On the exploitation of a population given by a system of linear equations with random parameters

We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.