On one system of ordinary differential equations of large dimension and a delay equation

We consider some system of ordinary differential equations modeling a multistage synthesis of a substance. We prove the global limit theorems that establish connections between the solutions to a system of ordinary differential equations of large dimension and the solutions to a delay equation. The obtained results enable us to find the approximate solutions to the systems under consideration of an arbitrarily high dimension on the whole half-axis. Some approximation estimates are established.