Transit times and mean ages for nonautonomous and autonomous compartmental systems
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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 ninedimensional 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.
Keywords
Carbon cycle CASA model Compartmental system Exponential stability Linear system McKendrick–von Förster equation Mean age Nonautonomous dynamical system Transit timeMathematics Subject Classification
34A30 34D051 Introduction
Compartment models play an important role in the modeling of many biological systems ranging from pharmacokinetics to ecology (Anderson 1983; Godfrey 1983; Jacquez and Simon 1993). Key values in understanding the dynamics of these systems are the transit time: the mean time a particle spends in the compartmental system measured as the mean time from entry into the system to leaving the system (Bolin and Rodhe 1973; Eriksson 1971), and the mean age: the mean age of particles still in the system (Bolin and Rodhe 1973; Eriksson 1971). It is well known that these quantities need not be the same (Bolin and Rodhe 1973; Eriksson 1971; Rothman 2015).
We are motivated by an interest in studying the dynamics of the terrestrial carbon cycle which is typically modeled as a number of discrete pools of carbon in plant biomass, litter and soil organic matter. Many of the best studied models of the dynamics of carbon are linear, which reflects the fact that changes in carbon pools are proportional to the pool size (Bolker et al. 1998). Additionally, most analyses make the further assumption that all parameters describing the dynamics (and the input fluxes) are constant in time, leading to a model in the form of an autonomous linear differential equation. In this autonomous case, it is possible to derive analytic formulae giving expressions for the transit time (GarciaMeseguer et al. 2003; Manzoni et al. 2009). These formulae for transit time are given in terms of (constant) transfer coefficients among compartments and analogous formulae are available for the mean age of particles in the system.
Many applications of models of terrestrial carbon relate to situations in which constant model parameters are replaced by timedependent functions. Perhaps the most wellknown examples are studies of how terrestrial carbon dynamics respond to climate change. In these, it is often assumed that the specific rates (per unit carbon) of carbon inputs and losses from the system change over time as a function of changes in climate, such as temperature. For example, increases in temperature are normally assumed to increase the rates of soil decomposition (Lloyd and Taylor 1994; Orchard and Cook 1983; Rothman 2015). As a consequence, the compartmental models of interest are nonautonomous, i.e. they depend on time (Luo et al. 2001, 2015; Xia et al. 2012). Nonautonomous compartmental systems are special cases of linear nonautonomous differential equations (Kloeden and Rasmussen 2011), which, in contrast to the linear autonomous case, cannot be solved analytically in general. Yet, both the mean age of particles in the system and the transit time remain of great interest for these timedependent systems, as both quantities can be potentially measured in the actual systems being modeled (Rothman 2015; Trumbore 2000).
In this paper, we develop a theory for transit times and mean ages of mass in nonautonomous compartmental systems. As noted in one of the first papers to study transit time (Bolin and Rodhe 1973), there is obviously a close connection between age distribution and transit time in compartment models. We will build on this relationship to develop an approach for understanding the definition of transit time. We define a timedependent version of transit time as the mean age of mass leaving the compartmental system. We use a timedependent version of the McKendrick–von Förster equation (Brauer and CastilloChavez 2012; McKendrick 1926; Thieme 2003), the classic firstorder partial differential equation describing age distributions, to prove that the mean age of mass satisfies an (inhomogeneous) linear nonautonomous differential equation. We show that under weak conditions, this equation is exponentially stable. Starting with demographic models highlights another important aspect of our approach. As is well known, solutions of demographic models depend on initial conditions, so quantities like the mean age and transit time also depend on initial conditions, but conventional definitions of these quantities ignore the influence of the initial conditions. For this reason, our nonautonomous approach also provides additional insight for autonomous compartmental systems that are not in equilibrium.
We apply the theory we have developed to numerically study transit times for a ninedimensional compartmental system model of the carbon cycle, which is a modified version of the Carnegie–Ames–Stanford approach (CASA) model (Buermann et al. 2007; Potter et al. 1993; Randerson et al. 1996). We compare our nonautonomous quantities to the classical notion of transit time for autonomous systems, where we freeze the nonautonomous system in time to obtain an autonomous system, and we assume that we are in equilibrium. Our simulations illustrate the different and sometimes diverging trajectories of the autonomous and nonautonomous quantities over time. Our results demonstrate the necessity of our theory for the computation of transit times in nonautonomous compartmental systems and in autonomous compartmental systems that are not in equilibrium.
