Estimation of average time profit for stochastic structured population

Yu. V. Masterkov ; L. I. Rodina

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 56 (2020), pp. 41-49.

Résumé

We investigate models of dynamics of the exploited population, given by the system with impulse influences, depending on random parametres. The considered population is structured, that is either consists of separate kinds $x_1, \ldots, x_n, $ or is divided on $n $ age groups. In particular, it is possible to investigate the population of $n $ various kinds of fishes between which there are competition relations for food or dwelling places. We assume, that in the absence of harvesting the population development is described by system of differential equations $\dot x =f (x),$ and in time moments $kd, $ $d> 0$ are taken some random share of a resource $ \omega (k),$ $k=1,2, \ldots,$ that leads to sharp (impulse) reduction of its quantity. It is possible to control gathering process so that not to extract more than it is necessary, if shares of an extracted resource for one or several kinds appear big enough; it is necessary that the certain part of a resource has remained for the purpose of increase in the size of following gathering. We received the estimation of average time profit from the resource extraction, executed with probability one, for the structured population in a case $n> 1.$ We described the way of extraction of a resource for a gathering mode in long-term prospect at which some part of population necessary for its further restoration constantly remains.

Détail

Keywords: structured population, average time profit
Mots-clés : optimal exploitation.

Bibliographie

[1] Reed W. J., “A stochastic model for the economic management of a renewable animal resource”, Mathematical Biosciences, 22 (1974), 313–337 | DOI | MR | Zbl

[2] Gleit A., “Optimal harvesting in continuous time with stochastic growth”, Mathematical Biosciences, 41 (1978), 111–123 | DOI | MR | Zbl

[3] Reed W. J., “Optimal escapement levels in stochastic and deterministic harvesting models”, Journal of Environmental Economics and Management, 6:4 (1979), 350–363 | DOI | MR | Zbl

[4] Ryan D., Hanson F. B., “Optimal harvesting of a logistic population in an environment with stochastic jumps”, Journal of Mathematical Biology, 24:3 (1986), 259–277 | DOI | MR | Zbl

[5] Weitzman M. L., “Landing fees vs harvest quotas with uncertain fish stocks”, Journal of Environmental Economics and Management, 43:2 (2002), 325–338 | DOI | Zbl

[6] 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

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

[8] Jensen F., Frost H., Abildtrup J., “Fisheries regulation: A survey of the literature on uncertainty, compliance behavior and asymmetric information”, Marine Policy, 81 (2017), 167–178 | DOI

[9] Abramova E. P., Perevalova T. V., “Influence of random effects on the equilibrium modes in the population dynamics model”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 55 (2020), 3–18 (in Russian) | DOI | MR | Zbl

[10] Rodina L. I., “Optimization of average time profit for a probability model of the population subject to a craft”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 28:1 (2018), 48–58 (in Russian) | DOI | MR | Zbl

[11] Rodina L. I., “Properties of average time profit in stochastic models of harvesting a renewable resourse”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 28:2 (2018), 213–221 (in Russian) | DOI | MR | Zbl