About the methods of renewable resourse extraction from the structured population

The problem of optimal extraction of a resource from the structured population consisting of individual species or divided into age groups, is considered. Population dynamics, in the absence of exploitation, is given by a system of ordinary differential equations and at certain time moments, part of the population, is extracted. In particular, it can be assumed that we extract various types of fish, each of which has a certain value. Moreover, there exist predator-prey interactions or competition relationships for food and habitat between these species. We study the properties of the average time benefit which is equal to the limit of the average cost of the resource with an unlimited increase in times of withdrawals. Conditions are obtained under which the average time benefit goes to infinity and a method for constructing a control system to achieve this value is indicated. We show that for some models of interaction between two species, this method of extracting a resource can lead to the complete extinction of one of the species and unlimited growth to the other. Therefore, it seems appropriate to study the task of constructing a control to achieve a fixed final value of the average time benefit. The results obtained here are illustrated with examples of predator-prey models and models of competition of two species and can be applied to other various models of population dynamics.