Static ecological system analysis

Abstract

In this article, a new mathematical method for static analysis of compartmental systems is developed in the context of ecology. The method is based on the novel system and subsystem partitioning methodologies through which compartmental systems are decomposed to the utmost level. That is, the distribution of environmental inputs and intercompartmental system flows as well as the organization of the associated storages generated by these flows within the system is determined individually and separately. Moreover, the transient and the static direct, indirect, acyclic, cycling, and transfer (diact) flows and associated storages transmitted along a given flow path or from one compartment, directly or indirectly, to any other are analytically characterized, systematically classified, and mathematically formulated. A quantitative technique for the categorization of interspecific interactions and the determination of their strength within food webs is also developed based on the diact transactions. The proposed methodology allows for both input- and output-oriented analyses of static ecological networks. The input- and output-oriented analyses are introduced within the proposed mathematical framework and their duality is demonstrated. Major flow- and stock-related concepts and quantities of the current static network analyses are also integrated with the proposed measures and indices within this unifying framework. This comprehensive methodology enables a holistic view and analysis of ecological systems.

Appendices

Appendix: Appendices

Analytic solutions to linear systems, subsystem flows and storages, the static diact flows and storages, the output-oriented ecological network analysis, the duality of the input- and output-oriented analyses, and additional method formulations are presented in this section.

Appendix A: Analytical solution to linear systems

In this section, we formulate analytical solutions to linear ecological systems with time-dependent input as proposed by Coskun (2019a, c).

The system partitioning methodology yields a linear system, if the original system is linear. That is, if Eq. 1 is linear, Eq. 3 takes the following linear form:

$$ \begin{array}{llll} \dot{X}(t) & = \mathcal{Z}(t) + A(t) X(t) , \quad X(t_{0}) = \mathbf{0} , \\ \dot{{x}}_{0}(t) & = A(t) x_{0}(t) , \quad \quad \quad \quad {x}_{0} (t_{0}) = x_{0} . \end{array} $$

(78)

Let V (t) be the fundamental matrix solution to the system Eq. 78 as formulated by Coskun (2019c). That is, V (t) is the unique solution of the system

$$ \begin{array}{llll} {\dot {V}}(t) & = A(t) {V}(t) , \quad {V}(t_{0}) = I . \end{array} $$

The solutions to Eq. 78 for substorage matrix, X(t), and initial substorage vector, x₀(t), in terms of V (t) become

$$ \begin{array}{@{}rcl@{}} X(t) &=& {\int}_{t_{0}}^{t} {V}(t) {V}^{-1}(s) \mathcal{Z}(s) \text{ds} \quad \text{and}\\ x_{0}(t) &=& {V}(t) {x}_{0} . \end{array} $$

(79)

Therefore, the solution to the original system, Eq. 1, becomes

$$ \begin{array}{llll} x(t) & = {V}(t) {x}_{0} + {\int}_{t_{0}}^{t} {V}(t) {V}^{-1}(s) z(s) \text{ds} . \end{array} $$

For the particular case of constant diagonalizable flow intensity matrix A, we have

$$ \begin{array}{llll} {V} (t) & = {\exp} \left( {\int}_{t_{0}}^{t} A \text{ds} \right) = {\mathrm{e}}^{\left( t-t_{0} \right) A} = {\Omega} {\mathrm{e}}^{(t-t_{0}) {\Lambda} } {\Omega}^{-1} \end{array} $$

(80)

where Ω is the matrix whose columns are the eigenvectors of A, and Λ is the diagonal matrix whose diagonal elements are the eigenvalues of A. For this particular case, Eq. 79 takes the following form:

$$ \begin{array}{@{}rcl@{}} X(t) &=& {\int}_{t_{0}}^{t} {\mathrm{e}}^{\left( t-s \right) A} \mathcal{Z}(s) \text{ds} \quad\text{and}\\ x_{0}(t) &=& {\mathrm{e}}^{\left( t-t_{0} \right) A} {x}_{0} . \end{array} $$

(81)

Consequently,

$$ \begin{array}{llll} x(t) & = {\mathrm{e}}^{\left( t-t_{0} \right) A} {x}_{0} + {\int}_{t_{0}}^{t} {\mathrm{e}}^{\left( t-s \right) A} z(s) \text{ds} . \end{array} $$

A subsystem scaling argument is proposed to analyze system behavior per unit input in Section “Subsystem scaling.” The scaled substorage matrix, $S(t) = X(t) \mathcal {Z}^{-1}$, can be expressed for constant invertible input matrix, $\mathcal {Z}(t) = \mathcal {Z} > {0}$, as follows:

$$ \begin{array}{@{}rcl@{}} S(t) & = & {\int}_{t_{0}}^{t} {V}(t) {V}^{-1}(s) \text{ds} = {\int}_{t_{0}}^{t} {\mathrm{e}}^{\left( t-s \right) A} \text{ds}\\ & = & {\Omega} \left( {\int}_{t_{0}}^{t} {\mathrm{e}}^{\left( t-s \right) {\Lambda} } \text{ds} \right) {\Omega}^{-1} \end{array} $$

(82)

using Eq. 81.

The output-oriented counterpart of S, $\bar {S}$, will be discussed further below in Appendix F. An example of the analytic solution to a linear system, Eq. 78, with time-dependent environmental input is presented in the Supplementary Materials (Section S5.1).

Appendix B: Subsystem flows and storages

The system partitioning methodology decomposes a system into mutually exclusive and exhaustive subsystems through a set of governing equations derived from subcompartmentalization and flow partitioning. This methodology enables the analysis of each subsystem generated by an environmental input individually and separately. The subsystem flows and storages in matrix form are formulated in this section.

The k th direct subflow matrix for the k th subsystem, $F_{k}= \left (f_{i_{k} j_{k}} \right )$, can be expressed as

$$ \begin{array}{llll} F_{k} = F \mathcal{X}^{-1} \mathcal{X}_{k}, \quad k=0,\ldots,n, \end{array} $$

(83)

where the k th substorage matrix$\mathcal {X}_{k} = \text {diag} \left ( [ x_{1_{k}}, \ldots , x_{n_{k}} ] \right )$ is the diagonal matrix of the substorage values in the k th subsystem. The k th input and output matrix functions are

$$ \mathcal{Y}_{k} = \mathcal{Y} \mathcal{X}^{-1} \mathcal{X}_{k} \quad \text{and} \quad \mathcal{Z}_{k} = \text{diag} \left( z_{k} \boldsymbol{e}_{k} \right) , $$

respectively, where e_k is the elementary unit vector whose components are all zero except the k th component, which is 1, and e₀ = 0. The k th direct subflow matrix, F_k, and the k th output and input vectors, $\hat {y}_{k} = \mathcal {Y}_{k} \boldsymbol {1}$ and $\check {z}_{k} = \mathcal {Z}_{k} \boldsymbol {1}$, are the counterparts for subsystem k of the direct flow matrix, F, and output and input vectors, y and z, for the original system. These matrices and vectors represent the subflow regime of the k th subsystem.

The solutions to the governing system, Eq. 9, for the substorage matrix, X(t), and the initial substorage vector, x₀(t), at steady state are given in Eq. 13. Using the notations and definitions of Eqs. 15 and 83, the diagonal k th inward and outward subthroughflow matrices, $\check {\mathcal {T}}_{k} = \text {diag}{\left ([\check {\tau }_{1_{k}},\ldots ,\check {\tau }_{n_{k}}] \right )}$ and $\hat {\mathcal {T}}_{k} = \text {diag}{ \left ( [\hat {\tau }_{1_{k}},\ldots ,\hat {\tau }_{n_{k}}] \right ) }$, for the k th subsystem can be expressed as follows:

$$ \begin{array}{llll} \check{\mathcal{T}}_{k} & = \mathcal{Z}_{k} + \text{diag}{ \left( F \mathcal{X}^{-1} \mathcal{X}_{k} \boldsymbol{1} \right) },\\ \hat{\mathcal{T}}_{k} & = \left( \mathcal{Y} + \text{diag} { \left( F^{T} \mathbf{1} \right) } \right) \mathcal{X}^{-1} \mathcal{X}_{k} = \mathcal{T} \mathcal{X}^{-1} \mathcal{X}_{k} . \end{array} $$

(84)

Due to the steady-state conditions, the k th inward and outward subthoroughflow matrices are equal, that is, ${\mathcal {T}}_{k} = \check {\mathcal {T}}_{k} = \hat {\mathcal {T}}_{k}$. The matrix ${\mathcal {T}}_{k}$ will be called the k th subthroughflow matrix. The steady-state conditions and Eq. 84 imply that

$$ \begin{array}{llll} \check{\mathcal{T}}_{\ell} \check{\mathcal{T}}_{k}^{-1} = \hat{\mathcal{T}}_{\ell} \hat{\mathcal{T}}_{k}^{-1} \quad \text{and} \quad \mathcal{T}^{-1} \hat{\mathcal{T}}_{k} = \mathcal{X}^{-1} \mathcal{X}_{k} \end{array} $$

(85)

for any subsystem k and ℓ, provided that ${\mathcal {T}}_{k}$ is invertible (Coskun 2019c). If some diagonal entries of ${\mathcal {T}}_{k}$ are zero, these relationships are still valid componentwise for its nonzero diagonal elements, as given in Eq. 103.

