48-58 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the model of population subject to a craft, in which sizes of the trade preparations are random variables. In the absence of operation the population development is described by the logistic equation $\dot x = (a-bx) x,$ where coefficients $a $ and $b $ are indicators of growth of population and intraspecific competition respectively, and in time moments $ \tau_k=kd$ some random share of a resource $\omega_k,$ $k=1,2, \ldots,$ is taken from population. We assume that there is a possibility to exert influence on the process of resource gathering so that to stop preparation in the case when its share becomes big enough (more than some value $u_k\in (0,1)$ in the moment $\tau_k$) in order to keep the biggest possible rest of a resource and to increase the size of next gathering. We investigate the problem of an optimum way to control population $ \bar u = (u_1, \dots, u_k, \dots)$ at which the extracted resource is constantly renewed and the value of average time profit can be lower estimated by the greatest number whenever possible. It is shown that at insufficient restriction of a share of the extracted resource the value of average time profit can be equaled to zero for all or almost all values of random parameters. We also consider the following problem: let a value $u\in (0,1)$ be given, by which we limit a random share of a resource $ \omega_k, $ extracted from population in time moments $\tau_k,$ $k=1,2, \ldots .$ It is required to find minimum time between neighboring withdrawals, necessary for resource renewal, in order to make it possible to do extractions until the share of the taken resource does not reach the value $u.$
Keywords: model of the population subject to a craft, average time profit
Mots-clés : optimal exploitation. 
@article{VUU_2018_28_1_a4,
     author = {L. I. Rodina},
     title = {Optimization of average time profit for a probability model of the population subject to a craft},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {48--58},
     year = {2018},
     volume = {28},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2018_28_1_a4/}
}
TY  - JOUR
AU  - L. I. Rodina
TI  - Optimization of average time profit for a probability model of the population subject to a craft
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2018
SP  - 48
EP  - 58
VL  - 28
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2018_28_1_a4/
LA  - ru
ID  - VUU_2018_28_1_a4
ER  - 
%0 Journal Article
%A L. I. Rodina
%T Optimization of average time profit for a probability model of the population subject to a craft
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2018
%P 48-58
%V 28
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2018_28_1_a4/
%G ru
%F VUU_2018_28_1_a4
L. I. Rodina. Optimization of average time profit for a probability model of the population subject to a craft. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 1, pp. 48-58. http://geodesic.mathdoc.fr/item/VUU_2018_28_1_a4/

[1] Bainov D. D., “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 | DOI | MR

[2] Dykhta V. A., Samsonyuk O. N., Optimal impulse control with applications, Fizmatlit, M., 2000, 256 pp.

[3] Belyakov A. O., Davydov A. A., Veliov V. M., “Optimal cyclic exploitation of renewable resources”, Journal of Dynamical and Control Systems, 21:3 (2015), 475–494 | DOI | MR

[4] Belyakov A. O., Davydov A. A., “Efficiency optimization for the cyclic use of a renewable resourse”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 22, no. 2 (2016), 38–46 (in Russian) | DOI

[5] Rodina L. I., “On the invariant sets of control systems with random coefficients”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 4, 109–121 (in Russian) | DOI

[6] Rodina L. I., Tyuteev I. I., “About asymptotical properties of solutions of difference equations with random parameters”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 26:1 (2016), 79–86 (in Russian) | DOI

[7] Rodina L. I., “On repelling cycles and chaotic solutions of difference equations with random parameters”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 22, no. 2, 2016, 227–235 (in Russian) | DOI

[8] Reed W. J., “Optimal escapement levels in stochastic and deterministic harvesting models”, Journal of Environmental Economics and Management, 6 (1979), 350–363 | DOI

[9] Kapaun U., Quaas M. F., Does the optimal size of a fish stock increase with environmental uncertainties?, Environmental and Resource Economics, 54:2 (2013), 293–310 | DOI | MR

[10] Clark C., Kirkwood G., “On uncertain renewable resourse stocks: Optimal harvest policies and the value of stock surveys”, Journal of Environmental Economics and Management, 13:3 (1986), 235–244 | DOI

[11] Reed W. J., Clarke H. R., “Harvest decisions and assert valuation for biological resources exhibiting size-dependent stochastic growth”, International Economic Review, 31 (1990), 147–169 | DOI | MR

[12] Weitzman M. L., “Landing fees vs harvest quotas with uncertain fish stocks”, Journal of Environmental Economics and Management, 43 (2002), 325–338 | DOI

[13] Shiryaev A. N., Probability, Nauka, M., 1989, 580 pp.

[14] Feller W, An introduction to probability theory and its applications, v. 1, Wiley, 1971 | MR