Estimation of average time profit for stochastic structured population

Abstract

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 x1,…,xn, 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 x˙=f(x), and in time moments kd, d > 0 are taken some random share of a resource ω(k), k=1,2,…, 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.