Unraveling emergent network indeterminacy in complex ecosystems: A random matrix approach

Significance

This study uses random matrix theory to provide mathematical criteria determining whether network indeterminacy—the unpredictability of perturbation outcomes via complex networks—emerges in ecological communities. The analysis demonstrates that large networks do not necessarily lead to greater unpredictability. Specifically, we found that indeterminacy is less likely in competitive and mutualistic communities than in top–down regulated food webs. We also note that predictable and unpredictable perturbations can coexist, with indeterminate outcomes more common in predator–prey networks. Therefore, our findings not only challenge the conventional understanding of the complexity–indeterminacy relationship but also suggest applications in other fields, such as microbial and medical sciences, enhancing our ability to predict perturbation outcomes of complex systems.

Abstract

Indeterminacy of ecological networks—the unpredictability of ecosystem responses to persistent perturbations—is an emergent property of indirect effects a species has on another through interaction chains. Thus, numerous indirect pathways in large, complex ecological communities could make forecasting the long-term outcomes of environmental changes challenging. However, a comprehensive understanding of ecological structures causing indeterminacy has not yet been reached. Here, using random matrix theory (RMT), we provide mathematical criteria determining whether network indeterminacy emerges across various ecological communities. Our analytical and simulation results show that indeterminacy intricately depends on the characteristics of species interaction. Specifically, contrary to conventional wisdom, network indeterminacy is unlikely to emerge in large competitive and mutualistic communities, while it is a common feature in top–down regulated food webs. Furthermore, we found that predictable and unpredictable perturbations can coexist in the same community and that indeterminate responses to environmental changes arise more frequently in networks where predator–prey relationships predominate than competitive and mutualistic ones. These findings highlight the importance of elucidating direct species relationships and analyzing them with an RMT perspective on two fronts: It aids in 1) determining whether the network’s responses to environmental changes are ultimately indeterminate and 2) identifying the types of perturbations causing less predictable outcomes in a complex ecosystem. In addition, our framework should apply to the inverse problem of network identification, i.e., determining whether observed responses to sustained perturbations can reconstruct their proximate causalities, potentially impacting other fields such as microbial and medical sciences.

Understanding how natural ecosystems respond to environmental changes is crucial, especially with escalating global warming and growing human impacts. A promising approach to tackling this issue is investigating species interaction relationships, which consist of direct causal links between species. Thus, many studies have undertaken empirical experiments and observations to investigate the impact of one species’ physiological, behavioral, and population state on others (1–5). However, in ecological communities where multiple species coexist, focusing solely on proximate, pairwise species interactions might be insufficient due to indirect effects—an emergent property of complex networks.

An indirect effect, any transmission of changes in a species’ state to the other via a chain of direct interaction links (6, 7), occurs in networks with more than two species. Importantly, depending on relative strengths of link connection, such indirect effects lead to outcomes unanticipated from direct species relationships (Box 1A). Furthermore, as the number of possible interaction chains exponentially increases indirect effects, regardless of their individual magnitude, might surpass the direct one (8, 9). Thus, indirect effects have traditionally been deemed a critical source for network indeterminacy (7, 10), i.e., a phenomenon that the perturbation outcomes in ecological networks are unpredictable from direct species relationships (Box 1). Several empirical studies have also found discrepancies between observed and experimentally/statistically estimated species effects, suggesting that indirect pathways in natural ecosystems obscure direct interspecies relationships (11–15). However, these studies also reported that a substantial fraction of the network is predictable from the known ecological information (11, 15), complicating the entire picture.

Box 1.

Network indeterminacy and press perturbation

Originally, the indeterminacy of ecological interactions is referred to a high variability of press perturbation outcomes to slight errors in direct interaction effects (10, 16). This concept is incorporated into our usage of network indeterminacy that an inverse interaction matrix cannot be approximated from the series expansion of sensitivity essence Φ, because the errors in A can be included in its random component E. Then, to provide a better intuitive understanding of network indeterminacy, we here take an intraguild predation (IGP) module as an example. This system is a simple, three-species ecosystem but contains various types of indirect interactions, such as trophic cascade, exploitative competition, and apparent competition, having the potential to show complex dynamics (17, 18).

A. Unexpected outcomes of press perturbations through indirect effects

This system consists of a predator–prey pair plus a third-party species consuming both species. Thus, the top predator (x₃) indirectly benefits the prey population (x₁), in contrast to its direct effect, by removing the intermediate predator (x₂) (18). This positive indirect effect of trophic cascade (dashed lines ending in a filled circle) can be larger than the negative direct predation (solid line arrows) if the predation efficiency of x₃ is higher for x₂ than for x₁. For example, such food-chain-like and competitive (i.e., dominance of an exploitative competition for the prey) IGPs can be parameterized in the GLV model with

$$\mathbf{A}=\left(\begin{array}{ccc}d& -0.9& -0.2\\ 0.6& d& -0.9\\ 0.1& 0.6& d\end{array}\right) \text{and} \left(\begin{array}{ccc}d& -0.9& -0.9\\ 0.6& d& -0.2\\ 0.6& 0.1& d\end{array}\right),$$

respectively, where d = −7/6. In these settings, sustained addition of x₃ increases the prey density in the food-chain-like IGP (Top of panel A) but decreases x₁ in the competitive IGP (Bottom of panel A).

$$\mathbf{A}=\left(\begin{array}{ccc}d& -0.9& -0.9\\ 0.6& d& -0.2\\ 0.6& 0.1& d\end{array}\right).$$

In summary, press perturbations are propagated via direct and indirect pathways of interaction, causing outcomes unexpected from direct species relationships.

B. Dynamics of press perturbations close to eigenvectors with different eigenvalues

The predictability can differ between press perturbations depending on which eigenvectors of the sensitivity essence Φ they are close to. For example, in the above food-chain IGP, the absolute eigenvalues of Φ are (1.003, 1.003, 0.066) and the eigenvector (denoted by the orange solid line) of |λ(Φ)| = 1.003 spans the direction in increasing the top-predator density x₃ (Top of panel B). Then, adding a press perturbation (arrow line) close to this eigenvector results in an outcome (black triangle), which is less consistent with the predictions (colored triangles) of the Â_k^–1 approximated with Eq. 2. In contrast, the press perturbation close to the eigenvector of |λ(Φ)| = 0.066 (along to the direction in increasing the prey density x₁) yields the outcome relatively predictable (Bottom of panel B).

Thus, under which ecological contexts do indirect effects render an interaction network indeterminate? Theoretical studies have addressed this issue by investigating interaction modules, three- or four-species systems with a specific network configuration (e.g., apparent competition, trophic cascade, and intraguild predation) and have elucidated the condition where each indirect path becomes more significant in the system dynamics than the direct links (17, 19–21). Regarding large ecological communities, in contrast, analytical interests have centered on the role of indirect effects in real, observed food webs (7, 10, 22, 23). Although a few mathematical investigations exist (16, 24, 25), their scope covers limited features of ecological networks (e.g., length of indirect paths or the number of feedback cycles). A unified theory exploring mechanisms driving network indeterminacy in complex ecosystems is still required.

This study aims to provide rigorous, mathematically sound criteria determining whether an interaction network is predictable from the direct species relationships for various ecological communities. To this end, we turn to the random matrix theory (RMT), which helps analyze the dynamical properties of complex ecosystems (26–29). Nevertheless, there are minor applications of RMT to the network indeterminacy issue due to problems with analyzing inverse community matrix in large complex ecosystems (see below). Thus, we develop a theoretical framework leveraging tools of linear algebra to solve the mathematical hurdles. The analysis of theoretical criteria would advance our understanding of how indirect effects cause emergent network indeterminacy in various ecological contexts.

