title: Transit times and mean ages for nonautonomous and autonomous compartmental systems.
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.
authors.affiliation: Department of Mathematics, Imperial College London, London, UK. m.rasmussen@imperial.ac.uk.
authors.forename: Martin
authors.initials: M
authors.lastname: Rasmussen
authors.affiliation: Department of Environmental Science and Policy, University of California, Davis, USA.
authors.forename: Alan
authors.initials: A
authors.lastname: Hastings
authors.affiliation: Computational Science Laboratory, Microsoft Research, Cambridge, UK.
authors.forename: Matthew J
authors.initials: MJ
authors.lastname: Smith
authors.affiliation: Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS, USA.
authors.forename: Folashade B
authors.initials: FB
authors.lastname: Agusto
authors.affiliation: Department of Mathematics, University of Texas, Arlington, TX, USA.
authors.forename: Benito M
authors.initials: BM
authors.lastname: Chen-Charpentier
authors.affiliation: Climate Change Science Institute, Oak Ridge National Laboratory, Oak Ridge, TN, USA.
authors.forename: Forrest M
authors.initials: FM
authors.lastname: Hoffman
authors.affiliation: Department of Microbiology and Plant Biology, University of Oklahoma, Norman, OK, USA.
authors.forename: Jiang
authors.initials: J
authors.lastname: Jiang
authors.affiliation: Department of Microbiology and Plant Biology, University of Oklahoma, Norman, OK, USA.|Microbiology, Biological Sciences Division, Pacific Northwest National Laboratory, Richland, WA, USA.
authors.forename: Katherine E O
authors.initials: KE
authors.lastname: Todd-Brown
authors.affiliation: Department of Mathematics, University of Oklahoma, Norman, OK, USA.
authors.forename: Ying
authors.initials: Y
authors.lastname: Wang
authors.affiliation: CSIRO Oceans and Atmosphere, Aspendale, VIC, Australia.
authors.forename: Ying-Ping
authors.initials: YP
authors.lastname: Wang
authors.affiliation: Department of Microbiology and Plant Biology, University of Oklahoma, Norman, OK, USA.
authors.forename: Yiqi
authors.initials: Y
authors.lastname: Luo
mesh_terms: D057486:Carbon Cycle; D008954:Models, Biological; D013997:Time Factors
keywords: CASA model; Carbon cycle; Compartmental system; Exponential stability; Linear system; McKendrick–von Förster equation; Mean age; Nonautonomous dynamical system; Transit time