Inferring ecosystem networks as information flows

Two species unidirectional coupling ecosystem

This bio-inspired ecosystem S((beta _{xy}=0), (beta _{yx})) describing the unidirectional coupling is run for 1000 time steps for reaching stationarity, generating a set of 1000 points long time-series dependent on (beta _{yx}). This means that species X has an increasing effect on Y with the increase of (beta _{yx}), but Y has no effect on X. Both CCM and the proposed OIF model are separately used to quantify the potential causality between species X and Y. The inferred causality dependent on (beta _{yx}) only (as a physical interaction) is shown in Fig. 2A. (beta), (rho) and TE have different units; specifically (beta) and (rho) are dimensionless while TE is measured in bits or nats (a logarithmic unit of information or entropy). Therefore, any comparison is done considering gradients of change when these variables vary together rather than making comparisons between absolute values which are meaningless. Figure 2A shows that under the condition of (beta _{xy})=0, results of “Y to X” (i.e. the estimated effect on Y on X) is close to 0 for the OIF model ((TE_{Y rightarrow X}(beta _{yx}))) that precisely describe the no-effect of Y on X. “X to Y” ((TE_{X rightarrow Y}(beta _{yx}))) well tracks the increasing strength of the effect of X on Y for increasing values of the physical interaction (beta _{yx}) embedded into the mathematical model. However, considering results of the CCM model, “Y to X” ((rho _{Y rightarrow X}(beta _{yx}))) presents non obvious (and likely wrong) non-zero values with higher fluctuations compared to (TE_{Y rightarrow X}(beta _{yx})) especially for lower values of (beta _{yx}). This erroneous estimates of CCM is likely related to the need of CCM for convergence. For CCM, “X to Y” (((rho _{X rightarrow Y}(beta _{yx})))) shows an increasing trend for increasing values of (beta _{yx}) and decreasing when (beta _{yx}) is greater than (sim)0.5 non-trivially. In consideration of these results for the unidirectional coupling ecosystem, the OIF model performs better over CCM in terms of unidirectional causality inference.

Two species bidirectional coupling ecosystem

In this case, the effect between two species is bidirectional. Species X has an effect on species Y and vice versa. The univariate dynamical systems S(0.2/0.5/0.8, (beta _{yx})) are run for 1000 time steps under the same conditions determined by (beta _{xy}). Certainly this situation is fictional since in real ecosystems the interaction strength is changing when other interacting species change their interactions.Thus, keeping one interaction fixed around one value is a strong unrealistic simplification (analogous of one-factor at-a-time sensitivity analyses) but it is a toy model that allows to verify the power of network inference models. These models generate three sets of 1000 points long time-series dependent of (beta _{yx}) for each fixed (beta _{xy}). OIF and CCM are used to infer “causality” between X and Y—in the form of (rho) and TE—and compare that against the real embedded interaction (beta _{yx}) and (beta _{xy}) shown in Fig. 2B,C,D. Considering all results of Fig. 2 corresponding to fixed (beta _{xy})s, the correlation coefficient (rho) yielded from CCM and TE from OIF are both able to track the strength of causal trajectories. However, TE seems to perform better in term of ability to infer fine-scale changes in interactions. In particular, considering Fig. 2D (right plot), higher (TE_{yx}) higher for low (beta _{yx}) makes sense because (beta _{xy} > beta _{yx}) that means Y has a larger influence on X than vice versa and then Y is able to predict X. Additionally, TE does not suffer of convergence problems; specifically, considering Fig. 2A (left plot), higher (rho) for small (beta _{yx}) is not sensical and that is likely related to convergence problems of CCM.

Considering all results of Fig. 2 corresponding to fixed (beta _{xy})s, the correlation coefficient (rho) yielded from CCM and TE from OIF are both able to track the strength of causal trajectories. Ideally, the causality from Y to X is a constant since (beta _{xy}) is a fixed value for each case. In this figure, the red curve in the right panel representing the OIF-inferred (TE-based) causality from Y to X is higher for greater (beta _{xy})s, while red curves representing CCM-inferred ((rho)) causality in the left panel present higher fluctuations especially for lower (beta _{yx}). For the causality from X to Y determined by (beta _{yx}) in the mathematical model, theoretically speaking, the causality from X to Y should monotonously grow when (beta _{yx}) increases from 0 to 1. In Fig. 2, blue curves in the right panel representing the OIF-inferred (TE-based) causality from X to Y present monotonously increasing features as a whole with the increasing (beta _{yx}), while those from CCM model ((rho)) do not and show considerable fluctuations.Therefore, OIF outperforms CCM in terms of the ability to infer the fine-scale changes in causality. In particular, considering Fig. 2D (right plot), higher (TE_{yx}) higher for low (beta _{yx}) makes sense because (beta _{xy} > beta _{yx}) that means Y has a larger influence on X than vice versa and then Y is able to predict X. Additionally, TE does not suffer of convergence problems; specifically, considering Fig. 2A (left plot), higher (rho) for small (beta _{yx}) is not sensical and that is likely related to convergence problems of CCM.