This paper is organized as follows. In Sect. 2, we first review the theory of transit times for autonomous compartmental systems, and we provide a heuristic derivation of the transit time formula. We then define nonautonomous compartmental systems in Sect. 3. In Sect. 4, we prove that under the assumption that the compartmental system is lower block triangular, and the diagonal blocks a diagonally dominant, the nonautonomous compartmental system is exponentially stable. In Sect. 5, we prove that the mean ages satisfy a linear nonautonomous differential equation, and we then use the stability criterion from Sect. 4 to prove exponential stability of the mean age equation. We define the concept of a transit time for nonautonomous compartmental systems in Sect. 6. In Sect. 7, we show that our nonautonomous theory is consistent with the autonomous case, in the sense that we get exactly the wellknown autonomous transit time formula when applying the nonautonomous transit time to an autonomous system. Finally, in Sect. 8, we apply the theory to compute transit times for a nonautonomous compartmental model of the carbon cycle, which is a simplified version of the Carnegie–Ames–Stanford approach (CASA) model.
2 Transit times and mean ages for autonomous compartmental systems

\(b_{ii} < 0\) for all \(i\in \{1,\ldots ,d\}\),

\(b_{ij} \ge 0\) for all \(i\not =j\in \{1,\ldots ,d\}\),

\(\sum _{i=1}^d b_{ij} \le 0\) for all \(j\in \{1,\ldots ,d\}\).
We assume that the homogeneous linear system \(\dot{x} = Bx\) is exponentially stable, i.e. all eigenvalues of B have negative real parts (this is fulfilled e.g. when the matrix B is strictly diagonally dominant). This means that (1) has the exponentially stable equilibrium \(x^* = B^{1}s\).
The concept of transit time for compartmental systems describes the mean time a particle spends in the compartmental system before it is released. There is a huge amount of literature on this topic, see e.g. Anderson (1983), Bolin and Rodhe (1973), Eriksson (1971), GarciaMeseguer et al. (2003), Manzoni et al. (2009), but to our knowledge, the following simple derivation of the transit time formula has not been written down before.
Example 1
3 Nonautonomous compartmental systems
In contrast to the autonomous case, both the coefficient matrix B and the input vector s of a nonautonomous compartmental system are allowed to depend on time.
Definition 1

\(b_{ii}(t) < 0\) for all \(i\in \{1,\ldots ,d\}\) and \(t\in I\),

\(b_{ij}(t) \ge 0\) for all \(i\not =j\in \{1,\ldots ,d\}\) and \(t\in I\),

\(\sum _{i=1}^d b_{ij}(t) \le 0\) for all \(j\in \{1,\ldots ,d\}\) and \(t\in I\).
Example 2
4 Exponential stability of nonautonomous compartmental systems
In this section, we provide a sufficient condition for global exponential stability of the nonautonomous compartmental system (6). This criterion will concern only the homogeneous part of (6), i.e. the matrixvalued function B, from which stability for the inhomogeneous equation follows. Since the result holds also for linear systems which are not compartmental systems, we formulate it more generally.
Theorem 1
 (i)
\((B_{nn}(t))_{ii} < 0 \) for all \(t\in I\) and \(i\in \{1,\ldots ,d_n\}\),
 (ii)
\((B_{nn}(t))_{ij} \ge 0 \) for all \(t\in I\) and \(i\not =j\in \{1,\ldots ,d_n\}\),
 (iii)
\(\sum _{j=1}^{d_n} (B_{nn}(t))_{ij} \le \delta \) for all \(t\in I\) and \(i\in \{1,\ldots ,d_n\}\).