Results

We first introduce an ecological network containing S interacting species as a continuous-time nonlinear system dx/dt = f(x), where f is a set of nonlinear functions describing each species’ population growth rate; an S-length vector x comprises species density x_i. This system is assumed to have a feasible equilibrium x* > 0, and a linearization around it yields the following equation:

$$\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}={\left.\frac{\partial \mathbf{f}}{\partial \mathbf{x}}\right|}_{{\mathbf{x}}^{\ast }}\mathbf{x}=\mathbf{M}\mathbf{x}.$$

M is a community matrix whose element M_ij represents the population-level direct effect of x_j on x_i (30, 31). The system’s response to instantaneous, infinitesimal perturbations (i.e., whether the system goes back to the equilibrium x* or not) can be evaluated by the eigenvalue distribution of the community matrix M. Thus, the RMT helps the stability analysis of M and has advanced the classic “complexity–stability” debate (26, 28, 29). However, the network indeterminacy issue requires a slightly different application of RMT, as described below.

Consider that an m-length input vector b = {b_i}, which consists of a slight and constant perturbation, is added to the system f at x*. This ‘press perturbation’ can shift each species’ growth rate via changes in any density, physiological, and behavioral state (32), moving the system to a different equilibrium x_b* satisfying f(x_b*, b) = 0 (Box 1). Thus, its dynamics can be described by considering the following chain rule:

$$\frac{\partial \mathbf{f}}{\partial \mathbf{b}}+{\left.\frac{\partial \mathbf{f}}{\partial \mathbf{x}}\right|}_{{\mathbf{x}}^{\ast }}\frac{\partial {\mathbf{x}}^{\ast }}{\partial \mathbf{b}}=0⟺\frac{\partial {\mathbf{x}}^{\ast }}{\partial \mathbf{b}}=-{\mathbf{M}}^{-1}\frac{\partial \mathbf{f}}{\partial \mathbf{b}},$$

where the existence of M^–1 is guaranteed by the local stability of M, and the long-term response of x* to b is determined by the elements of –M^–1, called the sensitivity matrix (33). That is, the sensitivity matrix contains information on how press perturbation affects the community equilibrium via possible direct/indirect pathways in ecological networks (10, 34) (Box 1).

The nonlinear system f can take various functional forms. Specifically, this study considers a generalized Lotka–Volterra (GLV) model (see SI Appendix, SI Text for extensions to other models), a fundamental one for ecological theory (26, 29, 35). Accordingly, the community matrix is M = X*A (Methods), and the sensitivity matrix is modeled as follows:

$$-{\mathbf{M}}^{-1}=-{\left({\mathbf{X}}^{\ast }\mathbf{A}\right)}^{-1}=-{\mathbf{A}}^{-1}{\left({\mathbf{X}}^{\ast }\right)}^{-1},$$ [1]

where X* is a diagonal matrix of {X*}_ii = x_i*. A ∈ ℝ^{S × S} is an interaction matrix whose off-diagonal element a_ij measures the individual-level direct effect of species j on i (31), thus can reflect the observed ecological structures such as the sparse-interaction networks (36), weak-interaction prevalence (37), and interaction-type (e.g., predator–prey, mutualism, and competition) mixing (5, 26).

In this study, the construction manner of A followed the previous ones (26, 38): 1) two species interact with a probability C (connectance); 2) each interaction pair (a_ij and a_ji) is randomly assigned from a bivariate distribution with a mean μ, variance σ², and correlation ρ; 3) the diagonal elements, representing the self-regulation intensity, are set to a negative value d to ensure the local stability/invertibility of A. For this community setting, the mean, variance, and correlation of off-diagonal entries of A become 𝔼[a_ij] = μC, V_A = 𝔼[a_ij²] − 𝔼²[a_ij] = C(σ² + μ²(1 – C)), and P_A = (𝔼[a_ija_ji] − 𝔼²[a_ij])/V_A = (ρσ² + μ²(1 – C))/(σ² + μ²(1 – C)), respectively (26, 27). These formulae can be interpreted ecologically as follows. Denser networks (i.e., C ≫ 0) with a positive interaction correlation ρ > 0, mutualisms are dominated when an interaction mean μ is positive, while competitive interactions tend to be frequent when μ < 0; for ρ < 0, the condition μ < 0 implies top–down regulated food webs, where the predation damages prey species suffer are more significant than the rewards the predator species receive, and otherwise stands for bottom–up regulated ones.

Eq. 1 also clarifies that indirect effects are analyzed via the inverse interaction matrix. However, it is generally difficult for large networks to obtain a closed-form solution of A^–1. A few mathematical studies used intricate transformation of the interaction matrix [calculating determinant and cofactor of A, (16, 25)], causing minor RMT applications to the network indeterminacy issue (but see ref. 39). Thus, this study adopted another approach combining several tools of linear algebra with the RMT to clear this hurdle. Specifically, the first preparation is a Neumann series, which expands an inverse into the geometric series of the original square matrix (40). That is, this expansion can directly relate the mathematical properties of a sensitivity matrix with its community matrix (9, 24).

To apply the series-expansion approach, we decompose an interaction matrix into the deterministic and random component as A = D + E (Fig. 1A). D is an S-dimensional deterministic matrix whose diagonal and off-diagonal entries are constant D_ii = d and D_ij = μC, respectively; as E = A − D, the diagonal entries E_ii = 0 and the off-diagonal ones are random variables with mean 0, variance V_E = V_A, and the correlation P_E = P_A. Thus, an inverse matrix can be transformed with the Neumann series expansion as follows:

$${\mathbf{A}}^{-1}={\left(\mathbf{D}+\mathbf{E}\right)}^{-1}={\left(\mathbf{D}\left(\mathbf{I}-\left(-\mathrm{\Phi }\right)\right)\right)}^{-1}=\left({\displaystyle \sum _{i=0}^{\infty }}{\left(-\mathrm{\Phi }\right)}^i\right){\mathbf{D}}^{-1},$$

Fig. 1.

A schematic of the ecological indeterminacy analysis. (A) Intermediate process of the ecological indeterminacy analysis: An interaction matrix A (Left) is decomposed into a determinant component D (Middle) and a random one E (Right). The colors in each matrix represent the value of i-j entries denoted by the color bars; the dark red in diagonal elements represents a strong negative value (a_ii = −1.383). (B) Neumann series expansion (Middle) approximates the off-diagonal entries of A^–1 (Right) based on the sensitivity essence Φ (Left). The colors in Φ correspond to the color bars in (A).

where Φ = D^–1E (hereafter, sensitivity essence). This transformation is possible whenever the maximum absolute eigenvalue satisfies the condition: γ_max = max(|λ(Φ)|) < 1 (λ(Φ) and γ denotes eigenvalues of Φ and its absolute, respectively) (40). In particular, the inverse interaction matrix can be approximated efficiently by the sum of the relatively lower terms as:

$${\mathbf{A}}^{-1}\approx {\widehat{\mathbf{A}}}_k^{-1}=\left({\displaystyle \sum _{i=0}^k}{\left(-\mathrm{\Phi }\right)}^i\right){\mathbf{D}}^{-1},$$ [2]

if γ_max ≪ 1 (e.g., for A⁻¹ with γ_max = 0.756, its off-diagonal entries are well approximated by Eq. 2 with k = 4, Fig. 1B). Therefore, the condition γ_max > 1 determines a criterion of ecological indeterminacy, where the outcome of any press perturbation is unpredictable.

