On properties of solutions to one system modeling a multistage substance synthesis

The Cauchy problem for a system of ordinary differential equations modeling a multistage substance synthesis is considered. We study properties of the last component of its solution, describing the concentration of the synthesis product, as a function of the parameter $\tau$ characterizing the total time of the synthesis process. The continuous dependence on $\tau$ is established, estimates for the continuity module are obtained. We prove the uniform convergence as $\tau \to 0$; moreover, the limit function is a solution to the Cauchy problem for one ordinary differential equation.