Fluctuating interaction network and time-varying stability of a natural fish community


Ecological theory suggests that large-scale patterns such as community stability can be influenced by changes in interspecific interactions that arise from the behavioural and/or physiological responses of individual species varying over time1,2,3. Although this theory has experimental support2,4,5, evidence from natural ecosystems is lacking owing to the challenges of tracking rapid changes in interspecific interactions (known to occur on timescales much shorter than a generation time)6 and then identifying the effect of such changes on large-scale community dynamics. Here, using tools for analysing nonlinear time series6,7,8,9 and a 12-year-long dataset of fortnightly collected observations on a natural marine fish community in Maizuru Bay, Japan, we show that short-term changes in interaction networks influence overall community dynamics. Among the 15 dominant species, we identify 14 interspecific interactions to construct a dynamic interaction network. We show that the strengths, and even types, of interactions change with time; we also develop a time-varying stability measure based on local Lyapunov stability for attractor dynamics in non-equilibrium nonlinear systems. We use this dynamic stability measure to examine the link between the time-varying interaction network and community stability. We find seasonal patterns in dynamic stability for this fish community that broadly support expectations of current ecological theory. Specifically, the dominance of weak interactions and higher species diversity during summer months are associated with higher dynamic stability and smaller population fluctuations. We suggest that interspecific interactions, community network structure and community stability are dynamic properties, and that linking fluctuating interaction networks to community-level dynamic properties is key to understanding the maintenance of ecological communities in nature.

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Figure 1: Reconstructed interaction network of a subset of the Maizuru Bay fish community.
Figure 2: Time-varying interspecific interactions in a subset of the Maizuru fish community.
Figure 3: Time-varying stability, interaction strength and Simpson’s diversity index.
Figure 4: Causal influences between dynamic stability, interaction strength and Simpson’s diversity index.


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We thank members of the Kondoh laboratory in Ryukoku University; F. Grziwotz, A. Telschow and T. Miki for discussions; S.-I. Nakayama for advice on the twin surrogate method; and T. Yoshida and M. Kasada for providing time series of the algae–rotifer system. This research was supported by CREST, grant number JPMJCR13A2, Japan Science and Technology Agency; KAKENHI grant number 15K14610 and 16H04846, Japan Society for the Promotion of Science; Foundation for the Advancement of Outstanding Scholarship (Ministry of Science and Technology, Taiwan); DoD-Strategic Environmental Research and Development Program 15 RC-2509; Lenfest Ocean Program 00028335; NSF DBI-1667584; NSF DEB-1655203; the McQuown Fund and the McQuown Chair in Natural Sciences (University of California, San Diego).

Author information




M.U., C.H. and M.K. designed the research programme; R.M. collected fish monitoring data; M.U. and G.S. conceived the idea of computing local Lyapunov stability from S-maps; M.U. performed analysis with help from C.H., E.R.D., H.Y. and C.-W.C.; M.U., C.H. and M.K. wrote the first draft of the paper; and all authors were involved in interpreting the results, and contributed to the final draft of the paper.

Corresponding authors

Correspondence to Masayuki Ushio or Michio Kondoh.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks J. Bascompte, U. Brose and K. McCann for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Effectiveness of the S-map method examined in two-species model systems and laboratory experiment systems.

a, Illustration of the unidirectional two-species model system. X has a direct influence on Y, but Y does not have an influence on X. b, An example of the dynamics of the two-species system. The interaction strength from X to Y was set at −0.31 in this example. c, The estimation of interaction strength using the S-map method. True interaction strength is −0.31, whereas the mean of the S-map coefficients is −0.309. The length of the time series used for the analysis was 1,000. d, Test of the S-map method in a two-species bidirectional system. Interaction strength from Y to X was fixed for each panel (as denoted in the header of each panel), and interaction strength from X to Y was changed (x axis). The length of the time series used for each analysis was 1,000 (see Methods). Dashed lines indicate 1:1 lines. Dynamics that show strong linearity (for example, limit cycle and equilibrium) were excluded from the analysis; that is, regions around the origin were excluded. e, Population dynamics of Didinium (predator) and Paramecium (prey). f, Estimation of interaction strength between Didinium and Paramecium. g, Population dynamics of the rotifer (predator) and two types of algae (prey). Inset illustrates the three-species experimental system. R, Ar and AK indicate rotifers, r-strategy algae and K-strategy algae, respectively. Units for the y axis are 106 cells per ml for the algae, and 10 individual females per ml for the rotifer. h, i, Estimation of pair-wise interaction strength among r-strategy algae, K-strategy algae and rotifers.

Extended Data Figure 2 Time series of dominant fish species and jellyfish in Maizuru Bay in Japan.

During a 12-year census (2002–2014), 285 surveys were conducted. The width of the grey region corresponds to a 1-year interval that runs from January to December (24 observations per year).

Extended Data Figure 3 Evaluation of the phase-lock twin surrogate method.

a, A false high cross-map skill and convergence, owing to seasonality. We set βxy = βyx = 0 (no causality between X(t) and Y(t)) and ax = ay = 0.3 (moderate seasonality). b, An example of the phase-lock twin surrogate time series. The original time series with strong seasonality is shown as a black solid line (Y(t); βxy = −0.3, βyx = 0, αx = 1.0 and αy = 1.0). The surrogate time series, with the same seasonality and nonlinearity as the original data, is shown as a solid red line. ch, Cross-map skill (terminal ρ) and terminal ρ −95% upper limit; ρ of 100 surrogate data by CCM between X and Y, when X and Y have no interaction (c, d), unidirectional interaction (e, f) and bidirectional interaction (g, h). The length of the time series used for the evaluation was 288 (equivalent to a 12-year census with 24 observations per year).