Applying the RMT to analyze γ_max requires characterizing the diagonal and off-diagonal elements of the sensitivity essence. To this end, the following decomposition of Φ, which is based on the structure of D and E (Methods), is helpful as:

$$\mathrm{\Phi }=\frac{1}{d-\mu C}\left(\mathbf{E}-\frac{\mu C}{\psi }\mathbf{u}{\mathbf{u}}^{\mathrm{T}}\mathbf{E}\right)=\frac{1}{d-\mu C}\left(\mathbf{E}-\frac{\mu C}{\psi }\mathbf{u}{\mathbf{v}}^{\mathrm{T}}\right),$$ [3]

where ψ = d + μC(S – 1), u is an S-length vector of all 1 (i.e., {u_i} = 1} and v = E^Tu. That is, the matrix E′ = E – ψ^–1μCuv^T is deemed the random matrix E plus a rank-1 perturbation of −ψ⁻¹μCuv^T. The RMT (41) tells that, for a random matrix with a low-rank perturbation, most eigenvalues λ(Φ) follow the elliptic law, but outlier eigenvalues may exist. These elliptic and outlier eigenvalues are correctly approximated by using tools of linear algebra (Eq. 11 in Methods), and the theoretical maximum of absolute eigenvalues is obtained as γ_max = max(γ₊, γ_o), where γ₊ and γ_o are the semimajor axis of the elliptic law and the absolute of outlier eigenvalues, respectively (the eigenvalue distributions can be visualized in the complex plane like SI Appendix, Fig. S2). Specifically, those values are expressed with the network structure of A such as variance and correlation as follows:

$${\gamma }_{+}=\frac{1+\left|P_{\mathbf{A}}\right|}{\left|d-\mu C\right|}\sqrt{S{V}_{\mathbf{A}}},$$ [4]

and

$$\begin{array}{c}{\gamma }_{\mathrm{o}}=\left\{\begin{array}{c}\left|\frac{\mu C}{\psi }\right|\frac{\sqrt{S\left(S-1\right)V_{\mathbf{A}}}}{\left|d-\mu C\right|}\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }+\frac{\psi }{\mu C}{P}_{\mathbf{A}}},\mu P_{\mathbf{A}}<0\hfill \\ \left|\frac{\mu C}{\psi }\right|\frac{\sqrt{S\left(S-1\right)V_{\mathbf{A}}}}{\left|d-\mu C\right|}\sqrt{\frac{2}{\pi }},\mu P_{\mathbf{A}}=0\\ \left|\frac{\mu C}{\psi }\right|\frac{\sqrt{S\left(S-1\right)V_{\mathbf{A}}}}{\left|d-\mu C\right|}\left(\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }}+\sqrt{-\frac{\psi }{\mu C}{P}_{\mathbf{A}}}\right),\mu P_{\mathbf{A}}>0\end{array}\right..\hfill \end{array}$$ [5]

Eq. 4 indicates that the semimajor axis γ₊ lengthens as positively and negatively correlated interactions increase (represented by larger |P_A|). In contrast, if the outlier eigenvalues exist, the value of γ_o depends on P_A differently for the competitive networks/bottom–up regulated food webs (μP_A < 0) and the mutualistic/top–down regulated food webs (μP_A > 0).

To verify this criterion, a numerical simulation was performed for multiple runs in which the interaction matrix A was randomly generated with different parameters (Methods). The results are summarized in Fig. 2A. The largest |λ(Φ)| calculated for each simulation run was consistent with its theoretical value γ_max under the given parameters, verifying the network criterion proposed. We also investigated the prediction skill ρ_k, which is calculated as the correlation coefficient between the off-diagonal elements {A^–1}_ij and its kth-order approximation obtained from Eq. 2. As shown in Fig. 2B, the prediction skill was relatively robust against an increase in the theoretical γ_max in its small value; when γ_max > 1, the ρ_k decreased sharply and approached zero. Furthermore, the criterion provided information on how the perturbation outcomes could be predicted from direct species relationships. For example, the ρ_k was relatively high by the first-order approximation (i.e., ρ₁) and additionally recovered as k increased for the conditions satisfying γ_max < 1 (the Left two panels in Fig. 2C). However, once the condition was broken, an increase in k caused a worsening of the prediction skill (the Right panel in Fig. 2C).

Fig. 2.

The relationship between the indeterminacy criterion and prediction skill of perturbation outcomes. (A) Correspondence of the theoretical γ_max and the maximum absolute eigenvalue max(|λ(Φ)|) calculated from 1,000 randomly generated interaction matrix A. The red line indicates max(|λ(Φ)|) = γ_max. (B) Prediction skill ρ_k of perturbation outcomes as a function of theoretical γ_max. ρ_k is calculated as a Pearson’s correlation coefficient between {A^–1}_ij and its kth-order approximation with Eq. 2 (here, k = 5). The red line represents the indeterminacy criterion γ_max = 1. (C) The effect of the approximation order k on the prediction skill ρ_k under different values of γ_max, which are represented at the top of each panel. The black dots are the results for each simulation, and the blue indicates the mean of 1,000 iterations.

The analysis of the theoretical criterion further provides deep insight into the conjecture on the ecological complexity as a source of network indeterminacy (7, 10, 24, 25). Specifically, we conducted the simulation investigating the effect of community size S and interaction connectance C on the maximum absolute eigenvalue. The results for negative interaction mean μ ≤ 0 (the Upper panels in Fig. 3) demonstrated that network complexity (i.e., larger S and C) increased theoretical γ_max and sampled max(|λ(Φ)|) similarly so that these values reached the threshold for P_A < 0. However, the pattern was different for positive interaction mean μ > 0 (the Bottom panels in Fig. 3). The γ_max slightly decreased as a function of S, especially for P_A = 0.0; an increase in C negatively impacted γ_max in relatively small networks (S < 100).

Fig. 3.

The complexity–indeterminacy relationships for finite-size ecological communities. The horizontal axis is community size S. The red, blue, and green colors indicate the results of connectance C = 1.0, 0.7, and 0.4, respectively. The values of μ and P_A are denoted at the top of each panel. The dots represent the maximum absolute eigenvalue, max(|λ(Φ)|), calculated from 100 randomly generated A, and the bold lines are the theoretical γ_max. The gray line is the indeterminacy criterion γ_max = 1.

These findings imply that the indeterminacy criterion depends on the parameters of ecological structures in a complicated manner. As explained in Methods, this prediction can be explored analytically in the limit S → ∞ as:

$$\begin{array}{c}\underset{S\to \infty }{\lim }{\gamma }_{\max }=\left\{\begin{array}{c}\frac{1+\left|P_{\mathbf{A}}\right|}{1+P_{\mathbf{A}}},\mu \le 0\hfill \\ \sqrt{\left(\frac{1+P_{\mathbf{A}}}{2\pi }-\frac{P_{\mathbf{A}}}{\mu C}\right)V_{\mathbf{A}}},\mu >0\cap P_{\mathbf{A}}<0\\ \sqrt{\frac{2}{\pi }{V}_{\mathbf{A}}},\mu >0\cap P_{\mathbf{A}}=0\\ \left(\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }}+\sqrt{\frac{P_{\mathbf{A}}}{\mu C}}\right)\sqrt{V_{\mathbf{A}}},\mu >0\cap P_{\mathbf{A}}>0\end{array}\right.,\hfill \end{array}$$ [6]

for μ < 0. Eq. 6 is derived from the elliptic law Eq. 4, and Eq. 7 are from the outlier eigenvalues Eq. 5. These formulae elucidate the following fact on the relationship between ecological complexity and indeterminacy, which is shown with a function of μ and ρ as proxies for P_A (Fig. 4 and SI Appendix, Fig. S3).

Fig. 4.

The relationship between ecological structures and network indeterminacy for infinite-size communities. The horizontal and vertical axes indicate the interaction mean μ and interaction correlation ρ, respectively. The connectance used in the analysis is denoted at the top of each panel. The red colors represent the area satisfying γ_max > 1.

Specifically, for negative networks (μ ≤ 0), the criterion depends only on the correlation of the off-diagonal elements of A, P_A. Thus, if all species are fully connected (C = 1.0, the Upper Left panel in Fig. 4), the press perturbation outcomes are essentially undetermined from the direct species relationships for P_A < 0 (the third quadrant in the μ-ρ plane) but not for P_A > 0 (the second quadrant). In contrast, for networks with μ > 0, the criterion depends on the interaction mean μ and the variance V_A as well as P_A; the region satisfying γ_max > 1 is more likely to occur in networks with P_A > 0 than those with P_A ≤ 0 (represented with the first and fourth quadrant in Fig. 4, respectively) due to the Cauchy–Schwarz inequality (a + b)^1/2 ≤ a^1/2 + b^1/2. Note that P_A = (ρσ² + μ²(1 – C))/(σ² + μ²(1 – C)), whose denominator is positive. Therefore, P_A is positive with the condition ρ > –(1 – C)(μ/σ)²: with decreasing C, γ_max is less likely to exceed the threshold for μ ≤ 0 and more likely to do for μ > 0 (Fig. 4).

