Model configuration

We estimated interaction strengths among phytoplankton and zooplankton guilds, environmental effects on phytoplankton and zooplankton abundance, plankton intrinsic growth rates, and plankton community stability in Lake Washington from 1962–1994 using multivariate autoregressive (MAR) models. MAR models are stochastic models describing changes in species abundance through time as a function of species interactions and environmental influences, while accounting for temporal autocorrelation in species abundances [20], [36], [37]. MAR models can also be used to estimate various metrics of community stability, such as return time to a stationary state following an ecosystem perturbation, or the distance away from a stationary state that an ecosystem can be pushed by a perturbation. Previous work has used MAR models to describe environmental effects on, and interactions among, lake phytoplankton and zooplankton [20], [22], [38], [39], effects of climate regime shifts on interactions among marine plankton [40], causes of estuarine fish declines [24], and effects of fishing on marine food webs [23]. Extended descriptions of MAR approaches to time-series data have been given previously [19], [20], [37], so we provide only a brief review of the model structure here.

MAR models are written in matrix form as:(Eq. 1)where, for p interacting species and q environmental covariates, X_t is a p×1 vector of species abundances (here, natural log-transformed) at time t; A is a p×1 vector of constants, representing intrinsic per-capita growth rates; B is a p×p species interaction matrix, with off-diagonal elements describing inter-specific interactions, and diagonal elements describing intra-specific interactions (i.e., density-dependence); C is a p×q matrix with elements describing environmental effects on species abundance; U_t is a q×1 vector of environmental covariates at time t; and E_t is a p×1 vector of process errors at time t, representing environmental variation not otherwise accounted for in the model. E_t is distributed as a multivariate Normal with mean 0 and a diagonal variance matrix ∑. Elements of B and C typically range from -1 to 1, with distance from 0 representing increasing negative or positive interaction strength. The diagonal elements of B typically range from 0 to 1, with values closer to 0 representing higher density dependence.

We also used MAR models to estimate community stability. Specifically, we estimated the rate at which the system returns to its stationary distribution following a disturbance by the maximum eigenvalue of the B matrix (that maximum eigenvalue is henceforth referred to as lambda, λ). Systems with values of λ closer to 0 are considered to be more stable because they tend to return to equilibrium conditions faster than systems with values of λ farther from 0 [21].

MAR models estimate mean intrinsic growth rates (captured by the A vector), community interactions (captured by the B matrix), environmental effects (captured by the C matrix), and community stability (captured by λ) across a given time series [20]. Here we use MAR models to quantify changes in interactions through time, by modeling community interactions for overlapping subsets of a time series, or moving “windows” of time, thereby estimating trends in MAR parameters. For a p×n matrix X of time series observations consisting of successive p×1 vectors X₁, X₂,…, X_n, and a moving window of size W<n, we estimated MAR parameters within n-W-1 successive windows. These windows contained data from X₂:X_W+1, X₃:X_W+2,…, X_n-W+1:X_n. Note that the time series starts at t = 2 to allow for the lag-1 effect in Eq. 1. The output of the mwMAR analysis is a new time series of estimated MAR parameters.

Lake Washington data and analysis

To investigate changes in interactions among zooplankton and phytoplankton guilds and the effects of environmental covariates in Lake Washington through time, we implemented the mwMAR approach using monthly plankton and environmental data from Lake Washington (Washington, U.S.A.) spanning 1962 to 1994 (396 timesteps; see Figure S1 for plankton time series). Our 33-year time series begins in the year of maximum sewage input (1962) when the lake experienced extensive nuisance algal blooms, especially of the cyanobacterium, Oscillatoria rubescens. Sewage diversion began the following year (1963), and was completed in 1968. Water clarity increased substantially by 1971 [12] and continued to improve through 1976, when the influential grazer Daphnia established in the lake [11] and Oscillatoria abundance decreased dramatically. Despite low abundances at times, and periods when they were not observe in samples, neither Daphnia nor Oscillatoria ever technically went extinct in Lake Washington. Before they begun to be observed at high abundances in 1973, Daphnia were observed every year but one (1971). Likewise, after their period of dominance ended in 1980, Oscillatoria continued (and continue) to appear in plankton samples, appearing in all but 3 years between 1980–1994.

The lake has additionally undergone significant warming throughout the historical record [10], which has altered the timing of zooplankton abundance cycles [14], [41]. Recent work, however, suggests species and nutrient (phosphorus) shifts related to the sewage effluent have had a stronger influence on the lake than shifts associated with warming [42]. These well-documented shifts in environmental drivers and plankton dynamics make Lake Washington an ideal ecosystem for evaluating the mwMAR model's sensitivity to non-stationary process. Indeed, the dominant environmental drivers and species interactions in Lake Washington are well-studied via observational [9], [12], experimental [43], [44] and traditional MAR approaches [39], [45], offering the necessary background to build informed community and environmental interaction matrices (B and C matrices, respectively).

