Transit times and mean ages for nonautonomous and autonomous compartmental systems (Q338332)

The authors consider the nonautonomous compartmental system \(\dot{x}=B(t)x+s(t)\), where \(B:I\to\mathbb{R}^{d\times d}\) is a bounded continuous function and \(s:I\to[0,\infty)^d\) is a bounded continuous function, \(I:=(\tau,\infty)\) with \(\tau\in \{-\infty\}\cup\mathbb{R}\), and the entries of the matrix \(B\) satisfy certain sign conditions. The following is taken from authors' abstract: We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick-von Förster 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 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.