The focus of the analysis thus far has been only on the maximum absolute eigenvalue. Note here that each eigenvalue of Φ has a corresponding eigenvector that determines the direction in which a perturbation will grow or decay at the rate specified by λ: an outcome of the press perturbation close to eigenvectors with |λ| < 1 should be predictable even though a system exhibits ecological indeterminacy (i.e., γ_max > 1). This prediction is demonstrated in the simple IGP module example (Box 1B). Then, more complex communities might also show different network estimation accuracy with low approximation orders, even though they have similar magnitudes of γ_max (Fig. 5A).

Fig. 5.

The relationships between the eigenspace exceeding the indeterminacy threshold and the press perturbation outcomes. (A) Histogram of the absolute eigenvalues |λ(Φ)| for 100 randomly generated A. The blue solid lines denote the probability density function for |λ(Φ)|. The Top- and Bottom-Left panels are the results with similar γ_max but different distributions. The Right panel demonstrates the log-scaled MAE (mean absolute error) between true {A^–1}_ij and its approximation with Eq. 2 for 250 randomly generated perturbations. (B) The Top and Bottom panels demonstrate the sampled volume |λ(Φ)| > 1 and the prediction skill ρ_k of {A^–1}_ij with Eq. 2 (k = 5) as a function of the theoretical volume Ξ, respectively. The red, blue, and green colors are the results for the γ_max = 1.2, 1.3, and 1.4, respectively.

One way to explore this intuition is to analyze the volume of eigenspace associating with the eigenvalues satisfying |λ(Φ)| > 1. As explained in Methods, such a measure, Ξ, can be calculated from the probability density function of |λ(Φ)| and the absolute outlier eigenvalues γ_o. Fig. 5B summarizes the results verifying the relationship between Ξ and ecological indeterminacy. Specifically, theoretical volume Ξ and the relative amount |λ| > 1 of randomly generated Φ were consistent; they had various magnitudes despite having the same γ_max under different parameter set (the Upper panel in Fig. 5B). Then, the prediction skill ρ_k = 5 of the randomly generated perturbations with Eq. 2 deteriorated with increased theoretical volume Ξ (the Bottom panel in Fig. 5B).

Discussion

Anticipating how natural ecosystems will respond to persistent perturbations is a pressing concern in the face of increasing anthropogenically induced environmental stresses. Nonetheless, the rise of indirect effects in complex networks may obscure observed, direct species effects, compounding this task (7, 10). In this study, we presented a pioneering theoretical framework to identify the conditions under which network indeterminacy, a phenomenon where the long-term consequences of press perturbations cannot be predicted solely from direct interactions, emerges in large ecological communities. Our analysis revealed that the network complexity does not necessarily result in unpredictability contrary to conventional thoughts (7, 10, 24, 25); rather, it depends on the other ecological structures, i.e., the property of species interaction.

As a starting point for discussion, consider the simple IGP module again (Box 1). The example of press perturbation dynamics demonstrated that, despite the same network topology, only a slight difference in the combination of each interaction strength yields a disparity in the predictability of the perturbation outcome (Box 1A). In addition, it has been theoretically shown that the efficiency difference in consuming the prey between the top predator and the intermediate one determines the relative importance of the indirect effect in this system (17, 18). These considerations stress the significance of interaction characteristics in the network indeterminacy. In fact, the formulation of indeterminacy criterion (Eqs. 4 and 5) conveys that the interaction strength (expressed by the variance σ) and the interaction-type parameters (i.e., interaction mean and correlation) determine whether the press perturbation outcome of finite networks (S ≪ ∞) is predictable from direct species relationships. This dependency can be easily illustrated with the network indeterminacy of infinite communities (Eq. 6) as described below.

Specifically, in communities where competitive or antagonistic interactions dominate (the second quadrant in Fig. 4), the outcomes of press perturbations will essentially be predictable from the direct interaction relationships even with the larger network size S; an increase in the link connection C reduces the possibility of satisfying the indeterminacy criterion γ_max > 1 in mutualistic networks (the first quadrant in Fig. 4). These findings renew the expected relationship between ecological complexity and indirect effects in previous studies of competitive and mutualistic systems (42, 43) as follows: complexity does augment indirect effects, but it does not make the perturbation outcomes entirely unpredictable. In contrast, larger networks with μ ≤ 0 and ρ ≤ 0 (the third quadrant in Fig. 4) inevitably meet the indeterminacy criterion, especially for fully connected networks. This result indicates that the response of top–down regulated food webs would innately be incomprehensible as a sum of direct predator–prey links, aligning with the line of the classical food-web indeterminacy studies (7, 10).

Why, then, does the likelihood of network indeterminacy vary significantly under different ecological structures? The RMT perspective would also shed light on this question as follows. The network indeterminacy criterion, γ_max, depends on the semimajor axis and the outlier eigenvalues of the ellipse of a random matrix E. These values are scaled by the self-regulation intensity d (Eqs. 4 and 5), which is set to maintain the stability of the original GLV system (Eq. 9), meaning that it balances the leading eigenvalue of the interaction matrix A. Note here that a theoretical study for the complexity–stability debate (26) demonstrated that an outlier eigenvalue and the right-most eigenvalue of the elliptic law becomes the leading one in networks with μ > 0 (i.e., the positive interaction mean of A) and those with μ < 0, respectively. These findings allow us to infer that the condition in which the semimajor axis is larger than the self-regulation is more likely to satisfy the network indeterminacy criterion. This situation occurs more frequently in trophic systems (i.e., networks with negative interaction correlation ρ < 0) because the prevalence of predator–prey interactions contract the growth rate of perturbation but enlarges its oscillation, squeezing the ellipse’s shape vertically (SI Appendix, Fig. S2). That is, the self-regulation intensity is a significant ecological property influencing both the network indeterminacy issue and the well-known complexity–stability relationship (26, 35, 38).

The RMT-based analysis further showed that ecological indeterminacy also depends on the volume of eigenspace where the eigenvalues meet the indeterminacy criterion. Specifically, the perturbation outcomes were less predictable for networks with more eigenvalues exceeding the indeterminacy criterion, even when they had the similar values of γ_max (Fig. 5B); this difference in the prediction accuracy between perturbations is less prominent in the higher-order approximation (Fig. 5A). As discussed above, such cases are expected to be more often in vertically expanded eigenvalue distributions (SI Appendix, Fig. S2). That is, this result conveys two points: 1) less/more predictable perturbations can coexist in the same ecological communities and 2) predicting the response to environmental changes is more challenging in networks where predator–prey relationships predominate than those with mutualistic and competitive interactions. Taken together, the combination of the RMT analysis and elucidating direct species relationships helps understand the types of environmental changes causing more unpredictable outcomes in a complex network, as well as the degree of network indeterminacy.

The above findings provide predictions that may be tested using empirical approaches. For example, an experimental study of the aphid-predator insect system showed that the addition of new predator species altering the system to an IGP-like module made poorer prediction of the predation strengths with the information on the predator/prey body-size ratio than the two-species case (14). This operation should be analogous to the indeterminate perturbation of the IGP example (see the Top panel in Box 1B). In contrast, it is expected that perturbations increasing the prey species (the Bottom panel in Box 1B) yield predation strengths more consistent to the model predictions, which can be tested by increasing the density of aphids in this system. For more complex networks, microbial community experiments may be helpful to investigate the predictions of network indeterminacy. Microbial–microbial relationships also cover a broad interaction spectrum (e.g., competition and mutualism), which can be identified by pairwise coculture experiments (44). Importantly, recent advances in experimental and sequencing technologies enable easy establishing and analyzing “synthetic” microbial communities (45, 46). Therefore, comparing the inverse of the direct microbial interaction matrix and perturbation outcomes of artificially created cocultures with different system sizes, interaction means, and correlations would allow for direct tests of the predicted relationships between network indeterminacy and interaction characteristics.