We define the decomposition and k th decomposition matrices, ${D} = (d_{i_{k}})$ and $\mathcal {D}_{k} = \text {diag}{([d_{1_{k}}, \ldots ,d_{n_{k}}])}$, as

$$ \begin{array}{@{}rcl@{}} {D} &=& \mathcal{X}^{-1} {X} = \mathcal{T}^{-1} {T}\quad\text{and}\\ \mathcal{D}_{k} &=& \mathcal{X}^{-1} \mathcal{X}_{k} = \mathcal{T}^{-1} {\mathcal{T}}_{k} . \end{array} $$

(86)

The second equalities in the definitions of D and $\mathcal {D}_{k}$ are due to Eqs. 15 and 84. Using Eq. 85, the k th direct subflow and substorage matrices, F_k and $\mathcal {X}_{k}$, can then be written in various forms as follows:

$$ \begin{array}{@{}rcl@{}} \mathcal{X}_{k} &=& \mathcal{R} \mathcal{T}_{k}\quad \text{and} \quad {F}_k = F \mathcal{D}_k = Q^\tau \mathcal{T}_k = Q^x \mathcal{X}_k, \end{array} $$

(87)

similar to the relationships formulated for the decomposed system in Eq. 34.

The decomposition and k th decomposition matrices, D and $\mathcal {D}_{k}$, decompose the compartmental throughflow matrix, $\mathcal {T}$, into the subthroughflow and the k th subthroughflow matrices, T and $\mathcal {T}_{k}$, as indicated in Eqs. 15 and 84. That is,

$$ \begin{array}{llll} T = \mathcal{T} D \quad \text{and} \quad \mathcal{T}_{k} = \mathcal{T} \mathcal{D}_{k} . \end{array} $$

(88)

The decomposition matrices can be considered as linear transformations as detailed in the next section.

The diagonal k th cumulative flow and storage distribution matrices can then be expressed as follows:

$$ \begin{array}{llll} \mathcal{N}_{k} = \mathcal{T}_{k} / z_{k} \text{and} \mathcal{S}_{k} = \mathcal{X}_{k} / z_{k} \text{with} \mathcal{S}_{k} = \mathcal{R} \mathcal{N}_{k} . \end{array} $$

(89)

The output-oriented subsystem flows and storages can be formulated using the substitutions defined in Eq. 151 for the corresponding input-oriented counterparts.

Appendix C: Matrix measures as linear transformations

The matrix measures introduced in the present manuscript can be considered as linear transformations. For example, Eqs. 19 and 20 indicate that the substorage and subthroughflow matrices, S and N, are linear transformations that map environmental input vector to storage and throughflow vectors. That is,

$$ \begin{array}{llll} S : z \longrightarrow {x} \quad & \text{and} \quad N : z \longrightarrow {\tau} . \end{array} $$

(90)

Similarly, we have

$$ \begin{array}{llll} S : \mathcal{Z} \longrightarrow {X} \quad & \text{and} \quad N : \mathcal{Z} \longrightarrow {T} . \end{array} $$

(91)

The treatment of cumulative distribution matrices as linear transformations can be extended to the other decomposition and distribution matrices introduced in the present manuscript.

The decomposition matrices $D = \mathcal {X}^{-1} X$ and $\mathcal {D}_{k} = \mathcal {X}^{-1} \mathcal {X}_{k}$ can also be expressed as linear transformations that map the system flows and throughflows to the subflows and subthroughflows, as indicated in Eqs. 15, 84, and 87. That is,

$$ \begin{array}{llll} \mathcal{D}_{k} : {F} \longrightarrow F_{k} , \quad {D} : {F} \longrightarrow \tilde{T} \quad \text{and}\\\mathcal{D}_{k} : \mathcal{T} \longrightarrow \mathcal{T}_{k}, \quad {D} : \mathcal{T} \longrightarrow {T} . \end{array} $$

(92)

Note that the decomposition matrices act by right multiplication on the system flow matrices.

The distribution matrices are also linear transformations that map transient storages and throughflows at step (m), X^(m) and T^(m), to transient storages or throughflows at the next step along all possible flow paths as formulated in Eqs. 33. That is,

$$ \begin{array}{@{}rcl@{}} \mathcal{R} &:& T^{(m)} \longrightarrow {X}^{(m)},\\ Q^{x} &:& {X}^{(m)} \longrightarrow {T}^{(m+1)},\\ L=\mathcal{R} Q^{x} &:& {X}^{(m)} \longrightarrow {X}^{(m+1)},\\ Q^{\tau} = Q^{x} \mathcal{R} &:& T^{(m)} \longrightarrow {T}^{(m+1)}. \end{array} $$

(93)

The direct storage distribution matrix S^d can also be considered as a linear transformation from T^(m) to X^(m+ 1):

$$ \begin{array}{llll} S^{\texttt{d}} : T^{(m)} \longrightarrow {X}^{(m+1)} . \end{array} $$

(94)

Moreover, the relationship $S^{\texttt {d}} = \mathcal {R} Q^{\tau } = L \mathcal {R}$ indicates that L and Q^τ are similar matrices with the same set of eigenvalues. If the distribution matrices are invertible, backtracking from the (m + 1)st to the (m)th step is possible using the relationships formulated in Eqs. 33. Note that we also have

$$ \begin{array}{llll} \mathcal{R} : T \longrightarrow {X} \quad & \text{and} \quad \mathcal{R} : \tau \longrightarrow {x} \end{array} $$

(95)

due to the relationships $X = \mathcal {R} T$ and $x = \mathcal {R} \tau$ as demonstrated by Coskun (2019d).

The flow- and storage-based diact distribution matrices in both input- and output-orientations respectively map compartmental throughflows and storages to the corresponding diact flows and storages, as listed in Table 1. Therefore, the diact distribution matrices can be expressed as linear transformations. For input-oriented analysis,

$$ \begin{array}{llll} N^{\texttt{*}} : \mathcal{T} \longrightarrow T^{\texttt{*}} \quad & \text{and} \quad S^{\texttt{*}} : \mathcal{T} \longrightarrow X^{\texttt{*}} \end{array} $$

(96)

similar to Eq. 90, where the superscript (^⋆) in each relationship stands for any of the diact symbols. The storage-based, diact flow and storage distribution matrices can also be interpreted as linear transformations that map compartmental storages to diact flows and storages, as listed in Table 2. That is,

$$ \begin{array}{llll} N^{\texttt{*},x} : \mathcal{X} \longrightarrow X^{\texttt{*}} \quad & \text{and} \quad S^{\texttt{*},x} : \mathcal{X} \longrightarrow T^{\texttt{*}} . \end{array} $$

(97)

Appendix D: Geometric series expansion of matrix measures

The invertibility of the substorage matrix X is proved in the most general nonlinear dynamic setting by Coskun (2019c). This implies that, S and N are also invertible due to Eqs. 23 and 26. Therefore, due to Eq. 22, N can be expressed as the sum of the following infinite geometric series:

$$ \begin{array}{@{}rcl@{}} N & = & (I - Q^{\tau})^{-1}\\ & = & I + Q^{\tau} + {(Q^{\tau})}^{2} + {(Q^{\tau})}^{3} + {\cdots} + {(Q^{\tau})}^{m} + \cdots \end{array} $$

(98)

The term (Q^τ)^m represents the regular matrix multiplication of Q^τm times by itself and, therefore, is the flow distribution matrix for the m^th step, and (Q^τ)⁰ = I.

Starting with environmental input, the flows generated at each step add up to the subthroughflow matrix, T, as formulated in Eq. 21. That is,

$$ \begin{array}{llll} T = N \mathcal{Z} = (I - Q^{\tau})^{-1} \mathcal{Z} = \left( \sum\limits_{m=0}^{\infty} (Q^{\tau})^{m} \right) \mathcal{Z} . \end{array} $$

(99)

The matrix S as formulated in Eq. 24 can also be expressed as an infinite series. The substorage matrix X can then be written as follows:

$$ \begin{array}{@{}rcl@{}} X & = & S \mathcal{Z} = \mathcal{R} (I - Q^{\tau})^{-1} \mathcal{Z}\\ & = & (I - L)^{-1} \mathcal{R} \mathcal{Z} = \left( \sum\limits_{m=0}^{\infty} L^{m} \right) \mathcal{R} \mathcal{Z} . \end{array} $$

(100)

Truncating the infinite geometric series in Eqs. 99 and 100 at m = M yields the cumulative transient subthroughflow and substorage matrices at step M, $T^{[M]} = \left ( {\sum }_{m=0}^{M} (Q^{\tau })^{m} \right ) \mathcal {Z}$ and $X^{[M]} = \left ( {\sum }_{m=0}^{M} L^{m} \right ) \mathcal {R} \mathcal {Z}$, respectively. Therefore,

$$ \begin{array}{llll} \lim\limits_{M \to \infty} X^{[M]} = \lim\limits_{M \to \infty} \left( \sum\limits_{m=0}^{M} L^{m} \right) \mathcal{R} \mathcal{Z} = X . \end{array} $$

(101)

Similarly, $\lim _{M \to \infty } T^{[M]} = T$.

