Transit times and mean ages for nonautonomous and autonomous compartmental systems

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

doi:10.1007/s00285-016-0990-8

Transit times and mean ages for nonautonomous and autonomous compartmental systems

Journal Article · Fri Apr 01 00:00:00 EDT 2016 · Journal of Mathematical Biology

DOI:https://doi.org/10.1007/s00285-016-0990-8· OSTI ID:1251563

Rasmussen, Martin ^[1]; Hastings, Alan ^[2]; Smith, Matthew J. ^[3]; Agusto, Folashade B. ^[4]; Chen-Charpentier, Benito M. ^[5]; Hoffman, Forrest M. ^[6]; Jiang, Jiang ^[7]; Todd-Brown, Katherine E. O. ^[8]; Wang, Ying ^[7]; Wang, Ying -Ping ^[9]; Luo, Yiqi ^[7]

Imperial College London, London (United Kingdom)
Univ. of California, Davis, CA (United States)
Microsoft Research, Cambridge (United Kingdom)
Univ. of Kansas, Lawrence, KS (United States)
Univ. of Texas, Arlington, TX (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Univ. of Oklahoma, Norman, OK (United States)
Univ. of Oklahoma, Norman, OK (United States); Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
CSIRO Oceans and Atmosphere, Aspendale, VIC (Australia)

In this study, 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.

View Accepted Manuscript (DOE)

Research Organization:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Organization:: USDOE

OSTI ID:: 1251563

Alternate ID(s):: OSTI ID: 1327769
OSTI ID: 1333444
OSTI ID: 1399520

Journal Information:: Journal of Mathematical Biology, Journal Name: Journal of Mathematical Biology Journal Issue: 6-7 Vol. 73; ISSN 0303-6812

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (29)