As explained, our framework is directly derived from the interaction matrix, thus being applicable to investigate the relationship between network indeterminacy and known ecological properties other than those studied here. For example, in this study, we considered species interactions occurring linearly (GLV model). However, natural organisms interact in various ways, which can be modeled as nonlinear functional responses (35, 47–49). That is, further analyses implementing each nonlinear interaction would be required (SI Appendix, SI Text), whereas additional investigation showed no qualitative difference in the conclusion in one of them (SI Appendix, Fig. S4). The variation in network topology is another ecological pattern to consider. Although we found that the interaction characteristics are determinants of indeterminacy among communities with a similar network configuration (Box 1A and Fig. 4), it was theoretically shown that the difference in network structure affects the ecological stability (28, 50). In addition, our study postulated random interactions, where any two species were randomly connected with no constraint. This should be unsuitable for natural settings, and nonrandom network configurations, such as nestedness of generalist/specialist species and compartmentalization of interaction webs, are frequently observed in natural ecological communities (51, 52). Fortunately, these properties have been introduced into RMT in the context of community stability (28, 50), being available to the indeterminacy issue (SI Appendix, SI Text and Fig. S5).

The present study advances our understanding of emergent network indeterminacy in complex ecosystems and further provides other benefits. For example, our results illustrate that decreasing connectance in positive networks and increasing negative correlations in negative ones enlarge the region satisfying the indeterminacy criterion (Fig. 4 and SI Appendix, Fig. S3). Thus, removing interaction links/other interaction types renders mutualistic/predator–prey relationships dominated networks unpredictable against press perturbations, implying the importance of their conservations. By elucidating direct species relationships, we can forecast which types of perturbations the ecological community is vulnerable in terms of network indeterminacy (Box 1B, Fig. 5). Furthermore, our framework of the network indeterminacy issue may apply to its inverse problem: Can the observed network of perturbation outcomes reconstruct direct causations? Such demands, defined as the problem of solving the inverse of association networks (53, 54), are significant in other fields, such as inferring gene regulatory systems (55, 56) and understanding microbial networks (57). Therefore, the theoretical framework developed here plays an important role not only in ecology but also in any area of human endeavor.

Methods

Sensitivity Matrix of the GLV Model.

In ecological contexts, the press perturbation can affect species growth rate via changes in any density, physiological, and behavioral state (32). In particular, we consider a GLV system with an S-length perturbation vector b as f_i(x, b) = x_i(r_i + Σ_ja_ijx_j + b_i), where f_i and r_i are the net population growth rate and the intrinsic growth rate of species i, respectively. The coefficient a_ij represents the response of species i’s per capita growth rate to a slight change in the interacting species (30, 31), and the parameter b_i stands for a perturbation specific to species i. With no perturbation condition (i.e., b = 0), the Jacobian elements are ∂f_i/∂x_i = a_iix_i + r_i + Σ_ja_ijx_j, ∂f_i/∂x_j = a_ijx_i. Therefore, combining the fact that r_i + Σ_ja_ijx_j* = 0 [the solution of f(x*, 0) = 0], the community matrix is obtained as M = X^*A, deriving the sensitivity matrix as Eq. 1.

Elliptic Law of the Interaction Matrix.

The diagonal element d of the interaction matrix A was determined to ensure the local stability of the community equilibrium x*. Thus, it required information on the distribution of the eigenvalues λ(A); this can be analytically obtained with the help of RMT as the following procedure. First, consider an S-dimensional real matrix Z, whose pairs of the off-diagonal elements (Z_ij, Z_ji) have mean 0, variance V_Z, and correlation P_Z. When the diagonal elements satisfy 𝔼[Z_ii] = 0, the RMT provides the “elliptic law” (41, 58, 59); that is, the eigenvalues λ(Z) = x + iy are expected to distribute uniformly (e.g., Upper panels in SI Appendix, Fig. S2) with a probability 1/(π(1 – P_Z²)) on an ellipse centered at (0, 0) as:

$$\frac{1}{S{V}_{\mathbf{Z}}}\left({\left(\frac{x}{1+P_{\mathbf{Z}}}\right)}^2+{\left(\frac{y}{1-P_{\mathbf{Z}}}\right)}^2\right)\le 1.$$ [7]

Thus, the maximum real value and the maximum absolute value of λ(Z) are expected to be the ellipse’s right edge γ_h = (1 + P_Z)(SV_Z)^1/2 and semimajor axis γ₊ = (1 + |P_Z|)(SV_Z)^1/2, respectively.

Then, the elliptic law can be extended to the nonzero mean of off-diagonal entries 𝔼[Z_ij] = μC. In this case, one of the eigenvalues λ(Z) converges to the expected value of row sums of Z, 𝔼[Σ_jZ_ij] = μC(S – 1), for large S due to the law of large numbers; the other (S – 1) eigenvalues and the center of the ellipse moves by –μC for holding the constraint Σ_iλ_i(Z) = Tr(Z) = 0 (26, 27). Therefore, an outlier eigenvalue, λ(Z) = μC(S – 1) will arise and become the dominant one if being larger than the right edge of the ellipse (1 + P_Z)(SV_Z)^1/2 – μC (26, 27). Combining these observations and the fact that a nonzero constant value of Z_ii linearly shifts the eigenvalues λ(Z), the diagonal entries of interaction matrix A were set to

$$d=-\left(\max \left(\left(1+P_{\mathbf{A}}\right)\sqrt{S{V}_{\mathbf{A}}}-\mu C,\mu C\left(S-1\right)\right)+1\right),$$ [8]

to satisfy the local stability of feasible equilibrium x* > 0 for all analyses.

Eigenvalue Distribution of Sensitivity Essence.

To analyze the absolute eigenvalues of the sensitivity essence, we first consider D^–1. As D is a deterministic matrix, the diagonal and off-diagonal entries are obtained as follows:

$${{\mathbf{D}}_{\mathit{ii}}}^{-1}=\frac{d+\mu C\left(S-2\right)}{\psi \left(d-\mu C\right)},$$

and

$${{\mathbf{D}}_{\mathit{ij}}}^{-1}=-\frac{\mu C}{\psi \left(d-\mu C\right)}.$$

Therefore, as E_ii = 0 and ψ = d + μC(S−1), the diagonal and off-diagonal entries of the sensitivity essence Φ = D^–1E can be represented by

$${\mathrm{\Phi }}_{\mathit{ii}}=\frac{\left(d+\mu C\left(S-2\right)\right){\mathbf{E}}_{\mathit{ii}}-\mu C{\sum }_{k\ne i}{\mathbf{E}}_{\mathit{ki}}}{\psi \left(d-\mu C\right)}=-\frac{\mu C{\sum }_k{\mathbf{E}}_{\mathit{ki}}}{\psi \left(d-\mu C\right)},$$

and

$${\mathrm{\Phi }}_{\mathit{ij}}=\frac{\left(d+\mu C\left(S-2\right)\right){\mathbf{E}}_{\mathit{ij}}-\mu C{\sum }_{k\ne i}{\mathbf{E}}_{\mathit{kj}}}{\psi \left(d-\mu C\right)}=\frac{\psi {\mathbf{E}}_{\mathit{ij}}-\mu C{\sum }_k{\mathbf{E}}_{\mathit{kj}}}{\psi \left(d-\mu C\right)},$$

respectively. For Φ_ij, the second term in the denominator, μCΣ_kE_kj, is constant within the same column. This indicates that the sensitivity essence Φ can further be decomposed as shown in Eq. 3, which is viewed as a random matrix with a rank-1 perturbation. In this case, two different mechanisms operate that may introduce outlier eigenvalues λ_o to λ(Φ) as described below.