Appendix E: The diact flows and storages

The input-oriented, simple and composite diact flows are defined, and the derivations for indirect flows and storages are presented in detail in Section “The diact flows and storages.” Parallel derivations for other diact flows and the associated storages generated by these flows are formulated in this section. The alternative path-based formulation of diact flows and storages is given in the Supplementary Materials (Section S3.1). The output-oriented, flow- and storage-based, simple and composite diact flows and storages are formulated further below and in the Supplementary Materials (Section S4.3).

The simplediactflows and storages are defined in Section “The diact flows and storages.” They are explicitly formulated in Section “The diact flows and storages” and further below. The simple diact flows can be listed as follows:

$$ \begin{array}{@{}rcl@{}} \tau^{\texttt{t}}_{i_k} = \tau_{i_k}-z_{i_k} = \tilde{\tau}_{i_k}, \quad \tau^{\texttt{d}}_{i_k} \!&=&\! f_{i_k k_k}, \quad \tau^{\texttt{i}}_{i_k} = \tau^{\texttt{t}}_{i_k} - \tau^{\texttt{d}}_{i_k},\\ \tau^{\texttt{c}}_{i_k} \!&=&\! \frac{\tau_{i_i}-z_{i_i}}{\tau_{i_i}} \tau_{i_k}, \quad \tau^{\texttt{a}}_{i_k} = \tau^{\texttt{t}}_{i_k} - \tau^{\texttt{c}}_{i_k}. \end{array} $$

(102)

Note that, the (i,k)—element of the simple transfer flow matrix is the intercompartmental subthroughflow at compartment i generated by the environmental input into k, z_k. That is, in matrix form, $\tilde {T}^{\texttt {t}} = \tilde {T}$. Let the superscript (^⋆) stand for any of the diact symbols. The (i,k)—elements of the other simple diac flow matrices, $\tilde {T}^{\texttt {*}} = ({\tau }^{\texttt {*}}_{i_{k}})$, represent the direct, indirect, non-cycling, and cycling segments of this simple transfer flow at the terminal compartment i. The associated storages generated by these simple diact flows are then represented by the (i,k)—elements of the corresponding simple diact storage matrices, $\tilde {X}^{\texttt {*}} = (x^{\texttt {*}}_{i_{k}})$.

The steady-state condition and the equivalence of the outward throughflow and subthroughflow intensities in the same direction imply the following proportionalities as formulated in Eq. 29, as well as, in matrix form, in Eq. 85:

$$ \frac{\hat{\tau}_{k_{\ell}} }{\hat{\tau}_{k_{k}} } = \frac{\check{\tau}_{k_{\ell}} }{\check{\tau}_{k_{k}} } \quad \text{and} \quad \frac{ {x}_{i_{\ell}} } { {\tau}_{i_{\ell}} } = \frac{ {x}_{i} } { {\tau}_{i} } . $$

(103)

Equation 103 implies that composite diact subflow ${\tau }^{\texttt {*}}_{i_{\ell } k_{\ell }}$ in subsystem ℓ parallel to simple diact subflow ${\tau }^{\texttt {*}}_{i_{k} k_{k}}$, from an input-receiving subcompartment k_k to i_k in subsystem k, and the associated substorage generated by diact subflow ${\tau }^{\texttt {*}}_{i_{\ell } k_{\ell }}$, ${x}^{\texttt {*}}_{i_{\ell } k_{\ell }}$, can be formulated as

$$ {\tau}^{\texttt{*}}_{i_{\ell} k_{\ell}} = {\tau}^{\texttt{*}}_{i_{k} k_{k}} \frac{ {\tau}_{k_{\ell}} } { {\tau}_{k_{k}} } \quad \text{and} \quad {x}^{\texttt{*}}_{i_{\ell} k_{\ell}} = {\tau}^{\texttt{*}}_{i_{\ell} k_{\ell}} \frac {{x}_{i} } { {\tau}_{i} } $$

(104)

for i,k = 1,…,n. By parallel subflows, we mean the flows that transit through different subcompartments of the same compartment at the same time. The diact flow and storage from compartment k to i then become

$$ \begin{array}{llll} {\tau}^{\texttt{*}}_{ik} = \sum\limits_{\ell=1}^{n} {\tau}^{\texttt{*}}_{i_{\ell} k_{\ell}} = n^{\texttt{*}}_{ik} \sum\limits_{\ell=1}^{n} {\tau}_{k_{\ell}} = n^{\texttt{*}}_{ik} {\tau}_{k} \quad \text{and}\\ {x}^{\texttt{*}}_{ik} = r_{i} {\tau}^{\texttt{*}}_{ik} = s^{\texttt{*}}_{ik} {\tau}_{k} \end{array} $$

(105)

where the diactflow and storage distribution factors are

$$ n^{\texttt{*}}_{ik} = \frac{ {\tau}^{\texttt{*}}_{i_{k} k_{k}} }{ {\tau}_{k_{k}} } = \frac{ n^{\texttt{*}}_{i_{k}} }{ n_{k_{k}} } \quad \text{and} \quad s^{\texttt{*}}_{ik} = r_{i} n^{\texttt{*}}_{ik} . $$

(106)

The scaled and nondimensionalized flows are defined as $n^{\texttt {*}}_{i_{k}} = {\tau }^{\texttt {*}}_{i_{k} k_{k}} / z_{k}$ and $n_{k_{k}} = {\tau }_{k_{k}} / z_{k}$. In matrix form, they can be expressed as $\tilde {N}^{\texttt {*}} = (n^{\texttt {*}}_{i_{k}})$ and $\mathcal {N} = \text {diag}{(N)}$. The scaled form in the second equality for $n^{\texttt {*}}_{ik}$ can be used when there is a zero environmental input and, therefore, $\tau _{k_{k}} = 0$ for some k, as discussed in Section “Subsystem scaling.” The diactflow and storage distribution matrices, ${N}^{\texttt {*}} = (n^{\texttt {*}}_{ik})$ and ${S}^{\texttt {*}} = (s^{\texttt {*}}_{ik})$, can then be written, using Eq. 106, as follows:

$$ \begin{array}{llll} {N}^{\texttt{*}} = \tilde{T}^{\texttt{*}} \mathsf{T}^{-1} = \tilde{N}^{\texttt{*}} \mathcal{N}^{-1} \quad \text{and} \quad {S}^{\texttt{*}} = \mathcal{R} {N}^{\texttt{*}} . \end{array} $$

(107)

The simplediactflows and storages can be expressed in terms of the composite diact flows and storages as

$$ \begin{array}{llll} {\tau}^{\texttt{*}}_{i_{k}} = {\tau}^{\texttt{*}}_{i_{k} k_{k}} \quad \text{and} \quad x^{\texttt{*}}_{i_{k}} = x^{\texttt{*}}_{i_{k} k_{k}} . \end{array} $$

(108)

That is, the simple diact flows are the composite diact flows from the input-receiving subcompartments to other subcompartments in the same subsystem. The simple diact flow and storage matrices can then be formulated, in terms of the distribution matrices:

$$ \begin{array}{llll} \tilde{T}^{\texttt{*}} = {N}^{\texttt{*}} \mathsf{T} \quad \text{and} \quad \tilde{X}^{\texttt{*}} = {S}^{\texttt{*}} \mathsf{T} = \mathcal{R} \tilde{T}^{\texttt{*}} . \end{array} $$

(109)

The ℓ th simplediactflow and storage matrices, $\tilde {\mathcal {T}}^{\texttt {*}}_{\ell }$ and $\tilde {\mathcal {X}}^{\texttt {*}}_{\ell }$, for the ℓ th subsystem will also be defined as the diagonal matrices whose diagonal elements are the ℓ th column vector of $\tilde {T}^{\texttt {*}}$ and $\tilde {X}^{\texttt {*}}$, respectively. That is, $\tilde {\mathcal {T}}^{\texttt {*}}_{\ell } = \text {diag}{(\tilde {T}^{\texttt {*}} \boldsymbol {e}_{\ell })}$ and $\tilde {\mathcal {X}}^{\texttt {*}}_{\ell } = \text {diag}{(\tilde {X}^{\texttt {*}} \boldsymbol {e}_{\ell })}$.

The ℓ th compositediactflow and storage matrices, $\mathcal {T}^{\texttt {*}}_{\ell } = ({\tau }^{\texttt {*}}_{i_{\ell } k_{\ell }})$ and $\mathcal {X}^{\texttt {*}}_{\ell } = (x^{\texttt {*}}_{i_{\ell } k_{\ell }})$, will be formulated as

$$ \begin{array}{llll} T^{\texttt{*}}_{\ell} = {N}^{\texttt{*}} \mathcal{T}_{\ell} \quad \text{and} \quad X^{\texttt{*}}_{\ell} = {S}^{\texttt{*}} \mathcal{T}_{\ell} = \mathcal{R} T^{\texttt{*}}_{\ell} , \end{array} $$

(110)

using Eq. 104. Therefore, the compositediactflow and storage matrices, ${T}^{\texttt {*}} = ({\tau }^{\texttt {*}}_{i k})$ and ${X}^{\texttt {*}} = (x^{\texttt {*}}_{i k})$, become

$$ \begin{array}{@{}rcl@{}} T^{\texttt{*}} &=& {N}^{\texttt{*}} \mathcal{T} = \sum\limits_{\ell = 1}^{n} T^{\texttt{*}}_{\ell} \quad \text{and}\\ X^{\texttt{*}} &=& {S}^{\texttt{*}} \mathcal{T} = \mathcal{R} T^{\texttt{*}} = \sum\limits_{\ell = 1}^{n} X^{\texttt{*}}_{\ell}, \end{array} $$

(111)

because of Eq. 105. Note that, the (i,k)—element of the composite transfer flow matrix, ${\tau }^{\texttt {t}}_{ik}$, is the total flow from compartment k transferred directly or indirectly through other compartments into i. The (i,k)—elements of the other composite diac flow matrices represent the direct, indirect, non-cycling, and cycling segments of this composite transfer flow at the terminal compartment i. The associated storages generated by these composite diact flows are then represented by the (i,k)—elements of the corresponding composite diact storage matrices, $x^{\texttt {*}}_{ik}$.

