On the positive solutions of a model system of nonlinear ordinary differential equations

This article investigates the properties of positive solutions of a model system of two nonlinear ordinary differential equations with variable coefficients. We found the new conditions on coefficients for which an arbitrary solution $(x(t), y(t))$ with positive initial values $x(0)$ and $y(0)$ is positive, nonlocally continued and bounded at $t>0$. For this conditions we investigated the question of global stability of positive solutions via method of constructing the guiding function and the method of limit equations. Via the method of constructing the guide function we proved that if the system of equations has a positive constant solution $(x_*, y_*)$, then any positive solution $(x(t), y(t))$ at $t\rightarrow +\infty$ approaches $(x_*, y_*)$. And in the case when the coefficients of the system of equations have finite limits at $t\rightarrow +\infty$ and the limit system of equations has a positive constant solution $(x_{\infty}, y_{\infty})$, via method of limit equations we proved that any positive solution $(x(t), y(t))$ at $t\rightarrow +\infty$ approaches $(x_{\infty}, y_{\infty})$. The results obtained can be generalized for the multidimensional analog of the investigated system of equations.