Estimates of harvesting characteristicsfor stochastic structured populations

Models of structured populations consisting of individual species $x_1,\ldots,x_n,$ or divided into $n$ age groups are considered. We assume that in the absence of exploitation, the dynamics of the population is given by a system of differential equations ${\dot x =f(x),}$ and at fixed moments of time, random fractions of the resource of each species are extracted from the population. A method is proposed for building control of the harvesting process, in which the amount of the extracted resource is limited in order to increase the size of the next collection. Estimates of the harvesting characteristics, such as the average time benefit and total income, including discounting, were obtained, performed with a probability of one.
Two methods are proposed to solve this problem. The first one can be applied to systems with the property of monotony of solutions with respect to initial conditions. In the second method, there are no restrictions on the properties of the system. It consists in constructing positively invariant sets in which the trajectories are located. The concept of Lyapunov functions with respect to a set, introduced by E. L. Tonkov, is used. Examples of estimation of the considered characteristics for models of interaction of two species, such as neutralism and competition, are given.