In matrix form, the simple and composite diact flows can be represented at the subcompartmental and compartmental levels as follows:

$$ \begin{array}{@{}rcl@{}} \begin{array}{llll} \tilde{\mathcal{T}}^{\texttt{*}}_{\ell} = \left[\begin{array}{lllllll} {\tau}^{\texttt{*}}_{1_{\ell}} & 0 & {\cdots} & 0 \\ 0 & {\tau}^{\texttt{*}}_{2_{\ell}} & {\cdots} & 0 \\ {\vdots} & & {\ddots} & {\vdots} \\ 0 & 0 & {\cdots} & {\tau}^{\texttt{*}}_{n_{\ell}} \end{array}\right] ,\\ \tilde{T}^{\texttt{*}} = \left[\begin{array}{lllllll} {\tau}^{\texttt{*}}_{1_{1}} & {\cdots} & {\tau}^{\texttt{*}}_{1_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{1_{n}} \\ {\tau}^{\texttt{*}}_{2_{1}} & {\cdots} & {\tau}^{\texttt{*}}_{2_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{2_{n}}\\ {\vdots} & {\ddots} & {\vdots} & {\ddots} & {\vdots} \\ {\tau}^{\texttt{*}}_{n_{1}} & {\cdots} & {\tau}^{\texttt{*}}_{n_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{n_{n}} \end{array}\right] ,\\ {T}^{\texttt{*}}_{\ell} = \left[\begin{array}{lllllll} {\tau}^{\texttt{*}}_{1_{\ell} 1_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{1_{\ell} k_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{1_{\ell} n_{\ell}} \\ {\tau}^{\texttt{*}}_{2_{\ell} 1_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{2_{\ell} k_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{2_{\ell} n_{\ell}}\\ {\vdots} & {\ddots} & {\vdots} & {\ddots} & {\vdots} \\ {\tau}^{\texttt{*}}_{n_{\ell} 1_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{n_{\ell} k_{\ell}} & {\cdots} & {\tau}^{\texttt{*}}_{n_{\ell} n_{\ell}} \end{array}\right] & ,\\ T^{\texttt{*}} = \left[\begin{array}{lllllll} {\tau}^{\texttt{*}}_{1 1} & {\cdots} & {\tau}^{\texttt{*}}_{1 k} & {\cdots} & {\tau}^{\texttt{*}}_{1 n} \\ {\tau}^{\texttt{*}}_{2 1} & {\cdots} & {\tau}^{\texttt{*}}_{2 k} & {\cdots} & {\tau}^{\texttt{*}}_{2 n}\\ {\vdots} & {\ddots} & {\vdots} & {\ddots} & {\vdots} \\ {\tau}^{\texttt{*}}_{n 1} & {\cdots} & {\tau}^{\texttt{*}}_{n k} & {\cdots} & {\tau}^{\texttt{*}}_{n n} \end{array}\right] , \end{array} \end{array} $$

(112)

where ${\tau }^{\texttt {*}}_{i_{\ell }} = {\tau }^{\texttt {*}}_{i_{\ell } \ell _{\ell }}$ and ${\tau }^{\texttt {*}}_{ik} = {\sum }_{k=1}^{n} {\tau }^{\texttt {*}}_{i_{\ell } k_{\ell }}$, as defined in Eqs. 108 and 111. The diact storage matrices can be represented similar to these diact flow matrices.

The difference between the composite and simple diact flows, $\tau ^{\texttt {*}}_{ik}$ and $\tau ^{\texttt {*}}_{i_{k}}$, and associated storages, $x^{\texttt {*}}_{ik}$ and $x^{\texttt {*}}_{i_{k}}$, is that the composite flow and storage from compartment k to i are generated by outward throughflow τ_k derived from all environmental inputs, and their simple counterparts from input-receiving subcompartment k_k to i_k are generated by outward subthroughflow $\tau _{k_{k}}$ derived from single environmental input z_k (see Fig. 4). In that sense, the composite and simple diact flows and storages measure the influence of one compartment on another induced by all and single environmental inputs, respectively.

Since the simple diact flows and storages are constituents of the corresponding composite diact flows and storages, we have as follows:

$$ \begin{array}{llll} T^{\texttt{*}} \geq \tilde{T}^{\texttt{*}} \Rightarrow \acute{T}^{\texttt{*}} = T^{\texttt{*}} - \tilde{T}^{\texttt{*}} \geq \textbf{0} \quad \text{and}\\ X^{\texttt{*}} \geq \tilde{X}^{\texttt{*}} \Rightarrow \acute{X}^{\texttt{*}} = X^{\texttt{*}} - \tilde{X}^{\texttt{*}} \geq \textbf{0} . \end{array} $$

(113)

These inequalities are defined componentwise. The elements of the k th column of $\acute {T}^{\texttt {*}}$ and $\acute {X}^{\texttt {*}}$ are the diact flows and storages from compartment k to the others generated by all environmental inputs but z_k. The matrices $\acute {T}^{\texttt {*}}$ and $\acute {X}^{\texttt {*}}$ will accordingly be called complementarydiactflow and storage matrices, respectively.

The simplediactthroughflow and compartmental storage matrices, $\tilde {\mathcal {T}}^{\texttt {*}}$ and $\tilde {\mathcal {X}}^{\texttt {*}}$, and vectors, $\tilde {\tau }^{\texttt {*}}$ and $\tilde {x}^{\texttt {*}}$, can be defined as

$$ \begin{array}{@{}rcl@{}} \tilde{\mathcal{T}}^{\texttt{*}} = \operatorname{diag}{(\tilde{T}^{\texttt{*}} \mathbf{1})} & \Rightarrow & \tilde{\tau}^{\texttt{*}} = \tilde{\mathcal{T}}^{\texttt{*}} \mathbf{1} \quad \text{and}\\ \tilde{\mathcal{X}}^{\texttt{*}} = \operatorname{diag}{(\tilde{X}^{\texttt{*}} \mathbf{1})} & \Rightarrow & \tilde{x}^{\texttt{*}} = \tilde{\mathcal{X}}^{\texttt{*}} \mathbf{1}. \end{array} $$

(114)

Similar relationships can also be formulated for compositediactthroughflow and compartmental storage matrices, $\mathcal {T}^{\texttt {*}}$ and $\mathcal {X}^{\texttt {*}}$, and vectors, τ^⋆ and x^⋆.

The relationship between the diact distribution matrices and the corresponding flow and storage matrices can be summarized as follows: by left multiplying the (sub)throughflow matrix, the diact flow and storage distribution matrices yield the corresponding diact (sub)flow and associated (sub)storage matrices. That is,

$$ \begin{array}{llll} {T}^{\texttt{*}} = {N}^{\texttt{*}} \mathcal{T} , \quad {X}^{\texttt{*}} = {S}^{\texttt{*}} \mathcal{T} \quad \text{and}\\ {T}^{\texttt{*}}_{\ell} = {N}^{\texttt{*}} \mathcal{T}_{\ell} , \quad {X}^{\texttt{*}}_{\ell} = {S}^{\texttt{*}} \mathcal{T}_{\ell} . \end{array} $$

(115)

Due to their construction, the diact distribution matrices and the corresponding flow and storage matrices are related as follows:

$$ {S}^{\texttt{*}} = \mathcal{R} {N}^{\texttt{*}} \Rightarrow {X}^{\texttt{*}} = \mathcal{R} {T}^{\texttt{*}} \quad \text{and} \quad {X}^{\texttt{*}}_{\ell} = \mathcal{R} {T}^{\texttt{*}}_{\ell} . $$

(116)

similar to Eq. 26. The simple counterparts of these relationships for the composite diact flow and storage distribution matrices can also be formulated, as listed in Tables 1 and 2. Note that, for the flow-based diact distribution matrices, the unit of S^⋆ is time [t], as the cumulative storage distribution matrix, S, and N^⋆ is dimensionless, as the cumulative throughflow distribution matrix, N.

It is worth emphasizing also that the methodology outlined above for derivations of diact flows and storages can be used to define new transaction types. New simple flow and storage matrices, the corresponding scaled and nondimensionalized flow and storage distribution matrices, as well as the composite flow and storage matrices can be formulated similar to the derivations of the diact flows and storages.

The input-oriented, flow- and storage-based, simple and composite diact flow and storage distribution matrices as well as the corresponding flow and associated storage matrices are listed in Tables 1 and 2 at both compartmental and subcompartmental levels. The storage-based diact flows and storages can also be formulated using Eq. 147. Additional formulations for each diact transaction type are separately presented below.