To understand the first mechanism, three tools of linear algebra are introduced. The first is the resolvent of a matrix E on z ∈ ℂ, defined as

$${\mathbf{G}}_{\mathbf{E}}\left(z\right)={\left(z\mathbf{I}-\mathbf{E}\right)}^{-1}.$$

The second is the Stieltjes transformation of E, defined as follows:

$$\begin{array}{c}{g}_{\mathbf{E}}\left(z\right)=\frac{1}{S}\mathrm{Tr}\left({\mathbf{G}}_{\mathbf{E}}\left(z\right)\right)=\frac{1}{\mathit{zS}}{\displaystyle \sum _i}\frac{1}{1-{\lambda }_i/z}=\frac{1}{\mathit{zS}}{\displaystyle \sum _i}{\displaystyle \sum _{k=0}^{\infty }}{\left(\frac{{\lambda }_i}{z}\right)}^k\hfill \\ ={\displaystyle \sum _{k=0}^{\infty }}\frac{\mathbb{E}\left[{\lambda }^k\right]}{z^{k+1}}.\hfill \end{array}$$

Thus, these two matrix representations enable relating any complex number to the eigenvalue distribution of E (41). Finally, the Sherman–Morrison determinant lemma is introduced as

$$\det \left(\mathbf{Y}+\stackrel{\sim }{\mathbf{u}}{\stackrel{\sim }{\mathbf{v}}}^{\mathrm{T}}\right)=\left(1+{\stackrel{\sim }{\mathbf{v}}}^{\mathrm{T}}{\mathbf{Y}}^{-1}\stackrel{\sim }{\mathbf{u}}\right)\det \left(\mathbf{Y}\right),$$

for any invertible square matrix Y (40, 41). Then, the eigenvalues of the matrix E′ are zeros of its characteristic polynomials as well as for the outlier one λ_o if it exists as

$$\det \left({\lambda }_{\mathrm{o}}\mathbf{I}-\mathbf{E}’\right)=\det \left({\lambda }_{\mathrm{o}}\mathbf{I}-\mathbf{E}+\frac{\mu C}{\psi }\mathbf{u}{\mathbf{v}}^{\mathrm{T}}\right)=0.$$

Thus, replacing Y = λ_oI – E, Y^–1 = G_E(λ_o), ũ |= u, ṽ = ψ^–1μCv, and v = E^Tu, the matrix determinant lemma of E′ becomes

$$\left(1+\frac{\mu C}{\psi }{\mathbf{u}}^{\mathrm{T}}\mathbf{E}{\mathbf{G}}_{\mathbf{E}}\left({\lambda }_{\mathrm{o}}\right)\mathbf{u}\right)\det \left({\lambda }_{\mathrm{o}}\mathbf{I}-\mathbf{E}\right)=0.$$

Then, the following relationship is immediately derived for G_E(z):

$$\left(z\mathbf{I}-\mathbf{E}\right){\mathbf{G}}_{\mathbf{E}}\left(z\right)=\mathbf{I}⟺\mathbf{E}{\mathbf{G}}_{\mathbf{E}}\left(z\right)=z{\mathbf{G}}_{\mathbf{E}}\left(z\right)-\mathbf{I}.$$

Thus, det(λ_oI – E) ≠ 0 because λ_o ∉ λ(E) from its definition, deriving the condition of E′ to have outlier eigenvalues as follows:

$$1+\frac{\mu C}{\psi }{\mathbf{u}}^{\mathrm{T}}\left({\lambda }_{\mathrm{o}}{\mathbf{G}}_{\mathbf{E}}\left({\lambda }_{\mathrm{o}}\right)-\mathbf{I}\right)\mathbf{u}=1+\frac{\mu C}{\psi }\left({\lambda }_{\mathrm{o}}{\mathbf{u}}^{\mathrm{T}}{\mathbf{G}}_{\mathbf{E}}\left({\lambda }_{\mathrm{o}}\right)\mathbf{u}-S\right)=0.$$ [9]

In addition, u^TG_E(λ_o)u is effectively approximated by Tr(G_E(λ_o) = Sg_E(λ_o) for the condition |λ_o| > |λ(E)| (SI Appendix, SI Text). Therefore, Eq. 10 can be transformed as follows:

$$1+\frac{\mu C}{\psi }\left({\lambda }_{\mathrm{o}}S{g}_{\mathbf{E}}\left({\lambda }_{\mathrm{o}}\right)-S\right)=1+\frac{\mu C}{\psi }S\left({\displaystyle \sum _{k=0}^{\infty }}\frac{\mathbb{E}\left[{\lambda }^k\right]}{{{\lambda }_{\mathrm{o}}}^k}-1\right)=0.$$

For a random matrix E, the higher terms in the geometric series can also be ignored due to |λ_o| > |λ(E)|. Therefore, as the up-to-second-order moments of the eigenvalues are 𝔼[λ⁰] = 1, 𝔼[λ¹] = Tr(E)/S = 0, and 𝔼[λ²] = Tr(E²)/S = (S – 1)P_AV_A, the first mechanism of the outlier eigenvalues becomes as follows:

$$\begin{array}{c}1+\frac{\mu C}{\psi }S\left(S-1\right)\frac{P_{\mathbf{A}}{V}_{\mathbf{A}}}{{{\lambda }_{\mathrm{o}}}^2}+O\left({{\lambda }_{\mathrm{o}}}^{-3}\right)\hfill \\ =0⟺{\lambda }_{\mathrm{o}}\approx \pm \sqrt{-\frac{\mu C}{\psi }S\left(S-1\right)P_{\mathbf{A}}{V}_{\mathbf{A}}}.\hfill \end{array}$$

The equation means that, as –ψ^–1CS(S – 1)V_A > 0, there will be two outliers along the real part direction for μP_A > 0 and those along the imaginary part direction for μP_A < 0 if it exists (Bottom panels in SI Appendix, Fig. S2).

The second mechanism affecting the outlier eigenvalues attributes to the structure of the matrix E′. Specifically, the rank-1 perturbation ψ^–1μCuv^T sets the trace of E′ to a real constant value κ_E′ as follows:

$${\kappa }_{{\mathbf{E}}^{’}}=\mathrm{t}\mathrm{r}\left(\mathbf{E}’\right)=-\frac{\mu C}{\psi }\sum _{i=1}\sum _{j=1}{\mathbf{E}}_{\mathit{ij}}=-\frac{\mu C}{\psi }\sum _{i=1}\sum _{j>\; >i}\left({\mathbf{E}}_{\mathit{ij}}+{\mathbf{E}}_{\mathit{ji}}\right)$$

From RMT (26, 27), the bulk eigenvalues fall within the ellipse centered at the origin (SI Appendix, Fig. S2), that is, Σ_iλ_i ≈ 0 for λ_i ∉ λ_o. Therefore, combining these mechanisms, the outlier eigenvalue(s) shift to satisfy the constraint Tr(E′) = Σλ(E′) as follows:

$$\begin{array}{c}{\lambda }_{\mathrm{o}}=\left\{\begin{array}{c}\frac{{\kappa }_{{\mathbf{E}}^{’}}}{2}\pm i\sqrt{\frac{\mu C}{\psi }S\left(S-1\right)V_{\mathbf{A}}{P}_{\mathbf{A}}},\mu P_{\mathbf{A}}<0\hfill \\ {\kappa }_{{\mathbf{E}}^{’}},\mu P_{\mathbf{A}}=0\\ \frac{{\kappa }_{{\mathbf{E}}^{’}}}{2}\pm \sqrt{-\frac{\mu C}{\psi }S\left(S-1\right)V_{\mathbf{A}}{P}_{\mathbf{A}}},\mu P_{\mathbf{A}}>0\end{array}\right..\hfill \end{array}$$ [10]

Then, the eigenvalues of the sensitivity essence λ(Φ) are obtained by scaling λ(E′) by (d−μC)⁻¹.

Criterion of Network Indeterminacy for Complex Ecological Communities.

The criterion is obtained as the condition γ_max = max(|λ(Φ)|) = max(γ₊, γ_o) > 1, where γ₊ and γ_o are the semimajor axis of the elliptic law and the absolute of outlier eigenvalues λ_o, respectively. The semimajor axis of the ellipse E is determined by the larger of the horizontal and vertical semiaxis of Eq. 8 with using the variance and correlation of A as follows:

$$\left({\gamma }_{\mathrm{h}},{\gamma }_{\mathrm{v}}\right)=\left(\left(1+P_{\mathbf{A}}\right)\sqrt{S{V}_{\mathbf{A}}},\left(1-P_{\mathbf{A}}\right)\sqrt{S{V}_{\mathbf{A}}}\right),$$ [11]

where γ_h and γ_v are the horizontal and vertical semiaxis, respectively. Scaling Eq. 12 with |d – μC|^–1 yields Eq. 4.

