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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 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. Lastly, 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:
 [1];  [2];  [3];  [4];  [5];  [6];  [7];  [8];  [7];  [9];  [7]
+ Show Author Affiliations
  1. Imperial College London, London (United Kingdom)
  2. Univ. of California, Davis, CA (United States)
  3. Microsoft Research, Cambridge (United Kingdom)
  4. Univ. of Kansas, Lawrence, KS (United States)
  5. Univ. of Texas, Arlington, TX (United States)
  6. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  7. Univ. of Oklahoma, Norman, OK (United States)
  8. Univ. of Oklahoma, Norman, OK (United States); Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
  9. CSIRO Oceans and Atmosphere, Aspendale, VIC (Australia)
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1251563
Alternate Identifier(s):
OSTI ID: 1327769
Grant/Contract Number:  
EP/I004165/1; W911NF-13-1-0305; AC05-00OR22725
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Mathematical Biology
Additional Journal Information:
Journal Volume: 73; Journal Issue: 6-7; Journal ID: ISSN 0303-6812
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; carbon cycle; CASA model; compartmental system; exponential stability; linear system; McKendrick-von Forster equation; mean age; nonautonomous; dynamical system; transit time; McKendrick–von Förster equation; nonautonomous dynamical system

Citation Formats

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, and Luo, Yiqi. Transit times and mean ages for nonautonomous and autonomous compartmental systems. United States: N. p., 2016. Web. https://doi.org/10.1007/s00285-016-0990-8.
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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. Transit times and mean ages for nonautonomous and autonomous compartmental systems. United States. https://doi.org/10.1007/s00285-016-0990-8
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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, and Luo, Yiqi. Fri . "Transit times and mean ages for nonautonomous and autonomous compartmental systems". United States. https://doi.org/10.1007/s00285-016-0990-8. https://www.osti.gov/servlets/purl/1251563.
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@article{osti_1251563,
title = {Transit times and mean ages for nonautonomous and autonomous compartmental systems},
author = {Rasmussen, Martin and Hastings, Alan and Smith, Matthew J. and Agusto, Folashade B. and Chen-Charpentier, Benito M. and Hoffman, Forrest M. and Jiang, Jiang and Todd-Brown, Katherine E. O. and Wang, Ying and Wang, Ying -Ping and Luo, Yiqi},
abstractNote = {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. Lastly, 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.},
doi = {10.1007/s00285-016-0990-8},
journal = {Journal of Mathematical Biology},
number = 6-7,
volume = 73,
place = {United States},
year = {2016},
month = {4}
}
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https://doi.org/10.1007/s00285-016-0990-8
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