E.1 Direct flow and storage

The composite direct flow is defined as $\tau ^{\texttt {d}}_{ij} = f_{ij}$ and, therefore, T^d = F. The proportionality given in Eq. 104 implies that

$$ \begin{array}{@{}rcl@{}} {x}^{\texttt{d}}_{i_{k} j_{k}} &=& {\tau}^{\texttt{d}}_{i_{k} j_{k}} \frac {{x}_{i} } { {\tau}_{i} } = {f}_{i_{k} j_{k}} \frac {{x}_{i} } { {\tau}_{i} }\quad \text{and}\\ {x}^{\texttt{d}}_{ij} &=& {\sum}_{k=1}^{n} {x}^{\texttt{d}}_{i_{k} j_{k}} = {f}_{ij} \frac {{x}_{i} } { {\tau}_{i} } . \end{array} $$

(117)

Defining the direct flow and storage distribution matrices as

$$ \begin{array}{@{}rcl@{}} N^{\texttt{d}} &=& F \mathcal{T}^{-1} = Q^{\tau}\quad\text{and}\\ S^{\texttt{d}} &=& \mathcal{R} N^{\texttt{d}} = \mathcal{R} Q^{\tau} = L \mathcal{R}, \end{array} $$

(118)

the ℓ th composite direct subflow and associated substorage matrices for the ℓ th subsystem, ${T}_{\ell }^{\texttt {d}}$ and ${X}_{\ell }^{\texttt {d}}$, can be expressed in matrix form as follows:

$$ \begin{array}{@{}rcl@{}} {T}_{\ell}^{\texttt{d}} &=& N^{\texttt{d}} \mathcal{T}_{\ell} = F \mathcal{T}^{-1} \mathcal{T}_{\ell} = F_{\ell}\quad\text{and}\\ {X}_{\ell}^{\texttt{d}} &=& S^{\texttt{d}} \mathcal{T}_{\ell} = \mathcal{R} {T}_{\ell}^{\texttt{d}} . \end{array} $$

(119)

The composite direct flow and storage matrices, T^d and X^d, then become

$$ \begin{array}{llll} {T}^{\texttt{d}} = N^{\texttt{d}} \mathcal{T} = F = {\sum}_{\ell=1}^{n} {T}_{\ell}^{\texttt{d}} \quad \text{and}\\ {X}^{\texttt{d}} = S^{\texttt{d}} \mathcal{T} = \mathcal{R} {T}^{\texttt{d}} = \mathcal{R} F = {\sum}_{\ell=1}^{n} {X}_{\ell}^{\texttt{d}} . \end{array} $$

(120)

The composite direct throughflow and compartmental storage matrices and vectors can also be formulated as

$$ \begin{array}{@{}rcl@{}} \mathcal{T}^{\texttt{d}} &=& \text{diag}{({T}^{\texttt{d}} \boldsymbol{1} )} = \mathcal{T} - \mathcal{Z} \!\Rightarrow\! \tau^{\texttt{d}} = \mathcal{T}^{\texttt{d}} \boldsymbol{1} = \tau - z \quad \text{and}\\ \mathcal{X}^{\texttt{d}} &=& \text{diag}{({X}^{\texttt{d}} \boldsymbol{1} )} \Rightarrow x^{\texttt{d}} = \mathcal{X}^{\texttt{d}} \boldsymbol{1} . \end{array} $$

(121)

E.2 Indirect flow and storage

The simple and composite indirect flows and storages are introduced in Section “The diact flows and storages.” The indirect flow and storage distribution matrices are

$$ \begin{array}{llll} N^{\texttt{i}} &=& \left( T - \mathcal{Z} - \tilde{T}^{\texttt{d}} \right) \mathsf{T}^{-1} \\&=& \left( N - I \right) \mathcal{N}^{-1} - F \mathcal{T}^{-1} \quad \text{and} \quad S^{\texttt{i}} = \mathcal{R} N^{\texttt{i}} . \end{array} $$

In component form, the indirect flow and storage distribution factors, can be written as

$$ n^{\texttt{i}}_{ik}\! =\! \frac{\tau_{i_{k}}\! -\! z_{i_{k}} - f_{i_{k} k_{k}}}{{\tau}_{k_{k}}} \! =\! \frac{ n_{i_{k}}\! -\! \delta_{ik} } {n_{k_{k}} } - \frac{ f_{ik} } {\tau_{k} } \quad \text{and} \quad s^{\texttt{i}}_{ik}\! =\! r_{i} n^{\texttt{i}}_{ik} . $$

The ℓ th composite indirect subflow and associated substorage matrices, ${T}^{\texttt {i}}_{\ell }$ and ${X}^{\texttt {i}}_{\ell }$, are formulated in Eq. 42 as follows:

$$ \begin{array}{llll} {T}_{\ell}^{\texttt{i}} &= {N}^{\texttt{i}} \mathcal{T}_{\ell} \quad \text{and} \quad {X}_{\ell}^{\texttt{i}} = S^{\texttt{i}} \mathcal{T}_{\ell} = \mathcal{R} {T}_{\ell}^{\texttt{i}} . \end{array} $$

The composite indirect flow and associated storage matrices given in Eq. 44 can then be expressed as

$$ \begin{array}{llll} {T}^{\texttt{i}} & = {N}^{\texttt{i}} \mathcal{T} = \sum\limits_{\ell=1}^{n} {T}_{\ell}^{\texttt{i}} \quad \text{and} \quad {X}^{\texttt{i}} = S^{\texttt{i}} \mathcal{T} = \mathcal{R} {T}^{\texttt{i}} = \sum\limits_{\ell=1}^{n} {X}_{\ell}^{\texttt{i}} . \end{array} $$

E.3 Acyclic flow and storage

The simple acyclic flow from an input-receiving subcompartment k_k to i_k in the k th subsystem can be formulated as

$$ \begin{array}{@{}rcl@{}} \tau^{\texttt{a}}_{i_{k}} & = & \tau^{\texttt{a}}_{i_{k} k_{k}} = {\tau}^{\texttt{t}}_{i_{k} k_{k}} - {\tau}^{\texttt{c}}_{i_{k}}\\ & = & (\tau_{i_{k}} - z_{i_{k}} ) - \left( \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \right) \tau_{i_{k}} . \end{array} $$

(122)

The composite subflow parallel to $\tau ^{\texttt {a}}_{i_{k} k_{k}}$ in subsystem ℓ then becomes

$$ {\tau}^{\texttt{a}}_{i_{\ell} k_{\ell}} = {\tau}^{\texttt{a}}_{i_{k} k_{k}} \frac{{\tau}_{k_{\ell}} }{{\tau}_{k_{k}} } = \left[ \frac{\tau_{i_{k}} - z_{i_{k}}} { {\tau}_{k_{k}} } - \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \frac{\tau_{i_{k}}} {{\tau}_{k_{k}}} \right] {\tau}_{k_{\ell}} $$

(123)

and, therefore, the acyclic flow and storage distribution factors are

$$ \begin{array}{@{}rcl@{}} n^{\texttt{a}}_{ik} & = & \frac{\tau_{i_{k}} - z_{i_{k}}} { {\tau}_{k_{k}} } - \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \frac{\tau_{i_{k}}} {{\tau}_{k_{k}}} = \frac{1}{n_{i_{i}}} n_{i_{k}} \frac{1}{n_{k_{k}}} - \frac{ \delta_{ik}}{n_{k_{k}}} \quad \text{and}\\ s^{\texttt{a}}_{ik} & = & r_{i} n^{\texttt{a}}_{ik} . \end{array} $$

(124)

The acyclic flow and storage distribution matrices can be formulated accordingly:

$$ \begin{array}{llll} N^{\texttt{a}} = \left( \mathcal{N}^{-1} N - I \right) \mathcal{N}^{-1} \quad \text{and} \quad {S}^{\texttt{a}} = \mathcal{R} {N}^{\texttt{a}} . \end{array} $$

(125)

The ℓ th composite acyclic subflow and associated substorage matrices can be written in matrix form as follows:

$$ \begin{array}{llll} {T}_{\ell}^{\texttt{a}} = {N}^{\texttt{a}} \mathcal{T}_{\ell} = \left( \mathcal{N}^{-1} N - I \right) \mathcal{N}^{-1} \mathcal{T}_{\ell} \quad \text{ and }\\ {X}_{\ell}^{\texttt{a}} = {S}^{\texttt{a}} \mathcal{T}_{\ell} = \mathcal{R} {T}_{\ell}^{\texttt{a}} . \end{array} $$

(126)

The composite acyclic flow and associated storage matrices become

$$ \begin{array}{@{}rcl@{}} {T}^{\texttt{a}} &=& {N}^{\texttt{a}} \mathcal{T} = \sum\limits_{\ell=1}^{n} {T}_{\ell}^{\texttt{a}}\quad\text{and}\\ {X}^{\texttt{a}} &=& S^{\texttt{a}} \mathcal{T} = \mathcal{R} {T}^{\texttt{a}} = \sum\limits_{\ell=1}^{n} {X}_{\ell}^{\texttt{a}} . \end{array} $$

(127)

E.4 Cycling flow and storage

The simple cycling subflow from an input-receiving subcompartment k_k to i_k in the k th subsystem can be formulated, using Eq. 135, as

$$ \tau^{\texttt{c}}_{i_{k}} = \tau^{\texttt{c}}_{i_{k} k_{k}} = \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \tau_{i_{k}} . $$

