TY - JOUR
T1 - Linking age, survival, and transit time distributions
AU - Calabrese, Salvatore
AU - Porporato, Amilcare Michele M.
N1 - Funding Information:
This work was partially funded through the Agriculture and Food Research Initiative of the USDA National Institute of Food and Agriculture (2011-67003-30222), the National Science Foundation through grants CBET-1033467, EAR-1331846, FESD-1338694, and EAR-1316258, and by the U.S. Department of Energy (DOE) through the Office of Biological and Environmental Research (BER) Terrestrial Carbon Processes Program (DE-SC0006967). We thank Anthony Parolari, Luca Ridolfi, Gianluca Botter, Arash Massoudieh, and one anonymous reviewer for useful comments. This paper is a theoretical study and hence no data were used.
Publisher Copyright:
©2015. American Geophysical Union. All Rights Reserved.
PY - 2015/10
Y1 - 2015/10
N2 - Although the concepts of age, survival, and transit time have been widely used in many fields, including population dynamics, chemical engineering, and hydrology, a comprehensive mathematical framework is still missing. Here we discuss several relationships among these quantities by starting from the evolution equation for the joint distribution of age and survival, from which the equations for age and survival time readily follow. It also becomes apparent how the statistical dependence between age and survival is directly related to either the age dependence of the loss function or the survival-time dependence of the input function. The solution of the joint distribution equation also allows us to obtain the relationships between the age at exit (or death) and the survival time at input (or birth), as well as to stress the symmetries of the various distributions under time reversal. The transit time is then obtained as a sum of the age and survival time, and its properties are discussed along with the general relationships between their mean values. The special case of steady state case is analyzed in detail. Some examples, inspired by hydrologic applications, are presented to illustrate the theory with the specific results.
AB - Although the concepts of age, survival, and transit time have been widely used in many fields, including population dynamics, chemical engineering, and hydrology, a comprehensive mathematical framework is still missing. Here we discuss several relationships among these quantities by starting from the evolution equation for the joint distribution of age and survival, from which the equations for age and survival time readily follow. It also becomes apparent how the statistical dependence between age and survival is directly related to either the age dependence of the loss function or the survival-time dependence of the input function. The solution of the joint distribution equation also allows us to obtain the relationships between the age at exit (or death) and the survival time at input (or birth), as well as to stress the symmetries of the various distributions under time reversal. The transit time is then obtained as a sum of the age and survival time, and its properties are discussed along with the general relationships between their mean values. The special case of steady state case is analyzed in detail. Some examples, inspired by hydrologic applications, are presented to illustrate the theory with the specific results.
KW - Kendrick-von Foerster
KW - age distribution
KW - input-output systems
KW - life expectancy
KW - survival time
KW - transit time
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U2 - 10.1002/2015WR017785
DO - 10.1002/2015WR017785
M3 - Article
AN - SCOPUS:84956763003
SN - 0043-1397
VL - 51
SP - 8316
EP - 8330
JO - Water Resources Research
JF - Water Resources Research
IS - 10
ER -