Transit times and mean ages for nonautonomous and autonomous compartmental systems

Martin Rasmussen; Alan Hastings; Matthew J. Smith; Folashade B. Agusto; Benito M. Chen-Charpentier; Forrest M. Hoffman; Jiang Jiang; Katherine E.O. Todd-Brown; Ying Wang; Ying Ping Wang; Yiqi Luo

doi:10.1007/s00285-016-0990-8

Transit times and mean ages for nonautonomous and autonomous compartmental systems

Martin Rasmussen, Alan Hastings, Matthew J. Smith, Folashade B. Agusto, Benito M. Chen-Charpentier, Forrest M. Hoffman, Jiang Jiang, Katherine E.O. Todd-Brown, Ying Wang, Ying Ping Wang, Yiqi Luo

Computational Earth Sciences Group

Research output: Contribution to journal › Article › peer-review

40 Scopus citations

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.

Original language	English
Pages (from-to)	1379-1398
Number of pages	20
Journal	Journal of Mathematical Biology
Volume	73
Issue number	6-7
DOIs	https://doi.org/10.1007/s00285-016-0990-8
State	Published - Dec 1 2016

Funding

Martin Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1 (2010–2015) and by funding from the European Union’s Horizon 2020 research and innovation programme for the ITN CRITICS under Grant Agreement Number 643073. Alan Hastings was supported by Army Research Office Grant W911NF-13-1-0305. Forrest M. Hoffman was supported by the Biogeochemistry–Climate Feedbacks Scientific Focus Area, which is sponsored by the Regional and Global Climate Modeling Program in the Climate and Environmental Sciences Division of the Biological and Environmental Research Program in the U.S. Department of Energy Office of Science. Oak Ridge National Laboratory is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. Katherine E. O. Todd-Brown is grateful for the support of the Linus Pauling Distinguished Postdoctoral Fellowship program which is funded under the Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy. Ying Wang was supported by a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities and by a Faculty Investment Program and a Junior Faculty Fellow Program grant from the Research Council and College of Arts and Sciences of the University of Oklahoma Norman Campus. Research in Yiqi Luo EcoLab was financially supported by U.S. Department of Energy grants DE-SC0006982, DE-SC0008270, DE-SC0014062, DE-SC0004601, and DE-SC0010715 and U.S. National Science Foundation (NSF) grants DBI 0850290, EPS 0919466, DEB 0840964, and EF 1137293. This work was assisted through participation of the authors in the working group Nonautonomous Systems and Terrestrial Carbon Cycle, at the National Institute for Mathematical and Biological Synthesis, an institute sponsored by the National Science Foundation, the US Department of Homeland Security, and the US Department of Agriculture through NSF award no. EF-0832858, with additional support from The University of Tennessee, Knoxville. The authors are grateful to two referees for useful comments that led to an improvement of this paper. Martin Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1 (2010â2015) and by funding from the European Unionâs Horizon 2020 research and innovation programme for the ITN CRITICS under Grant Agreement Number 643073. Alan Hastings was supported by Army Research Office Grant W911NF-13-1-0305. Forrest M. Hoffman was supported by the BiogeochemistryâClimate Feedbacks Scientific Focus Area, which is sponsored by the Regional and Global Climate Modeling Program in the Climate and Environmental Sciences Division of the Biological and Environmental Research Program in the U.S. Department of Energy Office of Science. Oak Ridge National Laboratory is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. Katherine E. O.Â Todd-Brown is grateful for the support of the Linus Pauling Distinguished Postdoctoral Fellowship program which is funded under the Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy. Ying Wang was supported by a Ralph E.Â Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities and by a Faculty Investment Program and a Junior Faculty Fellow Program grant from the Research Council and College of Arts and Sciences of the University of Oklahoma Norman Campus. Research in Yiqi Luo EcoLab was financially supported by U.S. Department of Energy grants DE-SC0006982, DE-SC0008270, DE-SC0014062, DE-SC0004601, and DE-SC0010715 and U.S. National Science Foundation (NSF) grants DBI 0850290, EPS 0919466, DEB 0840964, and EF 1137293. This work was assisted through participation of the authors in the working group Nonautonomous Systems and Terrestrial Carbon Cycle , at the National Institute for Mathematical and Biological Synthesis , an institute sponsored by the National Science Foundation, the US Department of Homeland Security, and the US Department of Agriculture through NSF award no.Â EF-0832858, with additional support from The University of Tennessee, Knoxville. The authors are grateful to two referees for useful comments that led to an improvement of this paper.

Funders	Funder number
Biological and Environmental Research program
College of Arts and Sciences of the University of Oklahoma Norman Campus	DE-SC0014062, DE-SC0010715, DE-SC0008270, DE-SC0004601, DE-SC0006982
Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory
National Science Foundation	DBI 0850290, EPS 0919466, DEB 0840964, EF 1137293
U.S. Department of Energy
U.S. Department of Homeland Security
Army Research Office	W911NF-13-1-0305
U.S. Department of Agriculture	0832858
Oak Ridge Associated Universities
Oak Ridge National Laboratory
University of Tennessee
Horizon 2020 Framework Programme	643073
UT-Battelle	DE-AC05-00OR22725
Engineering and Physical Sciences Research Council	EP/I004165/1
Norges Forskningsråd

Keywords

CASA model
Carbon cycle
Compartmental system
Exponential stability
Linear system
McKendrick–von Förster equation
Mean age
Nonautonomous dynamical system
Transit time

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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 F{\"o}rster 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.",

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AU - Rasmussen, Martin

AU - Hastings, Alan

AU - Smith, Matthew J.

AU - Agusto, Folashade B.

AU - Chen-Charpentier, Benito M.

AU - Hoffman, Forrest M.

AU - Jiang, Jiang

AU - Todd-Brown, Katherine E.O.

AU - Wang, Ying

AU - Wang, Ying Ping

AU - Luo, Yiqi

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - CASA model

KW - Carbon cycle

KW - Compartmental system

KW - Exponential stability

KW - Linear system

KW - McKendrick–von Förster equation

KW - Mean age

KW - Nonautonomous dynamical system

KW - Transit time

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JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

IS - 6-7

ER -

Transit times and mean ages for nonautonomous and autonomous compartmental systems

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Keywords

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