Additionally, (rho _{Y rightarrow X}(beta _{yx})) shows higher fluctuations on average especially for the condition of lower (beta _{yx})s compared to (TE_{Y rightarrow X}(beta _{yx})). When considering the effect of X on Y that is a function of (beta _{yx}) for CCM, (rho _{X rightarrow Y}) reaches an extreme value at around (beta _{yx}) = 0.5 and then declines for larger values of (beta _{yx}). This is not consistent with the expected effect of X on Y that should be proportional to (beta _{yx}) embedded into the mathematical model. The ability of (rho) to reflect the proportional relationship between the effect of X on Y (manifested by (beta _{yx})) vanishes for high (beta _{xy})s due to unexpected and somewhat inconspicuous changes in (rho _{X rightarrow Y}) for larger (beta _{yx}). In simple words, the expected increasing trend of (rho) is lost for larger (beta _{xy}) that is counterintuitive. On the other side, (TE_{X rightarrow Y}(beta _{yx})) invariably maintains an increasing trend for increasing values of (beta _{xy}). OIF is also performing better than CCM when predicting higher average values of (TE_{Y rightarrow X}) for increasing values of (beta _{xy}) (red curves in Fig. 2A–D, right plots) as expected by the fixed effect in the mathematical model of Y on X. These results suggest that when compared to (rho) of CCM, TE can track well the causal interactions over (beta _{yx}) with higher performance and without considering the convergence requirement of CCM. CCM needs to consider the length of time series that makes (rho _{X rightarrow Y}(beta _{yx})) convergent to a stable value, but uncertain for large differences in time-series length of (X,Y) and sensitive to short time series.

In more realistic settings for real ecosystems (and in analogy to global sensitivity analyses) when (beta _{xy}) and (beta _{yx}) are both considered as arguments of the two-variable (X,Y) bio-inspired model, the simulated ecosystem becomes a truly bivariate system, yet yielding complexity but more interest into the causality inference (Fig. 3). The dynamical system S((beta _{xy}), (beta _{yx})) was generated for 800 time steps under the same conditions mentioned above. We generated the datasets that allowed us to study linear and non-linear predictability indicators for inferring the embedded physical interactions. Specifically, we measure undirected linear correlation coefficient (corr_{X;Y}(beta _{xy},beta _{yx})), non-linear undirected mutual information (MI_{X;Y}(beta _{xy},beta _{yx})), directed non-linear correlation coefficient (rho _{X rightarrow Y}(beta _{xy},beta _{yx})) and (rho _{Y rightarrow X}(beta _{xy},beta _{yx})), and non-linear directed transfer entropy (TE_{X rightarrow Y}(beta _{xy},beta _{yx})) and (TE_{Y rightarrow X}(beta _{xy},beta _{yx})) as shown in Fig. 3. These 2D phase-space maps in Fig. 3 show strikingly similar patterns for classical linear correlation coefficients, MI, (rho) of CCM and TE of OIF which underline the fact that all methods are able to infer the interdependence patterns of interacting variables explicitly defined by (beta _{xy}) and (beta _{yx}). The color of phase-space maps is proportional to the inferred interaction between X and Y when the mutual physical interactions are varying according to the mathematical model in Eq. (1). In Fig. 3, even though phase-space maps of undirected (corr_{X;Y}(beta _{xy},beta _{yx})) and (MI_{X;Y}(beta _{xy},beta _{yx})) present similar patterns (in value organization and not value range) to those of directed (rho) and TE, neither (corr_{X;Y}((beta _{xy},beta _{yx})) and (MI_{X;Y}((beta _{xy},beta _{yx})) provide information about the direction of causality. As expected MI shows the opposite pattern of the average TE due to the fact that MI is the amount of shared information (or similarity) versus the amount of divergent information (divergence and asynchronicity) between X and Y.