(128)

The composite subflow parallel to $\tau ^{\texttt {c}}_{i_{k} k_{k}}$ in subsystem ℓ then becomes

$$ {\tau}^{\texttt{c}}_{i_{\ell} k_{\ell}} = {\tau}^{\texttt{c}}_{i_{k} k_{k}} \frac{{\tau}_{k_{\ell}} }{{\tau}_{k_{k}} } = \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \frac{\tau_{i_{k}}} {{\tau}_{k_{k}}} {\tau}_{k_{\ell}} $$

(129)

and, therefore, the cycling flow and storage distribution factors are

$$ \begin{array}{@{}rcl@{}} n^{\texttt{c}}_{ik} &=& \frac{\tau_{i_{i}} - z_{i_{i}}}{ \tau_{i_{i}}} \frac{\tau_{i_{k}}} {{\tau}_{k_{k}}} = \frac{ n_{i_{k}} }{n_{k_{k}}} - \frac{1}{n_{i_{i}}} n_{i_{k}} \frac{1}{n_{k_{k}}}\quad\text{and}\\ s^{\texttt{c}}_{ik} &=& r_{i} n^{\texttt{c}}_{ik} . \end{array} $$

(130)

The cycling flow and storage distribution matrices can be formulated accordingly:

$$ \begin{array}{llll} {N}^{\texttt{c}} = (N - \mathcal{N}^{-1} N) \mathcal{N}^{-1} \quad \text{and} \quad {S}^{\texttt{c}} = \mathcal{R} {N}^{\texttt{c}} . \end{array} $$

(131)

The ℓ th composite cycling subflow and substorage matrices can be expressed in matrix form as

$$ \begin{array}{llll} {T}_{\ell}^{\texttt{c}} = {N}^{\texttt{c}} \mathcal{T}_{\ell} = (N - \mathcal{N}^{-1} N) \mathcal{N}^{-1} \mathcal{T}_{\ell} \quad \text{and}\\ {X}_{\ell}^{\texttt{c}} = {S}^{\texttt{c}} \mathcal{T}_{\ell} = \mathcal{R} {T}_{\ell}^{\texttt{c}} . \end{array} $$

(132)

The composite cycling flow and storage matrices then become

$$ \begin{array}{@{}rcl@{}} {T}^{\texttt{c}} &=& {N}^{\texttt{c}} \mathcal{T} = \sum\limits_{\ell=1}^{n} {T}_{\ell}^{\texttt{c}}\quad\text{and}\\ {X}^{\texttt{c}} &=& {S}^{\texttt{c}} \mathcal{T} = \mathcal{R} T^{\texttt{c}} = \sum\limits_{\ell=1}^{n} {X}_{\ell}^{\texttt{c}} . \end{array} $$

(133)

The simple cycling subflow from an input-receiving subcompartment i_i to itself can, alternatively, be formulated as follows:

$$ \begin{array}{llll} {\tau}^{\texttt{c}}_{i_{i}} = \tau^{\texttt{i}}_{i_{i} i_{i}} = \tau^{\texttt{t}}_{i_{i} i_{i}} = {\sum}_{{j=1}}^{n} f_{i_{i} j_{i}} = \check{\tau}_{i_{i}} - z_{i} . \end{array} $$

(134)

The matrix form of this relationship is given in Eq. 66. Note that this relationship is equivalent to Eq. 128 for k = i. The simple cycling flow at subcompartment i_k, ${\tau }^{\texttt {c}}_{i_{k}}$, parallel to ${\tau }^{\texttt {c}}_{i_{i}}$ in the k^th subsystem then becomes

$$ {\tau}^{\texttt{c}}_{i_{k}} = \left( \check{\tau}_{i_{i}} - z_{i} \right) \frac{\hat{\tau}_{i_{k}} }{\hat{\tau}_{i_{i}} } . $$

(135)

Consequently, the simple cycling flow at compartment i becomes

$$ \begin{array}{@{}rcl@{}} {\tau}^{\texttt{c}}_{i} & = & \sum\limits_{k=1}^{n} {\tau}^{\texttt{c}}_{i_{k}} = \sum\limits_{k=1}^{n} \left( \tau_{i_{i}} - z_{i} \right) \frac{{\tau}_{i_{k}} }{{\tau}_{i_{i}} }\\ & = & \frac{ \tau_{i_{i}} - z_{i} } {{\tau}_{i_{i}} } \sum\limits_{k=1}^{n} {\tau}_{i_{k}} = \frac{ \tau_{i_{i}} - z_{i} } {{\tau}_{i_{i}} } {\tau}_{i} = n^{\texttt{c}}_{i} {\tau}_{i} \end{array} $$

(136)

where the simple cycling flow and storage distribution factors are

$$ n^{\texttt{c}}_{i} = \frac{ \tau_{i_{i}} - z_{i} } {{\tau}_{i_{i}} } = \frac{ n_{i_{i}} - 1 } {n_{i_{i}} } \quad \text{and} \quad s_{i}^{\texttt{c}} = r_{i} n_{i}^{\texttt{c}}, $$

(137)

where ${\tau }_{i_{i}} \neq 0$. The reflexivity of the simple cycling subflow formulated in Eq. 134 is also manifested in the distribution factors: $n^{\texttt {c}}_{i} = n^{\texttt {i}}_{ii} = n^{\texttt {t}}_{ii}$, as also discussed in Eq. 65 (see Fig. 10).

The new diagonal simple cycling flow and storage distribution matrices, $\mathcal {N}^{\texttt {c}} = \text {diag}{([ n^{\texttt {c}}_{1},\ldots ,n^{\texttt {c}}_{n}])}$ and $\mathcal {S}^{\texttt {c}} = \text {diag}{([s^{\texttt {c}}_{1},\ldots ,s^{\texttt {c}}_{n}])}$, then become

$$ \begin{array}{@{}rcl@{}} \mathcal{N}^{\texttt{c}} &=& \left( \mathsf{T} - \mathcal{Z} \right) \mathsf{T}^{-1} = \left( \mathcal{N} - I \right) \mathcal{N}^{-1}\quad\text{and}\\ \mathcal{S}^{\texttt{c}} &=& \mathcal{R} \mathcal{N}^{\texttt{c}} . \end{array} $$

(138)

The ℓ^th diagonal simple cycling subflow and associated substorage matrices, $\tilde {\mathcal {T}}_{\ell }^{\texttt {c}} = \text {diag}{([\tau ^{\texttt {c}}_{1_{\ell }},\cdots ,\tau ^{\texttt {c}}_{n_{\ell }}])}$ and $\tilde {\mathcal {X}}_{\ell }^{\texttt {c}} = \text {diag}{([x^{\texttt {c}}_{1_{\ell }},\cdots ,x^{\texttt {c}}_{n_{\ell }}])}$, can be expressed in the following matrix form:

$$ \begin{array}{@{}rcl@{}} \tilde{\mathcal{T}}_{\ell}^{\texttt{c}} &=& \mathcal{N}^{\texttt{c}} \mathcal{T}_{\ell}\quad\text{and}\\ \tilde{\mathcal{X}}_{\ell}^{\texttt{c}} &=& \mathcal{S}^{\texttt{c}} \mathcal{T}_{\ell} = \mathcal{N}^{\texttt{c}} \mathcal{X}_{\ell} = \mathcal{R} \tilde{\mathcal{T}}_{\ell}^{\texttt{c}} . \end{array} $$

(139)

The simple cycling subflow and associated storage matrices can then be formulated as

$$ \begin{array}{llll} \tilde{T}^{\texttt{c}} = \mathcal{N}^{\texttt{c}} T \quad \text{and} \quad \tilde{X}^{\texttt{c}} = \mathcal{S}^{\texttt{c}} T = \mathcal{N}^{\texttt{c}} X . \end{array} $$

(140)

Note that the two equivalent versions of the single cycling flow and storage distribution matrices are related as follows:

$$ \begin{array}{llll} \tilde{T}^{\texttt{c}} = \mathcal{N}^{\texttt{c}} T = {N}^{\texttt{c}} \mathsf{T} \quad \text{and} \quad \tilde{X}^{\texttt{c}} =\mathcal{S}^{\texttt{c}} T = {S}^{\texttt{c}} \mathsf{T} . \end{array} $$

(141)

Moreover,

$$ \begin{array}{@{}rcl@{}} \tilde{\mathcal{T}}^{\texttt{c}}_{\ell} & = & \operatorname{diag}{(\tilde{T}^{\texttt{c}} \mathbf{e}_{\ell})} = \operatorname{diag}{( {T}^{\texttt{c}}_{\ell} )} \quad\text{and}\quad \\ \tilde{\mathcal{X}}^{\texttt{c}}_{\ell} & = & \operatorname{diag}{(\tilde{X}^{\texttt{c}} \mathbf{e}_{\ell})} = \operatorname{diag}{( {X}^{\texttt{c}}_{\ell} )} . \end{array} $$

E.5 Transfer flow and storage

The simple transfer subflow from an input-receiving subcompartment k_k to i_k can be expressed as follows:

$$ \tau^{\texttt{t}}_{i_{k}} = \tau^{\texttt{t}}_{i_{k} k_{k}} = \sum\limits_{{j=1}}^{n} f_{i_{k} j_{k}} = \check{\tau}_{i_{k}} - z_{i_{k}} . $$

(142)

Therefore, the transfer flow and storage distribution factors become

$$ \begin{array}{@{}rcl@{}} n^{\texttt{t}}_{ik} = \frac{{\tau}_{i_{k}} - z_{i_{k}} }{\tau_{k_{k}}} = \frac{ n_{i_{k}} - \delta_{ik} }{n_{k_{k}}} \quad \text{and} \quad s^\texttt{t}_{ik} = r_{i} n^\texttt{t}_{ik}. \end{array} $$

(143)

Note that, although the derivation rationales, notations, and terminologies are different, the alternative simple cycling and transfer flow distribution factors, $n^{\texttt {c}}_{i}$ and $n^{\texttt {t}}_{ik}$, formulated in Eqs. 137 and 143, in terms of the elements of N, are equivalent to those defined by Finn (1980) and Szyrmer and Ulanowicz (1987), respectively.