Proof
5 The mean age system
We prove in this section that the mean ages of mass in a nonautonomous compartmental system are solutions of a linear nonautonomous differential equation, which we call the mean age system. We derive this result from the evolution of age distributions, given by the wellknown McKendrick–von Förster equation (McKendrick 1926; Brauer and CastilloChavez 2012; Thieme 2003), which is a linear first order partial differential equation. We also prove that the mean age system is exponentially stable under additional weak assumptions, by applying the theory developed in Sect. 4.
The mean age system is pivotal for the analysis of transit times for nonautonomous compartmental systems, since in order to compute the average time the mass spends in the system, we do not need to look at the full age distribution of ages, but only at the mean ages.
We are particularly interested in the transit time of (6) at a particular time t, which corresponds to the mean age of mass leaving the system at time t. For this purpose, we do not need the full age distribution determined by (13), since the situation is fully described by the mean age of mass in pool i, denoted as \({\bar{a}}_i (t)\). The following theorem says that the evolution of the mean ages is determined by an ordinary differential equation.
Theorem 2
Proof
We will show now that under additional weak assumptions, the mean age equation is exponentially stable.
Theorem 3
 (a)
\(s_i(t) \ge \delta \) for all \(t\in I\) and \(i\in \{1,\ldots ,d_1\}\), and
 (b)
for all \(n\in \{2,\ldots ,m\}\) and \(i\in \{1+\sum _{k=1}^{n1}d_k,2+\sum _{k=1}^{n1}d_k,\ldots ,\sum _{k=1}^{n}d_k\}\), there exists a \(j\in \{1,\ldots ,\sum _{k=1}^{n1}d_k\}\) such that \(b_{ij}(t) \ge \delta \) for all \(t\in I\).
Proof
We show now that the three conditions (i)–(iii) of Theorem 1 are satisfied with \(\delta \) replaced by \(\delta \min \{1, \min _{t\in I,i\in \{1,\ldots ,d\}} x_i(t)\}\). Note first that the matrix A(t, x(t)) has the same block decomposition as the matrix B(t), which is described in (9).
Condition (i) of Theorem 1 follows from (a) (in case of \(n=1\)) or (b) (in case \(n>1\); note that the sum of the entries in the ith row of the matrix A(t, x(t)) equals to \(s_i(t)\), and (b) guarantees that the diagonal entry is negative even though \(s_i(t)\) might be zero). Condition (ii) of Theorem 1 follows from the fact that the original system (6) is a compartmental system, and the solution x(t) of (6) has positive entries. Finally, condition (iii) of Theorem 1 follows from fact that the sum of the ith row of the matrix A(t, x(t)) equals to \(s_i(t)\), and the positive contribution of at least \(b_{ij}(t)x_j(t)\ge \delta \min _{t\in I,i\in \{1,\ldots ,d\}} x_i(t)\), with i and j chosen as in (b), will not be considered in the sum in condition (iii) of Theorem 1 and for this reason contributes negatively to this sum.
A natural choice for the solution \(t\mapsto x(t)\) in the above theorem is the exponentially stable solution defined in (12) if the interval I is unbounded below. If the interval I is unbounded below, this will be the only bounded solution of the system, i.e. the norm of all other solutions converges to \(\infty \) in the limit \(t\rightarrow \infty \), so the solution (12) is the only solution to which the theorem can be applied. However, if the interval I is bounded below, then all solutions of the nonautonomous compartmental system (6) are bounded and exponentially stable, and they are also bounded away from zero due to assumption (a) of Theorem 3.
6 Nonautonomous transit times
We define transit time as the mean age of mass leaving the system at a particular time t. Note that in our nonautonomous context, this quantity depends on the actual time t. We also provide a formula that corresponds to the mean age of mass currently residing in the compartmental system.
Definition 2
The transit time \(R_t\) is the mean age of carbon leaving the system at time t, where as the mean age \(M_t\) is the mean age of carbon in the system at time t.
Note that, in general, \(R_t\) and \(M_t\) are different, see Example 1 for the autonomous case. In the following example, we show that transit times and mean ages are the same for onedimensional compartmental systems.
Example 3
7 Consistency with the autonomous case
In this section, we derive simple expressions for the transit time and mean age from Definition 2 in the special case of an autonomous compartmental system. The expression for the autonomous transit time coincides with the heuristically obtained formula (2), and we confirm the expression for the mean ages stated in (3).