In a biological sense TE should be interpreted as the probability of likely uncooperative dynamics (leading to or driven by environmental or biological heterogeneity) while MI as the probability of cooperative dynamics (leading to or driven by homogeneity). Here we refer to cooperative and uncooperative interactions based on the similarity or dissimilarity in pair dynamics manifested by species abundance fluctuations. For instance divergence and asynchronicity (that define TE) in pair species dynamics manifest uncooperative interactions. The balance of cooperative and uncooperative interactions can result into net interactions at the ecosystem scale manifesting neutral patterns, or net interactions may lead to niche patterns biased toward strong environmental or biological factors⁵². Certainly, cooperation in a biological sense should be interpreted on a case by case basis. In a broader uncertainty propagation perspective⁴⁹, “cooperation” between variables means that variables contribute similarly to the uncertainty propagation, while “competition” means that one variable is predominant over the other in terms of magnitude of effects since TE is proportional to the magnitude rather than the frequency of effects. For the former case the total entropy of the system is higher than the latter case. Interestingly, correlation (text{ corr }(beta )), (rho) and TE show similar patterns in both organization and value range (but not in singular values of course), which sheds some important conclusions about the similarity and divergence of these methods as well as their capacity and limitations in characterizing non-linear systems.

When comparing the phase-space patterns from CCM and OIF (displaying (rho) and TE) a more colorful and informative pattern is revealed by OIF. This means that TE gives a better gradient when tracking the increasing strength of causality for increasing values of (beta _{xy}) and (beta _{yx}). When comparing the phase-space patterns for the two causal directions of (“X rightarrow Y”) and (“Y rightarrow X”), phase-space maps from CCM are very similar, while those from TE present apparent differences in the strength of effects for the two opposite direction of interaction. Therefore, OIF is more sensitive to the direction of interaction compared to CCM when detecting directional causality.

These results imply that TE performs better to distinguish directional embedded physical interactions (that are dependent on direct interactions (beta)-s, species growth rate (r_x) and (r_y), and contingent values X(t) and Y(t) determining the total interaction as seen in the model of Eq. (1)) in the species causal relationships. It should be emphasized how all linear and non-linear interaction indicators are inferring the total interaction and not only those exerted by (beta)-s. In a broad uncertainty purview⁴⁹ the importance of these three factors ((beta)-s, r-s and X(t)/Y(t)) depends on their values and probability distributions that define the dynamics of the system; dynamics such as defined by the regions identified by patterns in Fig. 3 for the predator-prey system in Eq. (1). In principle, the higher the difference between these three interaction factors in the species considered, the higher the predictability and sensitivity of OIF. Figure 4 highlights three different dynamics corresponding to the TE blue, green and red regions in Fig. 3.

In all dynamical states represented by Fig. 3, species are interacting with different magnitudes and this defines distinct network topologies. Three prototypical dynamics are shown in Fig. 4 with colors representative of (rho) and TE in Fig. 3. The “blue” deterministic dynamics has very high synchronicity and no divergence considering variable fluctuation range (the gap is deterministic and related to the numerically imposed (u=1)), as well as no linear correlation between non-lagged variables. In perfect synchrony one would have one point in the phase-space. Thus, absence of correlation does not imply complete decoupling of species but it can be a sign of small interactions. The “green” dynamics shows a relatively high synchronicity and medium divergence. In the phase-space of synchronous values of X and Y a correlation is observed with relatively small fluctuations because the divergence is small. Lastly, the “red” dynamics shows a relatively high asynchronicity and divergence. The stochasticity is higher than previous dynamics and the “mirage correlation” in the phase space has higher variance. Time-dependent mirage correlations in sign and magnitude mean that correlation (that may suggest common dynamics in a linear framework) does not imply similarity in dynamics for the two species. Non-linearity is higher from blue to red dynamics as well as predictability but lower absolute information entropy. Then, it is safe to say that linear dynamics (or small stochasticity) does not imply higher predictability.

Real-world sardine–anchovy-temperature ecosystem

CCM and proposed OIF model are also used for a real-world fishery ecosystem to infer potential causal interactions between Pacific sardines (Sardinops sagax) landings, Northern anchovies (Engraulis mordax) and sea-surface temperature (SST) recorded at Scripps Pier and Newport Pier, California. Sardines and anchovies do not interact physically (or the interaction is low in number), while both of them are influenced by the external environmental SST that is the external forcing. To quantify the likely causal interactions between species and SST based on real data, we use CCM considering the length of time series for convergence of (rho), as well as OIF considering a set of time delays for acquiring stable values of inferred interactions TEs.

