Properties of average time profit in stochastic models of harvesting a renewable resource

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 <(x)over dot> = g(x), which has the asymptotic stable solution phi(t) = K, K > 0. We assume that the lengths of the intervals theta(k) = tau(k) - tau(k-1) 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,... 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 (u) over bar = (u(1), ... ,u(k), ...), 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(k0), 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 long-term 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.