For our analyses we aggregated physical, chemical and plankton community data, which were collected at various intervals, into monthly means. Previous analyses of the Lake Washington plankton community interactions identified a simplified food web containing species that demonstrated strong roles in structuring the community [39], [45]. We targeted the most strongly-interacting taxa of this simplified food web with the present analysis, to determine how the dominant interactions changed through time. While weak species interactions can be important in structuring food webs, we chose to focus on the dominant taxa and interactions as a first test of this new method. These taxa were pooled into four taxonomic groups: diatoms and green algae – “DG,” both palatable food for grazing zooplankton; Oscillatoria – known to suppress Daphnia [44]; Daphnia; and non-daphnid and non-cladoceran crustaceans – “NDC,” comprised of non-daphnid cladocerans, Cyclops, and Diaptomus. Group abundance data were log-transformed to better capture non-linearities [20]. A more complete description of the data is available in Hampton et al. [39], and the raw data are available in Appendix S1.

We included as covariates in the mwMAR model surface temperature and total phosphorus, because they were previously identified as the strongest environmental drivers of plankton abundance in the lake [39], [45]. However, rather than simply use temperature as a covariate by itself, we instead used the data to estimate (1) a mean monthly signal indicative of long-term seasonal forcing, and (2) monthly deviations from the mean to capture short-term anomalies (e.g., a particularly warm July) or long-term trends (e.g., an overall increase). To ease comparison of effect sizes across all environmental covariates, we standardized all covariate data to a mean of 0 and a standard deviation of 1.

For our environmental covariate matrix (C) we included a priori only biologically meaningful interactions based on established environmental relationships: we assumed total phosphorus could not directly affect Daphnia or other zooplankton taxa. We expected shifts in mwMAR coefficients to lag behind known dates of change in the biotic community, sewage diversion and water clarity because our moving window size (7 years) is much larger than the timescale of most known changes. We graphically present all data at the end year of the moving window; thus, in our figures, results based on data from 1963–1970 would appear on the x-axis at year 1970.

Sensitivity analysis

The accuracy and precision of parameter estimates by the mwMAR model, as with other statistical methods, are sensitive to and affected by multiple factors, including food web configuration (i.e., the number of interacting species and covariates), window size, the variance structure of the process errors, and outliers in the data (see Appendix S2 for discussion and additional model validation). We conducted several sensitivity analyses to ensure such factors were not influencing the mwMAR model estimates. For example, the Lake Washington dataset is of high quality, and our outlier inspection showed no influence of outliers on the final results. In addition, because there is a tradeoff between precision of parameter estimates and accuracy of those estimates that is defined by window size, we conducted tests using simulated time series based on the Lake Washington food web configuration, to determine the appropriate window size for analysis of the Lake Washington dataset. In those simulations, the parameter estimation accuracy decreased sharply at window sizes smaller than 75 time steps (see Figure S1), and therefore we use a window size of 84 (the next factor of 12 larger than 75, given the monthly time step in the Lake Washington data). We also conducted simulations to determine what bias, if any, exist in parameter estimates during periods when a system is undergoing transition between states, for example between a eutrophic and clear-water state as was the case with Lake Washington. Last, to ensure that the mwMAR model was capable of capturing shifts in species interactions and environmental conditions outside of the Lake Washington case study, we fit mwMAR models to simulated time series with known interactions (see Appendix S2).

Statistical programming

MAR and mwMAR modeling was done in MATLAB (2007, The MathWorks), using the open-source program LAMBDA ([46]; freely available from http://conserver.iugo-cafe.org/user/e2holmes/LAMBDA) with additional programming by the authors. The coefficients of the A, B and C matrices were estimated using conditional least squares (CLS), and confidence intervals around each coefficient were established using 2,000 bootstrapped data sets. Each bootstrapped data set was generated by creating random E matrices and fitting the rest of the parameters using CLS (see [21] for details).

