Compartment Stock-and-Flow Models Revisited: DaD-Unstable Pine Forest Svicobia.ns Need a New Stability Concept

1. Introduction

In our recent publication (Logofet and Maslov, 2026), we analyzed Lyapunov stability within a maximally simple model comprising just three compartments in the stock-and-flow diagram (SFD, Figure 1). We found that applying stability conditions to the empirical data (representing what the original authors considered the ecosystem's climax state, i.e., an equilibrium within the dynamic system) revealed a failure to satisfy the mass-balance conditions. This finding prompted searching for the causes of the discrepancy specifically through field studies. The model has so reached its methodology purpose in spite of its deliberate simplicity.

At the same time, we understood that the appropriate degree of model simplification can hardly be formalized, so that the 5-compartment model (Figure 2) needed to be tested in a similar way. However, the authors we criticized did understand this as well, and they directed their efforts toward refining the data collected earlier.

“The bottom line” of this entire off-stage discussion was our realizing the issue of sensitivity to the data precision as a subject to be studied mathematically, and the analysis of DaD-stability (Logofet, 1997, 1998) seems to be a highly suitable tool for this purpose.

A rigorous definition of DaD-stability will be given below in Section 2; for now, suffice it to say that DaD-stable schemes are far less sensitive to errors in empirical data than those merely stable in the Lyapunov sense.

Section 3 reduces the analysis of DaD-stability in SFDs to that in Svicobians. Section 4 presents the criterion of DaD-stability in Svicobians and applies it to the SFD shown in Figure 2. The outcome of this application is discussed in Section 5, and the conclusions are presented in Section 6.

2. DaD-Stability in General

Recall certain tenets of the classical Lyapunov stability (Wikipedia, 2026). If we have a system of n ≥ 3 (nonlinear in general) ODEs

$${\displaystyle \frac{d\mathit{x}}{\mathit{dt}}}=\mathit{f}(\mathit{x},t),t\ge 0,$$

(1)

with a sufficiently smooth right-hand side vector function f(…) and an equilibrium point x^⋇ ∈ ℝⁿ, then it is stable in the linear approximation when the entire spectrum of the Jacobian matrix

$$\mathit{J}={\displaystyle \frac{\partial {\mathit{f}}_i\left({\mathit{x}}^{\divideontimes }\right)}{\partial x_j}}(i,j=,\dots ,n)$$

(2)

locates to the left of the imaginary axis in the complex plane:

Re λ_j(J) < 0, j = 1, …, n.

(3)

Such a matrix is also called stable.

Definition 2.1.

A real n×n matrix A is called D-stable if DA is stable for any positive diagonal D = diag{d₁, d₂, …, d_n}, d_j > 0 ∀j = 1, …, n (Quirk and Ruppert, 1965; Johnson, 1974; Datta, 1978).

Definition 2.2.

A real n×n matrix A is called additively D-stable, or aD-stable, if A − D is stable for any nonnegative diagonal D = diag{d₁, d₂, …, d_n}, d_j ≥ 0 ∀j = 1, …, n (Togawa, 1980; Johnson et al.,1985; Romanishin and Sinitskii, 2002; Kosov, 2010).

The original concept of matrix DaD-stability employs the notions of D- and aD-stability.

Definition 2.3.

A real n × n matrix is called DaD-stable if it is both D- and aD-stable (Logofet, 1997, 1998, 2005).

The utility of this concept has revealed in applications such as Lotka–Volterra population equations (Volterra, 1931; Logofet, 1993, 2018) or compartment models of biogeochemical cycles (Logofet, 1997), which are considered in the next Section.

3. DaD-Stability in SFDs

When the task is to construct a mathematical model with an equilibrium state that corresponds to the given SFD, the solution to this problem depends naturally on how the flows depend on the stocks that are connected by them. The traditional representation of flows as linear functions of stocks (Cullen, 1985) fails to qualitatively reproduce all the variety of regimes required to capture ecological dynamics. Yuri M. Svireshev was the first who expanded the class of traditional linear flow descriptions up to nonlinear ones such as the so-called "Volterra" formulation, namely, to consider the flow to be proportional to the product of the compartment stocks the flow connects (Svirezhev, 1997). His approach mirrors how predator–prey interactions are described in the well-known Lotka–Volterra population equations (Volterra, 1931; Scudo, Ziegler, 1978) in addition to the first two traditional linear functions of stocks, i.e., donor-controlled or recipient-conrolled ones.