For the outlier eigenvalues, λ_o, is determined by Eq. 11, in which κ_E′ is a random variable of the sum of E_ij; given that the mean and the variance of E_ij + E_ji are zero and 2V_A(1 + P_A), respectively, κ_E′ is expected to follow a normal distribution

$$p\left({\kappa }_{{\mathbf{E}}^{’}}\right)\sim N\left(0,\sqrt{{\left(\frac{\mu C}{\psi }\right)}^{2}S\left(S-1\right)V_{\mathbf{A}}\left(1+P_{\mathbf{A}}\right)}\right).$$

Then, the expected absolute value of κ_E′ becomes

$$\mathbb{E}\left[\left|{\kappa }_{{\mathbf{E}}^{’}}\right|\right]={\int }_{-\infty }^{\infty }\left|\kappa \right|p\left(\kappa \right)d\kappa =\left|\frac{\mu C}{\psi }\right|\sqrt{\frac{2}{\pi }S\left(S-1\right)V_{\mathbf{A}}\left(1+P_{\mathbf{A}}\right)}.$$

Substituting this equation into Eq. 11 yields the expectation of r_o = |λ_o| as follows:

$$\begin{array}{c}{\gamma }_{\mathrm{o}}=\left\{\begin{array}{c}\left|\frac{\mu C}{\psi }\right|\sqrt{S\left(S-1\right)V_{\mathbf{A}}}\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }+\frac{\psi }{\mu C}{P}_{\mathbf{A}}},\mu P_{\mathbf{A}}<0\hfill \\ \left|\frac{\mu C}{\psi }\right|\sqrt{S\left(S-1\right)V_{\mathbf{A}}}\sqrt{\frac{2}{\pi }},\mu P_{\mathbf{A}}=0\\ \left|\frac{\mu C}{\psi }\right|\sqrt{S\left(S-1\right)V_{\mathbf{A}}}\left(\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }}+\sqrt{-\frac{\psi }{\mu C}{P}_{\mathbf{A}}}\right),\mu P_{\mathbf{A}}>0\end{array}\right..\hfill \end{array}$$ [12]

Therefore, empirical and theoretical γ_o of the sensitivity essence are obtained by scaling Eqs. 10 and 12 by |d−μC|⁻¹.

These equations can be simplified in the limit S → ∞. Specifically, from the analytical settings and Eq. 8, the detailed condition for the outlier eigenvalues to be dominant becomes

$$\begin{array}{c}\mu C\left(S-1\right)>\left(1+P_{\mathbf{A}}\right)\sqrt{S{V}_{\mathbf{A}}}-\mu C⟺\mu \hfill \\ >\sigma \sqrt{\frac{\sqrt{SC\left(SC-4\left(1-C\right)\left(1-{\rho }^2\right)\right)}-\left(SC-4\left(1-C\right)\left(1+\rho \right)\right)}{2\left(1-C\right)\left(SC-4\left(1-C\right)\right)}}.\hfill \end{array}$$

As the right-hand side of this inequality converges into zero in the limit S → ∞, the diagonal element d and the accompanied model variable ψ is

$$\begin{array}{c}\underset{S\to \infty }{\text{lim}}d=\left\{\begin{array}{c}-\left(1+P_{\mathbf{A}}\right)\sqrt{S{V}_{\mathbf{A}}},\mu \le 0\hfill \\ -\mu SC,\mu >0\end{array}\right.,\hfill \end{array}$$

and

$$\begin{array}{c}\underset{S\to \infty }{\text{lim}}\psi =\underset{S\to \infty }{\text{lim}}\left(d+\mu C\left(S-1\right)\right)=\left\{\begin{array}{c}\mu SC,\mu \le 0\hfill \\ -1,\mu >0\end{array}\right.,\hfill \end{array}$$

respectively. Therefore, the semimajor axis of the ellipse becomes as follows:

$$\begin{array}{c}\underset{S\to \infty }{\text{lim}}{\gamma }_{+}=\underset{S\to \infty }{\text{lim}}\frac{1+\left|P_{\mathbf{A}}\right|}{\left|d-\mu C\right|}\sqrt{S{V}_{\mathbf{A}}}=\left\{\begin{array}{c}\frac{1+\left|P_{\mathbf{A}}\right|}{1+P_{\mathbf{A}}},\mu \le 0\hfill \\ \frac{1+\left|P_{\mathbf{A}}\right|}{\mu C}\sqrt{\frac{V_{\mathbf{A}}}{S}}\to 0,\mu >0\end{array}\right..\hfill \end{array}$$

Then, we consider the outlier eigenvalues. For μ ≤ 0, the above arguments yield the following equation:

$$\underset{S\to \infty }{\lim }\left|\frac{\mu C}{\psi }\right|\frac{\sqrt{S\left(S-1\right)V_{\mathbf{A}}}}{\left|d-\mu C\right|}=\frac{1}{\left(1+P_{\mathbf{A}}\right)\sqrt{S}}.$$

Thus, Eq. 5 is transformed as follows:

$$\begin{array}{c}\underset{S\to \infty }{\text{lim}}{\gamma }_{\mathrm{o}}=\left\{\begin{array}{c}\frac{1}{\left(1+P_{\mathbf{A}}\right)\sqrt{S}}\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }+\frac{\psi }{\mu C}{P}_{\mathbf{A}}}\to \frac{\sqrt{-P_{\mathbf{A}}}}{1+P_{\mathbf{A}}},P_{\mathbf{A}}<0\hfill \\ \frac{1}{\left(1+P_{\mathbf{A}}\right)\sqrt{S}}\sqrt{\frac{2}{\pi }}\to 0,P_{\mathbf{A}}=0\\ \frac{1}{\left(1+P_{\mathbf{A}}\right)\sqrt{S}}\left(\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }}+\sqrt{-\frac{\psi }{\mu C}{P}_{\mathbf{A}}}\right)\to \frac{\sqrt{P_{\mathbf{A}}}}{1+P_{\mathbf{A}}},P_{\mathbf{A}}>0\end{array}\right..\hfill \end{array}$$

This equation means that γ_o < γ₊ for any P_A in the condition μ ≤ 0. For μ > 0, the following equations are similarly derived:

$$\underset{S\to \infty }{\lim }\left|\frac{\mu C}{\psi }\right|\frac{\sqrt{S\left(S-1\right)V_{\mathbf{A}}}}{\left|d-\mu C\right|}=\sqrt{V_{\mathbf{A}}},$$

and

$$\begin{array}{c}\underset{S\to \infty }{\text{lim}}{\gamma }_{\mathrm{o}}=\left\{\begin{array}{c}\sqrt{V_{\mathbf{A}}}\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }-\frac{P_{\mathbf{A}}}{\mu C}},P_{\mathbf{A}}<0\hfill \\ \sqrt{V_{\mathbf{A}}}\sqrt{\frac{2}{\pi }},P_{\mathbf{A}}=0\\ \sqrt{V_{\mathbf{A}}}\left(\sqrt{\frac{1+P_{\mathbf{A}}}{2\pi }}+\sqrt{\frac{P_{\mathbf{A}}}{\mu C}}\right),P_{\mathbf{A}}>0\end{array}\right..\hfill \end{array}$$

Thus, γ_o is always larger than γ₊ for any P_A in positive networks μ > 0. From these findings, the criterion of ecological indeterminacy in the limit S → ∞ can be obtained as Eq. 6.

Volume of Eigenspace Causing Network Indeterminacy.

