Comparison Theorems for Systems of Differential Equations and Their Application to Solving Applied Problems.

Three classes of ordinary differential equations systems are highlighted, and properties possessed by the solutions of each class are described. It is shown that the solutions for the first and second class systems satisfy a monotonicity property with respect to initial data, and systems of two differential equations of all three considered classes cannot have periodic solutions. The obtained conditions for the absence of periodic solutions for autonomous second-order systems complement the known Bendixson criteria. One of the variants of a comparison theorem for systems of ordinary differential equations is formulated.
The issue of estimating the average temporal advantage of resource extraction for a structured population consisting of separate species or divided into age groups is considered. It is demonstrated that using the comparison theorem allows one to find estimates of the average temporal advantage in cases where analytical solutions of the corresponding systems are unknown. The obtained results are illustrated with models of interactions between two species, such as symbiosis and competition. It is shown that for symbiosis and neutralism models, the greatest value of the average temporal advantage is achieved when resources of both species are exploited simultaneously. For populations experiencing 'competition' type interactions, cases are identified in which it is advisable to extract resources from only one species or from both species.