The transfer flow and storage distribution matrices, $N^{\texttt {t}} = \left ( n^{\texttt {t}}_{ik}\right )$ and $S^{\texttt {t}} = \left ( s^{\texttt {t}}_{ik}\right )$, can be expressed as

$$ \begin{array}{@{}rcl@{}} N^{\texttt{t}} &=& \left( N - I \right) \mathcal{N}^{-1}\quad\text{and}\\ S^{\texttt{t}} &=& \mathcal{R} N^{\texttt{t}} = \mathcal{R} \left( N - I \right) \mathcal{N}^{-1} . \end{array} $$

(144)

The ℓ th composite transfer subflow and substorage matrices${T}^{\texttt {t}}_{\ell }$ and ${X}^{\texttt {t}}_{\ell }$ will then be formulated as

$$ \begin{array}{llll} {T}_{\ell}^{\texttt{t}} = N^{\texttt{t}} \mathcal{T}_{\ell} \quad \text{and} \quad {X}_{\ell}^{\texttt{t}} = S^{\texttt{t}} \mathcal{T}_{\ell} = \mathcal{R} {T}_{\ell}^{\texttt{t}}. \end{array} $$

(145)

The composite transfer flow and associated storage matrix measures then become

$$ \begin{array}{@{}rcl@{}} {T}^{\texttt{t}} &=& N^{\texttt{t}} \mathcal{T} = \sum\limits_{\ell=1}^{n} {T}_{\ell}^{\texttt{t}}\quad\text{and}\\ {X}^{\texttt{t}} &=& S^{\texttt{t}} \mathcal{T} = \mathcal{R} {T}^{\texttt{t}} = \sum\limits_{\ell=1}^{n} {T}_{\ell}^{\texttt{t}} . \end{array} $$

(146)

The simple transfer flow and storage matrices, $\tilde {T}^{\texttt {t}}$ and $\tilde {X}^{\texttt {t}}$, are also defined as follows:

$$ \begin{array}{llll} \tilde{T}^{\texttt{t}} = {N}^{\texttt{t}} \mathsf{T} = \tilde{T} = T - \mathcal{Z} \quad \text{and}\\ \tilde{X}^{\texttt{t}} = {S}^{\texttt{t}} \mathsf{T} = \mathcal{R} \tilde{T}^{\texttt{t}} = \tilde{X} = X - \mathcal{R} \mathcal{Z} . \end{array} $$

E.6 Storage-based diact transactions

Using the relationships formulated in Eq. 26 and the fact that

$$ \begin{array}{llll} \mathcal{S} = \mathcal{R} \mathcal{N} \Rightarrow \mathcal{N}^{-1} = \mathcal{S}^{-1} \mathcal{R} \quad \text{and}\\\mathcal{X} = \mathcal{R} \mathcal{T} \Rightarrow \mathcal{T}^{-1} = \mathcal{X}^{-1} \mathcal{R} , \end{array} $$

(147)

the diact flows and storages formulated in Table 1 can be expressed in terms of storage distribution matrix, S. The composite acyclic flow and storage distribution matrices, for example, become

$$ \begin{array}{llll} {N}^{\texttt{a}} = (\mathcal{S}^{-1} S - I) \mathcal{S}^{-1} \mathcal{R} \quad \text{and}\\{S}^{\texttt{a}} = \mathcal{R} {N}^{\texttt{a}} = \mathcal{R} (\mathcal{S}^{-1} S - I) \mathcal{S}^{-1} \mathcal{R} . \end{array} $$

(148)

Therefore,

$$ \begin{array}{@{}rcl@{}} {T}^{\texttt{a}} &=& {N}^{\texttt{a}} \mathcal{T} = (\mathcal{S}^{-1} S - I) \mathcal{S}^{-1} \mathcal{R} \mathcal{T} = {S}^{\texttt{a},x} \mathcal{X} \quad \text{and}\\ {X}^{\texttt{a}} &=& {S}^{\texttt{a}} \mathcal{T} = {N}^{\texttt{a},x} \mathcal{X} , \end{array} $$

(149)

where the storage–based composite acyclic flow and storage distribution matrices can respectively be defined as follows:

$$ \begin{array}{llll} {S}^{\texttt{a},x} = (\mathcal{S}^{-1} S - I) \mathcal{S}^{-1} \quad \text{and} \quad {N}^{\texttt{a},x} = \mathcal{R} {S}^{\texttt{a},x} . \end{array} $$

(150)

All the other input-oriented, storage-based diact flow and storage distribution matrices as well as the corresponding simple and composite flow and storage matrices can be formulated similarly, as listed in Table 2.

Appendix F: Output-oriented system analysis

The output-oriented system analysis is introduced by Leontief (1936) and Augustinovics (1970) in economics. Compartmental systems can be analyzed based on both environmental inputs and outputs. In the context of the proposed methodology, the output-oriented system analysis and the duality of the input- and output-oriented analyses through novel similarity relationships are introduced in this section. The output-oriented analysis backtracks outputs instead of forward tracking of inputs as the input-oriented analysis requires. The details of the derivation of output-oriented analysis are presented in the Supplementary Materials (Section S4).

The main difference between input- and output-oriented analysis in the proposed framework is that, in the latter case, the flow regime is conceptually reversed. That is, the system now becomes “driven” backward in time by outputs rather than forward in time by inputs (consider all arrows reversed in Fig. 1). Therefore, the terms input and inward in the setting of the input-oriented analysis of the original system need to be interpreted as output and outward in this case. More specifically, flow regime of the output-oriented analysis can be expressed as follows:

$$ \begin{array}{llll} \bar F = F^{T}, \quad \bar{\mathcal{Z}} = {\mathcal{Y}}, \quad \text{and} \quad \bar{\mathcal{Y}} = {\mathcal{Z}} . \end{array} $$

(151)

To distinguish input- and output-oriented quantities, we use the bar notation over the output-oriented ones.

The output-oriented analysis in reference to the input-oriented analysis is an inverse problem formulation in mathematical terms and is not well-defined for all classes of dynamical systems. Using the proposed system partitioning formulation, however, it is straightforward to formulate the output-oriented analysis at steady state. In this section, we will develop the output-oriented counterparts of the input-oriented measures and, more importantly, will also show the duality of the input- and output-oriented analyses through similarity relationships between pairs of corresponding matrix measures.

At steady state, inward and outward throughflows at each compartment are equal, so the reversal of the flow regime does not change neither the throughflows nor the associated storages generated by the throughflows. We, therefore, have the following relationships at steady state, in addition to the ones given in Eq. 14:

$$ \begin{array}{llll} \bar \tau = \check{\bar \tau} = \hat{\bar \tau} \quad & \text{and} \quad {\bar T} = \check{\bar T} = \hat{\bar T} \\ \tau = \bar \tau \quad \quad & \Longrightarrow \mathcal{T} = \operatorname{diag}(\tau) = \operatorname{diag}(\bar \tau) = \mathcal{\bar T} \\ x = \bar x \quad \quad & \Longrightarrow \mathcal{X} = \operatorname{diag}(x) = \operatorname{diag}(\bar x) = \mathcal{\bar X} \end{array} $$

(152)

where $\check {\bar \tau }$ and $\hat {\bar \tau }$ are the output-oriented inward and outward throughflow vectors, respectively. These relationships also imply that

$$ \begin{array}{llll} \bar{\mathcal{R}} = \bar{\mathcal{X}} \mathcal{\bar T}^{-1} = \mathcal{X} \mathcal{T}^{-1} = \mathcal{R} . \end{array} $$

(153)

On the other hand, since environmental inputs and outputs are not equal in general, their distribution and organization within the system, that is the input- and output-oriented subthroughflow and substorage matrices, are not the same. Therefore,

$$ \begin{array}{llll} X \neq \bar X \quad \text{and} \quad T \neq \bar T \quad \text{if} \quad z \neq y . \end{array} $$

(154)

Note, however, that

$$ x = X \boldsymbol{1} = \bar{X} \boldsymbol{1} = \bar{x} \quad \text{and} \quad \tau = T \boldsymbol{1} = \bar{T} \boldsymbol{1} = \bar{\tau} , $$

as also formulated in Eq. 152. These relationships imply that,

$$ \begin{array}{llll} \sigma^{x} = \boldsymbol{1}^{T} x = \boldsymbol{1}^{T} X \boldsymbol{1} = \boldsymbol{1}^{T} \bar{x} = \bar{\sigma}^{x} \quad \text{and}\\ \sigma^{\tau} = \boldsymbol{1}^{T} \tau = \boldsymbol{1}^{T} T \boldsymbol{1} = \boldsymbol{1}^{T} \bar{\tau} = \bar{\sigma}^{\tau} \end{array} $$

(155)

where σ^x and σ^τ will be called the total system storage and throughflow, respectively. The total system storage and throughflow are the same in either orientation, as indicated in Eq. 155.