For networks following the elliptic law, the value of eigenvalues larger than the criterion, that is, |λ| > 1, can be evaluated by the probability density function of the ellipse. The probability density function is the derivative of its cumulative distribution. For an S-dimensional real matrix Z, whose eigenvalues follow the elliptic law of Eq. 8, λ(Z)s are distributed on the ellipse with an equal probability of an inverse of the ellipse’s area as (πγ_hγ_v)^–1. Thus, the cumulative distribution function is obtained as the range within the ellipse below a given positive value of 0 ≤ γ ≤ γ₊. Such an area is a circle with a radius of γ < γ_– = min(γ_h, γ_v). Otherwise, the area is an intersection of the ellipse and the circle, which is decomposed into elliptic sectors with angle θ (0 < θ < π/2) and circular sectors with angle π/2 – θ. From Eq. 8, their geometry depends on the value of P_A. We considered the case of –1 < P < 0 because the ellipse of 0 ≤ P < 1 coincides with a π/2 rotation of –P. That is, the area of the circular sector, S_C(γ, θ) and that of the elliptic sector, S_E(γ, θ), which is calculated by scaling S_C(γ_–, tan^–1(γ_–/γ₊tanθ) by a factor γ_–/γ₊, are obtained as follows:

$$S_{\mathrm{C}}\left(\gamma ,\theta \right)=\frac{{\gamma }^2}{2}\left(\frac{\pi }{2}-\theta \right),$$

and

$$S_{\mathrm{E}}\left(\gamma ,\theta \right)=\frac{{\gamma }_{-}{\gamma }_{+}}{2}{\text{tan}}^{-1}\left(\frac{{\gamma }_{-}}{{\gamma }_{+}}\text{tan}\theta \right),$$

respectively. In addition, inserting the intersection point between the circular and elliptic sectors, (x, y) = (γcosθ, γsinθ), into Eq. 8, yields the following relationship between γ and θ,

$$\gamma =\frac{{\gamma }_{-}{\gamma }_{+}}{\sqrt{{{\gamma }_{+}}^2{\text{cos}}^2\theta +{{\gamma }_{-}}^2{\text{sin}}^2\theta }}⟺\theta ={\text{cos}}^{-1}\left(\frac{{\gamma }_{-}}{\gamma }\sqrt{\frac{{{\gamma }_{+}}^2-{\gamma }^2}{{{\gamma }_{+}}^2-{{\gamma }_{-}}^2}}\right).$$

Therefore, replacing the angle θ occurring in S_C and S_E, the cumulative distribution function of |λ(Z)| is P(|λ| ≤ γ) = γ²(γ_–γ₊)^–1 for 0 ≤ γ ≤ γ_– and, for γ_– ≤ γ ≤ γ₊, is obtained as follows:

$$\begin{array}{c}P\left(\left|\lambda \right|\le \gamma \right)=\frac{4\left(S_{\mathrm{C}}\left(\gamma \right)+S_{\mathrm{E}}\left(\gamma \right)\right)}{\pi {\gamma }_{-}{\gamma }_{+}}\hfill \\ =\frac{{\gamma }^2}{{\gamma }_{-}{\gamma }_{+}}\left(1-\frac{\pi }{2}{\text{cos}}^{-1}\left(\frac{{\gamma }_{-}}{\gamma }\sqrt{\frac{{{\gamma }_{+}}^2-{\gamma }^2}{{{\gamma }_{+}}^2-{{\gamma }_{-}}^2}}\right)\right)\hfill \\ +\frac{\pi }{2}{\tan }^{-1}\left(\sqrt{\frac{{\gamma }^2-{{\gamma }_{-}}^2}{{{\gamma }_{+}}^2-{\gamma }^2}}\right).\hfill \end{array}$$

By differentiating these equations by γ, the probability density function of |λ| is obtained as follows:

$$\begin{array}{c}P\left(\gamma \right)=\left\{\begin{array}{c}\frac{2\gamma }{{\gamma }_{-}{\gamma }_{+}},0\le \gamma <{\gamma }_{-}\hfill \\ \frac{2\gamma }{{\gamma }_{-}{\gamma }_{+}}\left(1-\frac{2}{\pi }{\text{cos}}^{-1}\left(\frac{{\gamma }_{-}}{\gamma }\sqrt{\frac{{{\gamma }_{+}}^2-{\gamma }^2}{{{\gamma }_{+}}^2-{{\gamma }_{-}}^2}}\right)\right),{\gamma }_{-}\le \gamma \le {\gamma }_{+}\end{array}\right..\hfill \end{array}$$

Combining these observations and the fact that the expected value of γ within (γ₁, γ₂) is calculated by

$$\mathbb{E}\left[{\gamma }_2>\; >\gamma >\; >{\gamma }_1\right]={\int }_{{\gamma }_1}^{{\gamma }_2}\gamma P\left(\gamma \right)\mathrm{d}\gamma ,$$

the theoretical volume of |λ| > 1, Ξ = 𝔼[γ₊ > γ > 1], is obtained as follows:

$$\begin{array}{c}\mathrm{\Xi }=\left\{\begin{array}{c}0,{\gamma }_{-}<{\gamma }_{+}<1\hfill \\ \frac{4{\gamma }_{-}}{3\pi }\left(E\left(\frac{\pi }{2},K\right)-E\left({\text{tan}}^{-1}{R}_1,K\right)-\frac{1}{{{\gamma }_{-}}^2{\gamma }_{+}}\left(\frac{\pi }{2}-{\text{cos}}^{-1}{R}_2\right)\right),{\gamma }_{-}<1<{\gamma }_{+}\\ \frac{4{\gamma }_{-}}{3\pi }\left(E\left(\frac{\pi }{2},K\right)-\frac{\pi }{2{{\gamma }_{-}}^2{\gamma }_{+}}\right),1<{\gamma }_{-}<{\gamma }_{+}\end{array}\right.,\hfill \end{array}$$ [13]

where the variables K, R₁, and R₂ are

$$\left\{\begin{array}{c}K=1-{\left(\frac{{\gamma }_{-}}{{\gamma }_{+}}\right)}^2\\ R_1=\sqrt{\frac{1-{{\gamma }_{-}}^2}{{{\gamma }_{+}}^2-1}}\\ R_2={\gamma }_{-}\sqrt{\frac{{{\gamma }_{+}}^2-1}{{{\gamma }_{+}}^2-{{\gamma }_{-}}^2}}\end{array}\right.,$$

and E(x, k) represents the second kind of elliptic integral as follows:

$$E\left(x,k\right)={\int }_0^k\sqrt{1-k^2{\sin }^2}\theta \mathrm{d}\theta .$$

If the outlier eigenvalues λ_o arise, Ξ is obtained by considering Eq. 11 into Eq. 14.

Simulation Analysis of the Indeterminacy Criterion for Various Ecological Communities.

To verify the ecological indeterminacy criterion Eq. 6, we first performed numerical simulation in which the interaction matrix A was constructed with different parameters (the results are summarized in Fig. 2). Specifically, the mean, variance, and correlation of linked interactions were randomly assigned as μ ~ U(–0.5, 0.5), σ ~ U(0.0, 1.0), and ρ ~ U(–0.5, 0.5), respectively (U(a, b) stands for a uniform distribution ranging from a to b for a < b). The off-diagonal pairs (a_ij, a_ji) were sampled from a bivariate normal distribution with these parameters, and the diagonal entries d were determined using Eq. 9. The community size and connectance were set to (S, C) = (250, 0.5). The simulation was iterated 1,000 times. With the randomly generated A, we investigated the relationship among the theoretical values of γ_max, max(|λ(Φ)|) for each run, and the correlation coefficient ρ_k between the i-j element of the inverse interaction matrix {A^–1}_ij, and its kth order approximation {Â^–1}_ij calculated using Eq. 2.

We further performed the effects of ecological complexity on γ_max. The interaction matrix was randomly generated by varying the community size S from 10 to 310 by 10 and from 900 to 1,000 by 50, and the connectance C with 0.4, 0.7, and 1.0 under different parameter sets of μ and P_A. The simulation was iterated 100 times, and we calculated max(|λ(Φ)|) for each run and compared its theoretical value γ_max (Fig. 3). Finally, the numerical simulations tested the relationship between the volume of eigenspace associated with the eigenvalues satisfying |λ(Φ)| > 1 and the prediction skill ρ_k of the perturbation outcomes. Specifically, an interaction matrix A was generated by randomly assigning parameters C, σ, μ, and ρ to satisfy the theoretical value of γ_max = 1.2, 1.3, and 1.4, and we compared the relative amount of |λ(Φ)| > 1 calculated for each run with its theoretical value Ξ. This procedure was iterated 100 times. Then, 250 randomly generated input vector b satisfying ||b||² = 1 were added to the GLV system, and the outcome of press perturbation was calculated by the true A^–1 and its approximation with Eq. 2 with order k = 5 (Fig. 5B). All simulation analyses were performed using R version 4.3.2 (R Development Core Team 2023); R source code used in this study is available in (60).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data and souce code are included in Zenodo (60).