Stability Coppel, W. A. Lecture Notes in Mathematics https://doi.org/10.1007/bfb0067781	book	January 1978
Mean Residence Times in Linear Compartmental Systems. Symbolic Formulae for their Direct Evaluation García-Meseguer, M. Bulletin of Mathematical Biology, Vol. 65, Issue 2 https://doi.org/10.1016/s0092-8240(02)00096-4	journal	March 2003
Terrestrial ecosystem production: A process model based on global satellite and surface data Potter, Christopher S.; Randerson, James T.; Field, Christopher B. Global Biogeochemical Cycles, Vol. 7, Issue 4 https://doi.org/10.1029/93gb02725	journal	December 1993
Roughness effects on fine-scale anisotropy and anomalous scaling in atmospheric flows Katul, G. G.; Porporato, A.; Poggi, D. Physics of Fluids, Vol. 21, Issue 3 https://doi.org/10.1063/1.3097005	journal	March 2009
Potential CO2-Enhanced Carbon Storage by the Terrestrial Biosphere Polglase, Pj; Wang, Yp Australian Journal of Botany, Vol. 40, Issue 5 https://doi.org/10.1071/bt9920641	journal	January 1992
The changing carbon cycle at Mauna Loa Observatory Buermann, W.; Lintner, B. R.; Koven, C. D. Proceedings of the National Academy of Sciences, Vol. 104, Issue 11 https://doi.org/10.1073/pnas.0611224104	journal	March 2007
A note on the concepts of age distribution and transit time in natural reservoirs Bolin, Bert; Rodhe, Henning Tellus, Vol. 25, Issue 1 https://doi.org/10.1111/j.2153-3490.1973.tb01594.x	journal	February 1973
Integral Manifolds for Carathéodory Type Differential Equations in Banach Spaces Aulbach, Bernd; Wanner, Thomas Six Lectures on Dynamical Systems https://doi.org/10.1142/9789812812865_0002	book	May 1996
The relationship between peak warming and cumulative CO₂emissions, and its use to quantify vulnerabilities in the carbon–climate–human system Raupach, Michael R.; Canadell, Josep G.; Ciais, Philippe Tellus B, Vol. 63, Issue 2 https://doi.org/10.3402/tellusb.v63i2.16196	journal	April 2011
System Structure and Connectivity Anderson, David H. Compartmental Modeling and Tracer Kinetics https://doi.org/10.1007/978-3-642-51861-4_11	book	January 1983
Mean Residence Times in Linear Compartmental Systems. Symbolic Formulae for their Direct Evaluation García-Meseguer, M. Bulletin of Mathematical Biology, Vol. 65, Issue 2 https://doi.org/10.1016/S0092-8240(02)00096-4	journal	March 2003
Criteria for exponential dichotomy for triangular systems Battelli, Flaviano; Palmer, Kenneth J. Journal of Mathematical Analysis and Applications, Vol. 428, Issue 1 https://doi.org/10.1016/j.jmaa.2015.03.029	journal	August 2015
Positive feedback between global warming and atmospheric CO ₂ concentration inferred from past climate change : GLOBAL WARMING FEEDBACK Scheffer, Marten; Brovkin, Victor; Cox, Peter M. Geophysical Research Letters, Vol. 33, Issue 10 https://doi.org/10.1029/2005GL025044	journal	May 2006
Analysis of soil carbon transit times and age distributions using network theories Manzoni, Stefano; Katul, Gabriel G.; Porporato, Amilcare Journal of Geophysical Research, Vol. 114, Issue G4 https://doi.org/10.1029/2009JG001070	journal	January 2009
Terrestrial ecosystem production: A process model based on global satellite and surface data Potter, Christopher S.; Randerson, James T.; Field, Christopher B. Global Biogeochemical Cycles, Vol. 7, Issue 4 https://doi.org/10.1029/93GB02725	journal	December 1993
Substrate limitations for heterotrophs: Implications for models that estimate the seasonal cycle of atmospheric CO ₂ Randerson, James T.; Thompson, Matthew V.; Malmstrom, Carolyn M. Global Biogeochemical Cycles, Vol. 10, Issue 4 https://doi.org/10.1029/96GB01981	journal	December 1996
Potential CO2-Enhanced Carbon Storage by the Terrestrial Biosphere Polglase, Pj; Wang, Yp Australian Journal of Botany, Vol. 40, Issue 5 https://doi.org/10.1071/BT9920641	journal	January 1992
Earth’s carbon cycle: A mathematical perspective Rothman, Daniel H. Bulletin of the American Mathematical Society, Vol. 52, Issue 1 https://doi.org/10.1090/S0273-0979-2014-01471-5	journal	September 2014
Predictability of the terrestrial carbon cycle Luo, Yiqi; Keenan, Trevor F.; Smith, Matthew Global Change Biology, Vol. 21, Issue 5 https://doi.org/10.1111/gcb.12766	journal	December 2014
The relationship between peak warming and cumulative CO₂ emissions, and its use to quantify vulnerabilities in the carbon-climate-human system Raupach, Michael R.; Canadell, Josep G.; Ciais, Philippe Tellus B: Chemical and Physical Meteorology, Vol. 63, Issue 2, p. 145-164 https://doi.org/10.1111/j.1600-0889.2010.00521.x	journal	January 2011
Qualitative Theory of Compartmental Systems Jacquez, John A.; Simon, Carl P. SIAM Review, Vol. 35, Issue 1 https://doi.org/10.1137/1035003	journal	March 1993
Six Lectures on Dynamical Systems Aulbach, B.; Colonius, F. World Scientific https://doi.org/10.1142/3012	book	May 1996
Compartment Models and Reservoir Theory Eriksson, E. Annual Review of Ecology and Systematics, Vol. 2, Issue 1 https://doi.org/10.1146/annurev.es.02.110171.000435	journal	November 1971
Linear Analysis of soil Decomposition: Insights from the Century Model Bolker, Benjamin M.; Pacala, Stephen W.; Parton, William J. Ecological Applications, Vol. 8, Issue 2 https://doi.org/10.1890/1051-0761(1998)008[0425:LAOSDI]2.0.CO;2	journal	May 1998
Age of soil Organic Matter and soil Respiration: Radiocarbon Constraints on Belowground c Dynamics Trumbore, Susan Ecological Applications, Vol. 10, Issue 2 https://doi.org/10.1890/1051-0761(2000)010[0399:AOSOMA]2.0.CO;2	journal	April 2000
On the Temperature Dependence of Soil Respiration Lloyd, J.; Taylor, J. A. Functional Ecology, Vol. 8, Issue 3 https://doi.org/10.2307/2389824	journal	June 1994
A note on the concepts of age distribution and transit time in natural reservoirs Bolin, Bert; Rodhe, Henning Tellus, Vol. 25, Issue 1 https://doi.org/10.3402/tellusa.v25i1.9644	journal	January 1973
Dichotomy spectra of triangular equations Pötzsche, Christian Discrete and Continuous Dynamical Systems, Vol. 36, Issue 1 https://doi.org/10.3934/dcds.2016.36.423	journal	June 2015
A semi-analytical solution to accelerate spin-up of a coupled carbon and nitrogen land model to steady state Xia, J. Y.; Luo, Y. Q.; Wang, Y. -P. Geoscientific Model Development, Vol. 5, Issue 5 https://doi.org/10.5194/gmd-5-1259-2012	journal	January 2012

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Transit times and mean ages for nonautonomous and autonomous compartmental systems

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