Shifting plankton dynamics in Lake Washington

We hypothesized that the mwMAR would show shifts in the interactions among the major taxa corresponding roughly with known periods of change in Lake Washington (e.g., years of and around 1968–1971, and 1976). For example, it has long been hypothesized that the highly-abundant Oscillatoria, owing to its low palatability, inhibited Daphnia before Daphnia's increase in Lake Washington in 1976 [11], during the period of time when the two species overlapped but Oscillatoria abundance was decreasing. These dynamics have been demonstrated experimentally [44], but our results are the first to corroborate this hypothesis using historical data. During the period of time between the peak in water quality (1971) and the dramatic increase in Daphnia abundance (1976) – the period of overlap between Oscillatoria and Daphnia and hypothesized inhibition of Daphnia by Oscillatoria – we found an increasingly negative effect of Oscillatoria on Daphnia. Once the mwMAR window included only dates following the large increase in Daphnia (i.e., 1976 and later), there was no detectable effect of Oscillatoria on Daphnia. The long, filamentous shape of Oscillatoria generally makes it inedible for Daphnia, which is one likely source of the negative per-capita effect estimated here during their period of overlap.

Oscillatoria also had a negative effect on diatoms and edible green algae, the main food source for Daphnia and other grazers in the lake. High intrinsic growth rates in edible phytoplankton estimated at the start of the time series decreased during the period when Oscillatoria was dominant. At the same time, density dependence in diatoms and green algae also decreased, suggesting inhibition in growth, possibly resulting from competition for limiting nutrients, or physical shading or toxic effects of excretions by Oscillatoria. Such inhibition of algae by Oscillatoria has also been demonstrated experimentally [44]. This apparent inhibition of phytoplankton by Oscillatoria rapidly decreased following an abrupt transition in the mid-1970s when the negative effect of Oscillatoria on DG decreased and disappeared (Figure 1). Coincident with these dynamics, the effect of Oscillatoria on Daphnia also weakened and the intrinsic growth rate of Daphnia increased from its minimum in 1972 to its peak in 1976 (Figure 4C). After 1976, Daphnia's intrinsic growth rate decreased and density dependence increased (Figure 3C) as the Daphnia population increased in abundance. In addition, while the result was not significant (95% CIs overlapped zero), DG may have had a bottom-up positive effect on Daphnia after being freed from inhibition by Oscillatoria, in the latter half of the time series (Figure S2). Taken together, these results corroborate the hypothesis that the establishment of Daphnia following the improvement of water quality in Lake Washington was impeded directly and indirectly by the cyanobacterium Oscillatoria.

Grazers are known to inhibit cyanobacteria under some environmental conditions [49], and our analysis found a negative effect of Daphnia on Oscillatoria coincident with Oscillatoria's decrease in abundance. In general, the frequency of cyanobacteria blooms is associated with the relationships between grazers and edible phytoplankton, such that when grazers and edible phytoplankton dynamics are stable (i.e., abundances do not undergo large, intrinsic oscillations), cyanobacteria are controlled by grazers [49]. These dynamics are often associated with phosphorus inputs to a lake. We observed a similar pattern in Lake Washington. As phosphorus inputs decreased, the grazing effect of Daphnia on edible phytoplankton increased concomitant with the inhibiting effect of Daphnia on Oscillatoria (Figures 1 and 2).

MAR coefficients have been shown previously to reflect changes in community dominance, when an increase in the abundance of one species or group coincides with a decrease in another [40], and therefore the negative effect of Daphnia on Oscillatoria may represent shifting dominance between the two taxa. The transition from Oscillatoria to Daphnia dominance was reflected in interactions among other plankton groups in the community. As the negative effect of Oscillatoria on Daphnia declined in the mid-1970s through to the early 1980s, and as Daphnia increased in abundance, Daphnia had stronger impacts on their main food source (DG) and competitors (NDC; Figure 2). At the same time, the strength of density-dependence (Figure 3) and density-independent growth rates increased for the grazing zooplankton groups (Figure 4), suggesting the release of the grazer community from inhibition by Oscillatoria. No previous work has shown an effect of Oscillatoria on other grazer groups beyond Daphnia, but the increase in population growth rates (A matrix elements) of the NDC group following Oscillatoria's decline suggests a possible negative interaction.

The negative effects of Daphnia on their food and competitors weakened towards the end of the time series, apart from an intensified grazing effect of Daphnia on DG at the very end. One potential explanation for the weakened grazing effect at the end of the time series relates indirectly to the warming of the lake during this time. Between 1962 and 2002, the lake surface temperature increased by 1.4°C during the stratified months, and associated with this warming was an advance in the spring phytoplankton bloom by 19 days [10]. Most of the warming and spring bloom advance occurred in the period 1962-1994. The weakening of the effect of total Daphnia on the phytoplankton group during that period, in the present analysis, could be a reflection of shifts in species-specific phenology and grazing characteristics [14], [41].

