Abstract

We consider models of harvesting a renewable resource given by differential equations with impulse action, which depend on random parameters. In the absence of harvesting the population development is described by the differential equation $ \dot x =g (x), $ which has the asymptotic stable solution $\varphi (t) \equiv K,$ $K> 0.$ We assume that the lengths of the intervals $ \theta_k =\tau_k\tau _ {k1} $ between the moments of impulses $ \tau_k $ are random variables and the sizes of impulse action depend on random parameters $v_k, $ $k=1,2, \ldots. $ It is possible to exert influence on the process of gathering in such a way as to stop preparation in the case where its share becomes big enough to keep some part of a resource for increasing the size of the next gathering. We construct the control $ \bar u = (u_1, \dots, u_k, \dots),$ which limits the share of an extracted resource at each instant of time $ \tau_k $ so that the quantity of the remaining resource, starting with some instant $ \tau _ {k_0}$, is no less than a given value $x> 0. $ We obtain estimates of average time profit from extraction of a resource and present conditions under which it has a positive limit (with probability one). It is shown that in the case of an insufficient restriction on the extraction of a resource the value of average time profit can be zero for all or almost all values of random parameters. Thus, we describe a way of longterm extraction of a resource for the gathering mode in which some part of population necessary for its further restoration constantly remains and there is a limit of average time profit with probability one.

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