Results from CCM in Fig. 5A (plots from top to bottom) show that no significant interaction can be claimed between sardines and anchovies, as well as from sardines or anchovies in the SST manifold which expectedly indicates that neither sardines nor anchovies affect SST. This latter results, considering its biological plausibility should be taken as one validation criteria of predictive models, or complimentary as a test for anomaly detection of spurious interactions. The reverse effect of SST on sardines and anchovies can be quantitatively detected with the correlation coefficient (rho) as well as TE. Although the calculated causations between SST and sardines or anchovies are moderate, CCM is able to provide a good performance in causality inference when the length of time series used is long enough due to convergence requirement.

Figure 5B shows OIF’s results of inferred causal interactions between sardines, anchovies and SST dependent on the time delay u. For sardines and anchovies, OIF exposes bidirectional interactions that are actually biologically plausible, especially when both populations coexist in the same habitat, versus the results of CCM that infer (rho =0). Ecologically speaking, even though fish populations do not directly influence sea temperature, we can find some clues about SST in fish populations influenced by SST. These clues can be interpreted as information of SST encoded in fish populations over abundance time records. So, observations of fish populations can be used to inversely predict the change of SST; this can be interpreted as “reverse predictability” (or “biological hindcasting”) in a similar way of when predicting historical climate change from ice cores. This information is captured by OIF, leading to nonzero values of TE from fish populations to SST. In this regard, we emphasize the distinction between direct and indirect (reverse) information flow, where direct information flow is most of the time larger and signifies causality (e.g. of SST for sardine and anchovies), and indirect (reverse) information flow that is typically smaller and signifies predictability (e.g. sardine and anchovies for ocean fluctuations). It is possible—especially for linear systems where an effect is observed immediately after a change—that information of SST encoded in fish populations is high if the interdependence, represented by the functional time delay u, of the environment-biota is small. However, for highly non-linear systems such as fishes and the ocean, changes in temperature may take a while before being encoded into fish population abundance⁵³. Thus, it is correct that the highest values of TE are for high u. Values of TE for small u-s are numerical artifacts related to systematic errors leading to overestimation of interactions that are time-delayed eventually. One way to circumvent this problem, largely present for short time series, would be to extend time series by conserving their dynamics (see Li and Convertino³⁹) or to bound the calculation of TE only for the u that maximizes the Mutual Information; this would provide an average u within a range where TE is approximately invariant. Thus, for the effect of external SST on sardines and anchovies, OIF model gives unstable causal interactions with bias for lower time delays due to known dependencies of TE on u (such as cross-correlation for instance) that establishes the temporal lag on which the dependency between X and Y is evaluated. In a sense, plots in Fig. 5B are like cross-variograms for the pairs of variables considered. TE becomes stable when the time delay is located in an appropriate range. It means that OIF requires an optimal time delay that makes results of the causality inference robust and that is related to optimal TEs (as highlighted in Li and Convertino³⁹ and Servadio and Convertino⁴⁹) that defines the most likely interdependency between variables for the u with the highest predictability. The fact that TE of sardine and anchovies to SST is high for same small ranges of u may be also a byproduct of data sampling, i.e., fish and SST sampling locations are different (fish abundance is actually about fish landings) and that can introduce spurious correlations/causation. Overall, these findings suggest that the OIF model provides more plausible results, but it requires careful selection of optimal time delays.

Figure S1 shows the relationships between normalized (rho) and TE estimated for all selected values of L and u of pairs in Fig. 5 (sardine-anchovy, sardine and SST, anchovy and SST). These plots show opposite results than the proportionality between (rho) and TE in Fig. 3 because non-optimal values are used, that is non-convergent (rho)-s and suboptimal TE during the interaction inference procedure (Fig. S1). TE for too small u-s determines overestimation of interactions due to the implicit assumptions that variables have an immediate effect on each other and that is not always the case as highlighted by the vast time-lagged determined non-linear regions in Fig. 3. If “transitory” values of (rho) for small L are disregarded, as well as TEs for small u-s, the relationship between (rho) and TE shows a correct linear proportionality.