Extended Data Figure 4 Sensitivity to the inclusion of subdominant species and observation errors.

ad, Relationship between the dynamic stability calculated from the community of 15 dominant species versus that of a 16-species community. A subdominant species (D. temminckii (a), P. cottoides (b), T. niphobles (c) or T. poecilonotus (d)) was added to the community of 15 dominant species, and the dynamic stability was calculated by the procedure described in the Methods. Inset shows the interaction network structures of the 16-species community. Solid black line indicates the 1:1 line. Red circle indicates the newly included subdominant species. Blue and red arrows indicate positive and negative time-averaged interactions, respectively, associated with the subdominant species. Grey arrows and circles indicate the edges and nodes, respectively, of the original community of 15 dominant species. ej, Effects of observation errors on the calculations of the dynamic stability. e, Observation errors were added to the original time series (see Methods), R2 was calculated between the original dynamic stabilities and those calculated from the time series with an added error. This procedure was repeated 100 times for each error magnitude (%). Midline, box limits, whiskers and points indicate median, upper and lower quartiles, 1.5× interquartile range and outliers, respectively (n = 100 for each box). fj, Examples illustrating the relationships between the original dynamic stabilities versus those calculated after the addition of 1% (f), 5% (g), 10% (h), 20% (i) and 30% (j) observation errors. The solid line indicates the 1:1 line. The dashed line indicates the dynamic stability = 1.0.

Extended Data Figure 5 Relationship between dynamic stability and coefficient of variation of fish abundance.

a, Time series of mean values of CV. CV was calculated using a moving window (window width = 6 time points; 3 months) for population dynamics of each fish species. Mean values of CV were then calculated by averaging CV values of the 15 fish species. b, Comparison of CV between stable and unstable periods (n = 56 for stable conditions and n = 203 for unstable conditions). Under stable conditions (dynamic stability < 1.0), the CV is significantly lower than it is under unstable conditions (P < 0.0001, two-sided t-test). Midline, box limits, whiskers and points indicate median, upper and lower quartiles, 1.5× interquartile range and outliers, respectively.

Extended Data Figure 6 CCM between dynamic stability and surface water temperature, species richness, total fish abundance and the s.d. and skewness of the interaction strength distribution.

ac, Time series of surface water temperature (a), richness of dominant fish species (b) and total abundance of dominant fish species (c). The width of the grey region corresponds to a 1-year interval (24 observations per year). df, Results of CCM analysis between dynamic stability and surface water temperature (d), species richness (e) and total fish abundance (f). gh, Results of CCM between the dynamic stability and s.d. of interaction strength (g) and skewness of the interaction strength distribution (h). Dark solid lines indicate cross-map skill (ρ) from dynamic stability to another variable. Shaded regions indicate 95% confidence intervals of 100 surrogate time series. Significant cross-map skills (ρ) are highlighted in red (dh). i, j, Correlations between median:maximum interaction strength (IS) (weak interaction index) and s.d. of interaction strength (i) and the skewness (j) (n = 261 for each panel). The dynamic stability is indicated in blue. The weak interaction index and s.d. and skewness of interaction strength were predominantly linearly correlated, which suggests that the s.d. and skewness of interaction strength are alternative representations of the weak interaction index in our data.

Extended Data Figure 7 Abundance-based stability index of the fish community.

a, Temporal dynamics of the abundance-based stability index. Euclidean distance between W(t + 1) and W(t) was calculated (see Methods for the definition of W(t)). Note that the abundance of each fish species was standardized before calculating the Euclidean distance. bg, Results of CCM between the abundance-based stability and interspecific interactions, species richness, diversity and surface water temperature. Dark solid lines indicate cross-map skill (ρ) from the abundance-based stability to another variable. Shaded regions indicate 95% confidence intervals of 100 surrogate time series. Significant cross-map skills (ρ) are highlighted in red.

Extended Data Figure 8 Results of quantile regressions between Simpson’s diversity index and properties of the distributions of interaction strengths.

ah, Quantile regressions and their regression coefficients of the mean IS (a, b), median:maximum interaction strength (c, d), skewness (e, f) and s.d. of interaction strength (g, h) were plotted against Simpson’s diversity index. The solid red line indicates the 50% quantile and the dashed black lines enclose the 2.5% and 97.5% quantiles (a, c, e, g; n = 261 for each panel). Regression coefficients (slopes) were plotted against quantiles (b, d, f, h), and show that all coefficients exhibit an increasing trend as the quantile increases.

Extended Data Table 1 Result of CCM for the dominant fish species in Maizuru Bay
Extended Data Table 2 Dimensionality and nonlinearity of the stability index and interaction strength

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Ushio, M., Hsieh, C., Masuda, R. et al. Fluctuating interaction network and time-varying stability of a natural fish community. Nature 554, 360–363 (2018). https://doi.org/10.1038/nature25504

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