As a result, the system of model equations takes on the following form:

$$\dot{\mathit{x}}=\mathit{q}+\mathit{Bx}+\mathbf{diag}{\left\{x\right.}_1,\dots ,\left.x_n\right\}\mathit{Cx},$$

(4)

where q represents a vector of external system inputs, while the elements of matrices B and C are determined by simple formulas based on the given SDG (Zavalishin and Logofet, 1997).

It turned out (ibidem) that for a sufficiently broad subset of situations within the Svirezhev class of functions – specifically, when all inter-compartmental flows entering and leaving a given component depend non-trivially on that component's stock – the Jacobian matrix of system (4), linearized at the equilibrium point, takes on the following invariant form:

A = (S − Q)D,

(5)

where:

matrix S = F^T – F is skew-symmetric by construction (^T means transposition);

matrix F = [f_ij] represents the intercompartment flows of the given SFD ( f_ij: i ⟹ j);

diagonal matrix Q = diag{q)₁, …, (q_n} contains the values, q_i ≥ 0, of permanent inputs;

$\mathit{D}=\mathbf{diag}\left\{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x_1^{*}$}\right.}\right.,\dots ,\left.{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x_n^{*}$}\right.}\right\}$ represents the inverse values of equilibrium stocks, $x_{\mathit{i}}^{*},i=1,\dots ,n$ (Zavalishin and Logofet, 1997).

It would be natural to require matrix A to be indecomposable (or irreducible in some texts) and pattern-nonsingular, the latter being understood as the existence of a nonzero term in the standard expansion of det(A), the matrix determinant. Both conditions can be verified in terms of the directed graph (digraph) associated with the matrix (Harary et al., 1965; Berman and Hershkowitz, 1985; Logofet, 1993) and correspond to a certain structural integrity of the original model.

Definition 3.1. A real, indecomposable, and pattern-nonsingular matrix A of the form (5) is called a Svicobian if S is a skew-symmetric matrix while the diagonal matrices Q and D have nonnegative and positive main diagonals, respectively (Logofet, 1998).

All the Svicobians are proven to be Lyapunov semi-stable, i.e., Re λ(A) ≤ 0 (Zavalishin and Logofet, 1997).

Now, following Definitions 2.1 and 2.2, we comprehend quite a special meaning the D-stability and additive D-stability concepts take on for Svicobians, quite important for the SFD modelling practice. A D-stable Svicobian maintains its stability at any equilibrium stock values, hence the model being invulnerable to any errors in their practical estimations. Similarly, an additively D-stable Svicobian means invulnerability to errors in external inputs. Finally, the ultimate dream for a modeller is a DaD-stable Svicobian (Definition 2.3) invulnerable to the both kinds of error.

4. DaD-Stability Criterion for Svicobians

When the task is to verify whether a particular SFD model yields a DaD-stable Svicobian, it is time to put into practice the coherent theory from the previous Section. Unexpected but good news is that there exists a characterization, i.e., a practically applicable criterion for a Svicobian to be DaD-stable. It is equally unexpected and also good that the criterion has come from a related area – the analysis of the so-called sign stability of matrices, i.e., when the matrix remains Lyapunov stabile in spite of any quantitative variations of its elements albeit preserving the matrix sign pattern (Jeffries, 1974; Jeffries et al., 1977; Logofet, 1993, 2005). The criterion is formulated in terms of the digraph.

Definition 4.1. An associate digraph is said to pass the color test if all its nodes can be colored black and white to meet the following 4 rules:

1) all the nodes corresponding to negative diagonal elements are black;

2) there are white nodes;

3) each white node is connected to at least one other white node;

4) if a black node is connected to a white one, then it is also connected to another one or more white nodes.

Figure 3 illustrates the rules 1)–4) in the same order.

Theorem 4.2. If the digraph of a nonsingular Svicobian cannot be colored by the rules of black-and-white test then this Svicobian is DaD-stable (Logofet, 1978, Theorem 2).

Example 4.1 For the diagram shown in Figure 2, formula (5) yields the following matrix:

$${\mathit{A}}_5=\left[\begin{array}{ccccc}0& {\displaystyle \raisebox{1ex}{${-f}_{12}$}\!\left/ \!\raisebox{-1ex}{$x_2^{*}$}\right.}& 0& 0& {\displaystyle \raisebox{1ex}{${(f_{51}-f}_{15})$}\!\left/ \!\raisebox{-1ex}{$x_5^{*}$}\right.}\\ {\displaystyle \raisebox{1ex}{$f_{12}$}\!\left/ \!\raisebox{-1ex}{$x_1^{*}$}\right.}& 0& 0& 0& {\displaystyle \raisebox{1ex}{${-f}_{25}$}\!\left/ \!\raisebox{-1ex}{$x_5^{*}$}\right.}\\ 0& 0& 0& 0& {\displaystyle \raisebox{1ex}{${-f}_{35}$}\!\left/ \!\raisebox{-1ex}{$x_5^{*}$}\right.}\\ 0& 0& 0& 0& {\displaystyle \raisebox{1ex}{${-f}_{45}$}\!\left/ \!\raisebox{-1ex}{$x_5^{*}$}\right.}\\ {\displaystyle \raisebox{1ex}{${(f_{15}-f}_{51})$}\!\left/ \!\raisebox{-1ex}{$x_1^{*}$}\right.}& {\displaystyle \raisebox{1ex}{$f_{25}$}\!\left/ \!\raisebox{-1ex}{$x_2^{*}$}\right.}& {\displaystyle \raisebox{1ex}{${-f}_{35}$}\!\left/ \!\raisebox{-1ex}{$x_3^{*}$}\right.}& {\displaystyle \raisebox{1ex}{$f_{45}$}\!\left/ \!\raisebox{-1ex}{$x_4^{*}$}\right.}& {\displaystyle \raisebox{1ex}{${-q}_5$}\!\left/ \!\raisebox{-1ex}{$x_5^{*}$}\right.}\end{array}\right],$$

(6)

where the flows f_ij, i, j = 1, …, 5, and equilibrium stocks x*_j , j = 1, …, 5, are positive quantities subject to empirical estimation.

Figure 4 presents the associate digraph of Svicobian (6) in possible colorings. Rule 4) is seen to be violated by nodes ③ and ④ in Figure 4a), hence Svicobian (6) might be DaD-stable according to Theorem 4.2. However, Figure 4b) shows an alternative coloring: rule 1) requires the nodes corresponding to negative diagonal elements to be black, but it does not prohibit the black color in other nodes. It means that Svicobian (6) does not possess DaD-stability. What it means for applications is discussed in the next Section.

5. Discussion

The “unexpected” application of the Jeffries color test in the analysis of Svicobian DaD-stability has actually a deep fundamental reason. The necessary conditions for an irreducible nonsingular matrix to be sign-stable preclude any cycles of length 3 or longer in its associate digraph (Quirk and Ruppert, 1965). As a result, the matrix was semi-stable in the sense that Re λ(A) ≤ 0 (Logofet, 1993), and Svicobians possess this exact same property (Zavalishin and Logofet, 1997). Robert M. May reformulated those necessary conditions in terms of community matrix (May, 1971, 1973,1974) and reduced species interaction types to prey–predator alone, hence to a skew-symmetric matrix.

In the linear approximation of a perturbed solution (e.g., ResearchGate, 2015), there may be exponentials with negative or purely imaginary exponents, and the color test provides the necessary conditions for the latter to exist. Therefore, if the test fails then Re λ(A) < 0, hence Lyapunov stable equilibrium remains the only option both for the community matrix and Svicobians.

Unfortunately, Svicobian (6) turned out to be DaD-unstable. Fortunately, the concept of DaD-stability allows inherently for any, whatever large, variations in stocks and flows as explained at the end of Section 3. In practice, however, experts might limit those variations to certain estimates, while matrix theory does offer a refinement of the DaD-stability concept, in particular, relaxing requirements regarding stock levels.

Definition 5.1. Consider a matrix box of the form:

Θ = diag{d_j}: 0 < θ1_j ⩽ d_j ⩽ θ2_j < +∞, j = 1, … , n.

A real n×n matrix A is called interval D-stable if DA is stable for every matrix D ∈ Θ (Kushel, 2019, 2025). An expert who has been studying a given object for a long enouh time may well assign the intervals, (θ1_j, θ2_j), of values for empirical estimates. Our practice now motivates developing a new concept of DaD stability that would narrow the intervals for the flow values.

6. Conclusions

Revisiting stock-and-flow models has revealed some new problems indeed. DaD-stability analysis of the SFDs has turned out reducible to the DaD-stability problem for Svicobians, while the latter proves to share a common criterion with the sign-stability of real n×n matrices, so-called the Jeffries’s color test. The criterion has proved the Pine Forest Svicobian to be DaD-unstable, thereby decreasing the model’s invulnerability to errors in the empirical data. In practice, however, this decrease is by no means crucial and can be expert-reduced to certain intervals for empirical assessments. The novel concept of matrix interval D-stability addresses this challenge with regard to estimating the magnitude of stocks, and the task for future research is to develop an analogous concept for flows, or for both.