Real-world multispecies ecosystem

Interactions between fish species living in the Maizuru bay are intimately related to external environmental factors of the ecosystem where they live, the number of species living in this region considering also the unreported ones and biological species interactions, which lead to a complex dynamical nonlinear system. In Fig. 6 the network of observed fish species (Table S1) is reported where only the interactions considered in Ushio et al.³⁷ for the CCM are reported. This is because the goal is to compare the CCM inferred network to the TE-based one based on abundance. Figure 7 shows the temporal fluctuations of abundance and the functional interaction matrices of (rho) and TE. In this paper we study and compare average ecosystem networks for the whole time period considered but dynamical networks can also be extracted via time-fluctuating (rho) and TE as shown in Fig. S3. These dynamical networks can be useful for studying how diversity is changing over time and ecosystem stability (Figs. S4, S6–S7) as well as understanding the relationship between (rho) and TE (Fig. S5). In the network of Fig. 6 the color and width of links are proportional to the magnitude of TE (Table S2); for the former a red-blue scale is adopted where the red/blue is for the highest/lowest TEs. The diameter is proportional to the Shannon entropy of the species abundance pdf (Table S3). The color of nodes is proportional to the structural node degree, i.e. how many species are interconnected to others. Therefore, the network in Fig. 6 is focusing on uncooperative species whose divergence and/or asynchronicity (that is a predominant factor in determining TE over divergence) is large. Yet, the connected species are rarely but strongly interacting in magnitude rather than frequently and weakly (i.e., cooperative or similar dynamics). Additionally, the species with the smallest variance in abundance are characterized by the smallest Shannon entropy (smallest nodes) and more power-law distribution although the latter is not a stringent requirement since both pdf shape and abundance range (in particular maximum abundance) play a role in the magnitude of entropy. Average entropy such as average abundance are quantities with limited utility in understanding the dynamics of an ecosystem as well as ecological function. Nonetheless, species with high average abundance (e.g. species 5) have a very regular seasonal oscillations and the largest number of interactions with divergent species. This dynamics is expected considering the population size of these dominant species and their synchrony with regular environmental fluctuations.

Figure S2 shows that the strongest linear correlation is for the most divergent and asynchronous species (from species 4–9) for which both (rho) and TE are the highest (Fig. 7B,C). This confirms the results of Fig. 3 and the fact that competition (or dynamical diversity more generally) increases predictability. This also highlights the fact that linear correlation among state variables does not imply synchronicity or dynamic similarity as commonly assumed. The interaction matrices in Fig. 7B, C confirm that TE has the ability to infer a larger gradient of interactions than (rho) and the total entropy of the TE matrix is lower than (rho). Pairwise the inferred interaction values by CCM and OIF are different but (rho) and TE patterns appear clearly similar and yet proportional to each other.

CCM and OIF models are applied to calculate the potential interactions between all pairs of species. Figure 7B,C show interaction matrices describing the normalized (rho) from CCM and TE from OIF model of all pairwise species, respectively. The greater the strength of likely interaction, the warmer the color. These results demonstrate that CCM and OIF model present similar patterns for the interaction matrices in terms of interaction distribution, gradient and magnitude in order of similarity. This indicates that both CCM and OIF are able to infer the potentially causal relationships between species. Compared to the CCM interaction heatmap the OIF heatmap presents larger gradients of inferred interactions that highlight the divergence and asynchrony in fish populations of species 4–9 from other species. This difference can be observed in Figure S2 that shows the strongest linear correlations for the most divergent and asynchronous species (4–9). It is worth noting that species 4-8 are all native species (See Table SS11). Therefore, despite the patterns of interactions of CCM and TE are similar, CCM allows one a better identification of clusters of species with similar or distinct interaction ranges. Additionally TE estimates some weak observed interactions such as of species 2 (E. japonicus) with others, while CCM essentially considers null interactions for these species.

Precisely, the most interacting species (4–9) are the most divergent and asynchronous species (with respect to the whole community) as well as diverse in terms of values of abundance; these species form the ”collective core” that is likely determining the stability of the ecosystem. Interestingly the number of these species is relatively small and it confirms results of other studies (see^{39, 54,55,56,57}) showing that the number of species with weak interactions is much larger. Theory suggests that this pattern promotes stability as weak interactors dampen the destabilizing potential of strong interactors⁵⁴. Mediated cooperation (e.g. by many “weakly” interacting competitors) as shown by Tu et al.⁵² promotes biodiversity and diversity increases stability. When considering abundance values (at same time steps) of collective core species (Fig. S2) these species are linearly related and this increases their mutual predictability by either using linear or non-linear models based on correlation coefficient and TE. This proves that non-linearity increases predictability.