Lemma 1
Proof
Proposition 1
Proof
Note that derivation of the autonomous quantities for transit time R and mean age M in Proposition 1 required the autonomous compartmental system (18) to be in equilibrium, and the classical approach to transit times, as outlined in Sect. 2, is not applicable for autonomous systems not in equilibrium. It is very important to note that Definition 2 is useful for autonomous systems also, since it is applicable to systems that are not in equilibrium. For such autonomous systems, transit times and mean ages will depend on time in general, and although they converge to R and M in the limit \(t\rightarrow \infty \), they might be very different to R and M.
8 Mean ages and transit times for the CASA model
Here we illustrate predicted changes in the mean age of carbon leaving and remaining in the system for a terrestrial carbon model under a climate change scenario. We consider a modification of the CASA model as used in Buermann et al. (2007) globally without resolving the spatial details of carbon pools using nine pools representing the global terrestrial carbon (e.g. three pools for plant biomass, or litter or soil organic matter). This caused the model to be precisely of the form of (6). Climate change was simulated by increasing atmospheric \(\mathrm {CO_2}\) over time, which affected both B(t) and s(t) in (6). Increased \(\mathrm {CO_2}\) directly increases carbon inputs s(t) through carbon dioxide fertilization. They also directly increase mean global temperatures. This increases the carbon loss rates from some of the carbon pools, changing components of B(t), and also has an effect on s(t). Thus increased \(\mathrm {CO_2}\) alters the input and loss rates of components of the terrestrial carbon cycle, making both the sign and magnitude of the net change in carbon storage dependent the sensitivity of carbon inputs and loss rates.
Perhaps surprisingly, the monotonic forcing of B(t) and s(t) translates into nonmonotonic effects on \(R_t\) and \(M_t\). A detailed mathematical investigation of this phenomenon is outside the scope of the present study.
The nonautonomous properties \(R_t\) and \(M_t\) show contrasting trajectories to the instantaneous properties R and M (which we computed according to Proposition 1, but note that, since the system is nonautonomous, the assumptions of this proposition are not fulfilled). For example the latter properties change monotonically over time. This must be because the long term outcome of an increase in the input rate of young carbon and an increase in the output rate of old carbon is a decrease in the age of carbon both leaving and remaining in the system. Over the course of the simulation the numerical values of the autonomous and nonautonomous properties become visibly different (Fig. 3). This is because it will take a long time for the values of \(R_t\) and \(M_t\) to approach R and M due to the small loss rate of the ninth soil pool.
9 Conclusions
Models for terrestrial carbon cycling have led to renewed interest in the properties of compartment models. Key quantities that have been studied over many years in compartment models with parameters fixed in time (Eriksson 1971; Bolin and Rodhe 1973; Anderson 1983) are the mean age of particles in the system and the transit time of particles leaving the system. Formulae for these quantities that give the mean age and transit time in terms of parameters of the system in the long time limit have led to insights, but cannot be applied to the case of changing parameters.
As parameters change, for example in a model of carbon cycling due to climate change, it is not correct to calculate the mean age or transit time from the instantaneous parameter values. Using the theory of nonautonomous differential equations as a tool, and beginning with time dependent age structured models, we are able to define and derive formulae for the transit time and mean age for particles in the case of temporally changing parameters. These definitions lead to quantities that reduce to the analogous formulae for the autonomous (constant parameter) case when parameters do not change in time. However, the formulae for the nonautonomous case also highlight the fact that even in the constant parameter case the transit time and mean age do depend on initial conditions; some of the standard formulae do not include this dependence.
The difference between a transit time or mean age that is computed based on the parameters at a given instant and the better approach of taking into account the history of the system can be substantial as we illustrate using a variant of the CASA model. Thus, the approach we develop here is not just of mathematical interest but is of substantial practical importance as well.
Notes
Acknowledgments
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 W911NF1310305. 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 UTBattelle, LLC under Contract No. DEAC0500OR22725 with the U.S. Department of Energy. Katherine E. O. ToddBrown 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 DESC0006982, DESC0008270, DESC0014062, DESC0004601, and DESC0010715 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. EF0832858, 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.
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