About one stochastic harvesting model of a renewed resourse

L. I. Rodina

Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 685-695

Résumé

We investigate the models of dynamics of the harvested population, given by the control systems with impulse influences depending on random parameters. We assume that in the absence of harvesting population development is described by system of the differential equations $\dot x =f (x)$ and in time moments $kd, $ $d> 0$ from population are taken some random share of a resource $ \omega (k) = (\omega_1 (k), \ldots, \omega_n (k)) \in \Omega, $ $k=1,2, \ldots, $ that leads to sharp (impulse) reduction of its quantity. Considered resource $x\in\mathbb R^n _ + $ is non-uniform, that is or it consists of separate kinds $x_1, \ldots, x_n, $ or it is divided on $n $ age groups. In particular, it is possible to assume that we make harvesting of $n $ various kinds of fishes between which there are competition relations for food or dwelling places. We describe the probability model of a competition of two kinds for which we receive the estimations of average time benefit from the resource extraction, fulfilled with probability one.

Détail

Keywords: model of the population subject to a craft, average time profit
Mots-clés : optimal exploitation.

Bibliographie

[1] A. Glait, “Optimal harvesting in continuous time with stochastic growth”, Mathematical Biosciences, 41 (1978), 111–123

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

[3] T. R. Lewis, “Exploitation of a renewable resourse under uncertaintly”, Canadian Journal of Economics, 14 (1981), 422–439

[4] D. Ryan, F. B. Hanson, “Optimal harvesting of a logistic population with stochastic jumps”, J. Math. Biol., 24 (1986), 259–277

[5] U. Kapaun, M. F. Quaas, “Does the optimal size of a fish stock increase with environmental uncertainties?”, Economics Working Paper, 9 (2012), 1–40

[6] L. G. Hansen, F. Jensen, “Regulating fisheries under uncertainty”, Resourse and Energy Economics, 50 (2017), 164–177

[7] F. Jensen, H. Frost, J. Abildtrup, “Fisheries regulation: A survey of the literature on uncertainty, compliance behavior and asymmetric information”, Marine Policy, 21 (2017), 167–178

[8] L. I. Rodina, “Optimization of average time profit for a probability model of the population subject to a craft”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 28:1 (2018), 48–58 (In Russian)

[9] L. I. Rodina, I. I. Tyuteev, “On Estimation of an Average Time Profit in Probabilistic Environmental and Economic Models”, Modeling and Analysis of Information Systems, 25:3 (2018), 257–267 (In Russian)

[10] A. N. Shiryaev, Probability, Nauka Publ., Moscow, 1980, 574 pp. (In Russian)

[11] G. Yu. Riznichenko, Lectures on Mathematical Models in Biology. Part 1, Scientific-Publishing Centre “Regular and Chaotic Dynamics”, Izhevsk, 2002, 232 pp. (In Russian)

[12] L. I. Rodina, “Properties of average time profit in stochastic models of harvesting a renewable resource”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 28:2 (2018), 213–221 (In Russian)