The choice of the optimal u that maximizes MI leads to the optimal TE model and resultant interaction network. The observed u over time is really small (Fig. S9) and this signifies how likely the ecosystem has small memory and responds quickly to rapid changes, or the information of change is carried over time by ecosystem’s interactions which lead to accurate short-term forecasting. In other words, temperature-induced changes may take long time but the information of change is replicated at short time periods. The chosen time delay (u =1) corresponds to the species sampling of two weeks. Note that values of u are also dependent on the data resolution and they are strongly related to fluctuations rather than absolute (alpha)-diversity value. Thus, while biodiversity may fluctuate rapidly in time, value of (alpha)-diversity for seasons or longer time periods can be more stable and manifesting higher memory (representative of u for the whole ecosystem) than the one between species pairs (related to pair’s u). Short-term catastrophic dynamics (for instance related to dramatic habitat change, sudden invasions, extinctions or rapid adaptations) may lead to irreversible shifts in interactions (strength and sign); this, in turn can affect biodiversity patterns that are completely uninformed by past dynamics. Thus, there is certainly a limit to predictability and to the validity of time delays which can change very rapidly. However, we insist in emphasizing that models are predictive tools and predictions are not necessarily causality reflecting the many and highly complex underlying processes. Yet, interpretation of results must be done with care.

We also study temporally dynamical networks for the fish ecosystem community (see “Real-world sardine–anchovy-temperature ecosystem” section). CCM and OIF model are applied to quantify the causality between all possible pairs of species at each time period by calculating (rho) and TE, respectively. Estimated effective (alpha)-diversity (Eq. S1.2) from CCM- and TE-based inferred networks at each time point can be obtained and then compared to the taxonomic (or ”real”) (alpha)-diversity. Results are shown in Fig. 8 and Fig. S6. In the whole time period, the estimated (alpha)-diversity from CCM is constant, whereas the global trend of the estimated (alpha)-diversity from OIF model slightly decreases over time that is consistent with the global trend of real (alpha)-diversity. CCM always predicts a non-zero interaction for all species (including negative values) whereas OIF predicts zero interactions for some species that are then not making part of the estimated effective (alpha)-diversity.

Figure 8 shows the effective (alpha) diversity from CCM and OIF for an optimal threshold of (rho) and TE (i.e., 0.2 and 0.3) that maximizes the correlation coefficient and Mutual Information (MI) between (alpha _{CCM}) or (alpha _{TE}) and the taxonomic (alpha), respectively. The maximization of the correlation coefficient and MI guarantees that the estimated effective (alpha) are the closest to the taxonomic (alpha). Figure S6 shows effective (alpha) for unthresholded interactions and other thresholds. Note that the threshold on TE does not coincide with the value of TE that maximizes the total network entropy (Fig. 8) and then some of the reported species may not be part of the ecosystem strongly. Thus, this threshold method is also useful to identify species that are truly forming local diversity versus transient species. Considering the pattern of fluctuations of effective (alpha)-diversity from CCM, they are poorly unrelated to the real (alpha)-diversity, while those from OIF are much more synchronous with seasonal fluctuations of real (alpha)-diversity. However, (alpha _{TE}) is a bit higher than the average taxonomic (alpha). Both CCM and TE predict a decrease in (alpha) in time that corresponds to an increase in SST. As shown in Fig S6, OIF is attributing higher sensitivity to SST for small interaction species because (alpha) fluctuations show seasonality that happens when species follow environmental dynamics closely. Vice versa, CCM predicts a broader sensitivity for all positively interacting species. These results reveal that OIF gives an effective tool to measure meaningful interdependence relationships between species for constructing temporally dynamical networks where the number of nodes over time [estimated (alpha (t))] can reflect closely the taxonomic (alpha)-diversity. This allows us to find more reliably how changes of environmental factors (e.g. SST) affect biodiversity in ecosystems. The establishment of thresholds on interactions is also useful for exploring ranges of interdependencies and associated effective (alpha)-diversity with respect to the average taxonomic diversity. (beta) effective diversity is another very important macroecological indicator informing about ecosystem changes; for instance in Li and Convertino³⁹ (beta)-diversity identified distinct ecosystem health states. However, (alpha) and (beta) (effective diversity) variability are highly linked to each other and yet looking into one or another would provide equivalent results. The difference between taxonomic and effective (beta)-diversity may provide some information about invasive or rare species have weak influence on the ecosystem since they are characterized by low TEs. Supplementary Information contains further elaborations on results.

Source: Ecology - nature.com