Transit times and mean ages for nonautonomous and autonomous compartmental systems

We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick-von F\"orster equation, we show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time and show that our nonautonomous theory generalises the autonomous case. We apply these results to study a nine-dimensional nonautonomous compartmental system modeling the terrestrial carbon cycle, which is a modification of the Carnegie-Ames-Stanford approach (CASA) model, and we demonstrate that the nonautonomous versions of transit time and mean age differ significantly from the autonomous quantities when calculated for that model.