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Linear autonomous compartmental models as continuous-time Markov chains: transit-time and age distributions

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Metzler,  Holger
Theoretical Ecosystem Ecology Group, Dr. Carlos Sierra, Department Biogeochemical Processes, Prof. S. E. Trumbore, Max Planck Institute for Biogeochemistry, Max Planck Society;
IMPRS International Max Planck Research School for Global Biogeochemical Cycles, Max Planck Institute for Biogeochemistry, Max Planck Society;

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Sierra,  Carlos
Theoretical Ecosystem Ecology Group, Dr. Carlos Sierra, Department Biogeochemical Processes, Prof. S. E. Trumbore, Max Planck Institute for Biogeochemistry, Max Planck Society;

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Citation

Metzler, H., & Sierra, C. (2018). Linear autonomous compartmental models as continuous-time Markov chains: transit-time and age distributions. Mathematical Geosciences, 50(1), 1-34. doi:10.1007/s11004-017-9690-1.


Cite as: http://hdl.handle.net/11858/00-001M-0000-002D-8F0F-A
Abstract
Linear compartmental models are commonly used in different areas of science, particularly in modeling the cycles of carbon and other biogeochemical elements. The representation of these models as linear autonomous compartmental systems allows different model structures and parameterizations to be compared. In particular, measures such as system age and transit time are useful model diagnostics. However, compact mathematical expressions describing their probability distributions remain to be derived. This paper transfers the theory of open linear autonomous compartmental systems to the theory of absorbing continuous-timeMarkov chains and concludes that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution. This probability distribution generalizes the exponential distribution from its application to one-compartment systems to multiple-compartment systems. Furthermore, this paper shows that important system diagnostics have natural probabilistic counterparts. For example, in steady state the system’s transit time coincides with the absorption time of a related Markov chain, whereas the system age and compartment ages correspond with backward recurrence times of an appropriate renewal process. These relations yield simple explicit formulas for the system diagnostics that are applied to one linear and one nonlinear carbon-cycle model in steady state. Earlier results for transit-time and system-age densities of simple systems are found to be special cases of probability density functions of phase-type. The new explicit formulas make costly long-term simulations to obtain and analyze the age structure of open linear autonomous compartmental systems in steady state unnecessary.