In parallel to the input-oriented static solutions given in Eq. 12, the output-oriented substorage matrix and initial stocks, $\bar X$ and $\bar x_0$, at steady state become

$$ \begin{array}{llll} \bar X &= - \bar A^{-1} \bar{\mathcal{Z}} = \bar{\mathcal{X}} \left( \mathcal{\bar T} - \bar{F} \right)^{-1} \bar{\mathcal{Z}} \\ &= - \bar A^{-1} \mathcal{Y} = {\mathcal{X}} \left( \mathcal{T} - F^{T} \right)^{-1} \mathcal{Y} \quad \text{and} \quad \bar x_{0} = \mathbf{0} \end{array} $$

(156)

where

$$ \begin{array}{llll} \bar A = \left( \bar{F} - \mathcal{\bar T} \right) \bar{\mathcal{X}}^{-1} = \left( F^{T} - \mathcal{T} \right) \mathcal{X}^{-1} = F^{T} \mathcal{X}^{-1} - {\mathcal{R}}^{-1} . \end{array} $$

The output-oriented substorage and subthroughflow matrices and initial subthroughflow vector can then be written as

$$ \begin{array}{@{}rcl@{}} \check{\bar{T}} &=& \mathcal{Y}+F^{T} \mathcal{X}^{-1} \bar{X} ,\\ \hat{\bar{T}} &=& \left( \mathcal{Z}+\operatorname{diag} \left( F \mathbf{1} \right) \right) \mathcal{X}^{-1} \bar{X} = \mathcal{T} \mathcal{X}^{-1} \bar{X} = {\mathcal{R}}^{-1} \bar{X} ,\\ \bar{\tau}_{0} &=& \check{\bar{\tau}}_{0} = \hat{\bar{\tau}}_{0} = \mathbf{0} . \end{array} $$

(157)

Therefore, the static output-oriented substorage and subthroughflow matrices, $\bar X$ and $\bar T$, obtained by the output-oriented system partitioning methodology can be expressed by the relationships given in Eqs. 156 and 157. The (i,k)—elements of these matrices, $\bar {x}_{i_k}$ and $\bar {\tau }_{i_k}$, represent the storage in and throughflow at compartment i destined to exit the system as output from compartment k. Therefore, the output-oriented system partitioning enables partitioning the composite compartmental flows and storages into constituent subcompartmental segments destined to exit the system as environmental outputs separately from each compartment.

Similar to the input-oriented scaling introduced in Section “Subsystem scaling,” all but the initial subsystems can be scaled by corresponding positive environmental output, y_k > 0 ($\mathcal {Y}$ is invertible), to analyze the system behavior per unit output. The output-oriented cumulative storage and throughflow distribution matrices, $\bar S$ and $\bar N$, can, therefore, be formulated as follows:

$$ \begin{array}{llll} \bar{S} = \bar{X} \mathcal{Y}^{-1} = \mathcal{X} \left( \mathcal{T} - F^{T} \right)^{-1} \quad \text{and}\\ \bar{N} = \bar{T} \mathcal{Y}^{-1} = \left( I - F^{T} \mathcal{T}^{-1} \right)^{-1} . \end{array} $$

(158)

The derivation of this formulation is presented in the Supplementary Materials (Section S4.1). All relationships formulated for the input-oriented analysis in the present paper can be extended to output-oriented analysis. The output-oriented counterparts of the relationships formulated in Eq. 26, for example, become

$$ \begin{array}{llll} {\bar T} = \mathcal{R}^{-1} \bar{X} \Rightarrow \bar{S} = \mathcal{R} \bar{N} . \end{array} $$

(159)

Appendix G: Output-oriented diact flows and storages

The output-oriented diact flow and storage distribution matrices and the simple and composite flows and storages are listed in Table 3 at both compartmental and subcompartmental levels. Similar to the input-oriented, storage-based diact flows and storages introduced in Section “E.6 Storage-based diact transactions,” the output-oriented storage-based transactions are also formulated in the Supplementary Materials (Section S4).

Some important relationships between the input- and output-oriented diact flows and storages are also presented in the Supplementary Materials. For example, it is shown that, in either orientation, the simple cycling throughflows and storages are the same, and the acyclic throughflows and storages are related as follows:

$$ \begin{array}{llll} \tilde{\tau}^{\texttt{c}} = \tilde{\bar{\tau}}^{\texttt{c}}, \quad \tilde{x}^{\texttt{c}} = \tilde{\bar{x}}^{\texttt{c}}\quad \text{and}\\ \tilde{\tau}^{\texttt{a}} + z = \tilde{\bar{\tau}}^{\texttt{a}} +y , \quad \tilde{x}^{\texttt{a}} + \mathcal{R} z = \tilde{\bar{x}}^{\texttt{a}} + \mathcal{R} y . \end{array} $$

(160)

In other words, the amount of environmental inputs that transit through the system compartments are the same in both orientations as well as the remaining portion of the inputs cycling within the system.

It is also discussed that the input-oriented transient subflows at the terminal and the output-oriented transient subflows at the initial compartment along a linear flow path are the same. Along all possible flow paths, it can then be shown also that

$$ \begin{array}{llll} T^{\texttt{d}} = { ( \bar{T}^{\texttt{d}} ) }^{T} \quad \text{and} \quad T^{\texttt{a}} = { ( \bar{T}^{\texttt{a}} ) }^{T} . \end{array} $$

(161)

These relationships, however, are not true for the corresponding storages, that is $X^{\texttt {d}} \neq { ( \bar {X}^{\texttt {d}} ) }^T$ and $X^{\texttt {a}} \neq { ( \bar {X}^{\texttt {a}} )}^T$, because of the differences in the compartmental residence times.

G.1 Duality of the input and output-oriented analyses

The following relationships between the input- and output-oriented flow and storage distribution matrices can be derived using Eqs. 22, 23, and 158:

$$ \begin{array}{llll} N \mathcal{T} = \mathcal{T} {\bar N}^{T} \quad \text{and} \quad S \mathcal{X} = \mathcal{X} {\bar S}^{T} . \end{array} $$

(162)

They can be expressed in component form as

$$ \frac{n_{i_{k}} }{ {\bar n}_{k_{i}} } = \frac{\tau_{i}}{\tau_{k}} \quad \text{and} \quad \frac{s_{i_{k}} }{ {\bar s}_{k_{i}} } = \frac{x_{i}}{x_{k}} . $$

(163)

In terms of linear algebra, Eq. 162 indicates that the pairs of matrices $(S,{\bar S}^T)$ and $(N, {\bar N}^T)$ are similar to each other, with the same set of eigenvalues. Using Eqs. 26 and 162, the similarity relationships for matrix measures X, ${\bar X}$, T, and ${\bar T}$ can also be expressed as follows:

$$ \begin{array}{llll} X \mathcal{X} = \mathcal{X} ( {\bar S}^{T} \mathcal{Z} ) &\quad \text{and} \quad {\bar X} \mathcal{X} = \mathcal{X} (S^{T} \mathcal{Y}) \\ T \mathcal{T} = \mathcal{T} ( {\bar N}^{T} \mathcal{Z} ) & \quad \text{and} \quad {\bar T} \mathcal{T} = \mathcal{T} (N^{T} \mathcal{Y}) . \end{array} $$

(164)

The formulas in Eq. 164 can be used to express the relationships between input- and output-oriented substorage and subthroughflow matrices:

$$ \begin{array}{llll} X = ( \mathcal{X} \mathcal{Y}^{-1} ) {\bar X}^{T} ( \mathcal{Z} \mathcal{X}^{-1} ) \quad \text{and}\\ T = ( \mathcal{T} \mathcal{Y}^{-1} ) {\bar T}^{T} ( \mathcal{Z} \mathcal{T}^{-1} ) . \end{array} $$

(165)

Due to these relationships, componentwise for subcompartments i_i, we have

$$ \frac{\tau_{i_{i}}}{\bar{\tau}_{i_{i}}} = \frac{z_{i}}{y_{i}} = \frac{x_{i_{i}}}{\bar{x}_{i_{i}}} , $$

(166)

which is consistent with Eqs. 153 and 163 as well.

The relationships introduced in this section imply that, two of the four matrix measures, S, $\bar S$, N, and $\bar N$ are redundant, as they can be derived from the other two (see Fig. 11). As examples:

$$ \begin{array}{llll} {\bar S} &= \mathcal{X} N^{T} \mathcal{T}^{-1} \quad \text{and} \quad {N} = {\mathcal{R}} \mathcal{X} {\bar S}^{T} \mathcal{X}^{-1} . \end{array} $$

(167)

Using the duality relationships, the output-oriented distribution matrices in Table 3 can also be written in terms of the input-oriented matrix measures.

Fig. 11

Relationships between the input- and output-oriented storage and flow distribution matrix measures. The numbers refer to the equation numbers formulating the relationships