On the properties of systems of differential equations and problems of optimal resource extraction

A classification of autonomous nonlinear systems of ordinary differential equations is proposed. The behavior of the trajectories of the systems of each class is studied and it is shown that solutions of systems of the first and second classes have the property of monotonicity of solutions with respect to the initial data. One version of the comparison theorem for these systems is given. Examples of systems of the second and third classes are considered for models of interactions of two biological species and their phase portraits are constructed. It is shown that problems of optimal extraction of a renewable resource are solved differently for systems of different classes. In particular, the question of whether it is advisable to extract one type of resource or two types is discussed.