The results presented here highlight opportunities to learn more from time series data about how species interactions shift with changes in the environment across ecosystem types, and how those changing food web dynamics are liable to affect community stability and resilience to further disturbance. Ecosystem-based approaches to management often include a focus on food web dynamics, but quantifying changes in species interactions, and how those changes map onto the environmental template, proves difficult. Linking shifts in species interactions to specific environmental drivers opens opportunities to focus efforts aimed at retaining resilience as ecosystems undergo rapid change.

Community stability and environmental covariates

The Lake Washington system has undergone major shifts in chemistry and ecology that are reflected in community stability. The peak of instability occurred in April 1973, (a window that included data from May 1966 – April 1973; Figure 5). Values of λ greater than 1 indicate an unstable system [21], and here λ exceeded 1 for windows ending in November 1970 – November 1974, representing the period of time between December 1963 – November 1974, inclusive. This is the time period that included major ecosystem shifts: high nutrient levels, sewage diversion, and nutrient reduction; high Oscillatoria abundance followed by its decline; and the first rare appearances of Daphnia. By the time of Oscillatoria's disappearance, maximum water clarity, and establishment of Daphnia in 1976, the community stability was increasing, and continued to increase until the end of the time series. Thus, the period of time the lake was undergoing the most substantial and dramatic shifts throughout the ecosystem, and before Daphnia gained a foothold, was the least stable period in the community as well.

We observed effects of monthly mean temperature on the abundance of all plankton guilds (Figure S3), which agrees with previous MAR analyses [39], [45], and with the MAR model estimated here from the whole Lake Washington time series (Table S1). Previous work has suggested that Lake Washington plankton phenology also responds to lake warming [10], [50], and that the relationships between temperature and plankton taxa are evidence of the potential influence of a warming lake on food web dynamics [39]. However, we found no significant effects of deviations from the long-term seasonal temperature patterns, suggesting that lake-warming effects are not detectable in the abundance of these plankton guilds.

Caveats and considerations

Our results suggest moving-window MAR models may be useful in systems with sufficient time-series data for understanding shifting abiotic and biotic dynamics. As with all statistical methods, however, practitioners must consider possible caveats and issues in advance of and throughout analyses. The data and ecosystem considerations applicable to prior MAR model applications also extend to our moving-window approach. Users must have sufficient time-series data for valid parameter estimation, which varies depending on the time scale of interactions in the system and frequency of observations. The moving-window MAR model imposes the further consideration of having sufficient time-series data for multiple windows and surrounding the event(s) of interest. Importantly, bias in model estimates shrinks as the ratio between window size and system transition period increases, and users are cautioned to interpret model estimates during system transitions with consideration of such bias. However, the window could be configured for different purposes: made smaller to detect changes before they occur, or sized to optimize detection of a change in a particular state variable.

Applications of this method will benefit from a priori knowledge of ecological interactions and drivers in the modeled system to build a robust MAR model. In our analysis of the Lake Washington plankton community, we simplified the plankton community based on previous work that highlighted the strongest food web interactions and key environmental covariates [39]. However, Hampton et al. [39] also pointed out the importance of other plankton taxa in driving the dynamics of the dominant species in Lake Washington, such as Cryptomonas, picoplankton and non-colonial rotifers. Therefore, it is possible that additional food web dynamics contribute to the interaction coefficients observed here, which could be highlighted by future analyses. Furthermore, if the model failed to include an influential environmental driver of Lake Washington plankton dynamics, the model results might be erroneously interpreted: if one plankton guild responds negatively to an unmeasured environmental variable, and another guild responds positively, this might incorrectly be interpreted as a negative interaction between the two guilds. In the Lake Washington case, years of experimental work and field observations have identified environmental variables that are robust driving signals. In addition, preliminary, exploratory MAR model runs were performed to screen a broad suite of potential drivers on plankton time series data. The analyses here rely heavily on those two approaches to validation, and potential users are advised to similarly behave as ecological detectives.

Additionally, as with prior MAR approaches, users must invest time in simulation modeling that allows them to test how the approach is likely to work with data similar to theirs. Simulation of data from a model with similar parameters to the study ecosystem helps identify the appropriate moving window size and, thus, estimate the precision associated with future predictions of system change. Because much of the MAR approach is based on iterative fitting approaches, creating and testing simulation data sets from known parameter values with similar lengths, covariate and taxa numbers, and variance, is critical to interpreting knowledge gained from MAR models. For the moving-window approach, users should carefully examine the effect of window size on their simulation datasets (see Appendix S2 for an example analysis using simulated datasets). A priori knowledge or hypotheses related to the resolution of data and interactions as well as the strength and timing of the predicted shift should be considered during the process of simulation modeling. Comparison of the mwMAR output with whole time-series MAR estimates is useful in assessing when the broad confidence intervals estimated with the mwMAR model are potentially masking significant interactions.