Theoretical illustration that enhanced synchrony can result from reinforcement between Moran drivers

We constructed a numerical example including frequency-specific synchrony arising from multiple drivers, as a conceptual illustration and for use in verifying the effectiveness of statistical methods to be used on data. The example is described below. By construction, the example includes two drivers of synchrony that can either reinforce or counteract one another in the same way as the drivers of PCI synchrony identified below. Thus the example also serves a pedagogical purpose by introducing, in simplified form, the kinds of relationships we later show to hold for plankton in British seas. The example makes use of phase-shifted relationships between variables and timescale-specific synchrony, and shows explicitly how interaction effects can increase or decrease synchrony. The approach also illustrates the use of the wavelet tools we apply to empirical data. Our example is demonstrated principally through Fig 1. By way of overview, Fig 1a, 1b and 1c show an artificial spatiotemporal ‘driver’, a wavelet measure of its synchrony, and the resulting synchrony of a ‘driven’ variable, respectively (details below). Fig 1d, 1e and 1f show a different driver, its synchrony, and the resulting synchrony of a second driven variable, respectively. Fig 1g, 1h and 1i describe the synchrony of a third driven variable that depends on both drivers.

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Fig 1. Synthetic example showing the effects of two different synchronizing factors, individually and in interactive combination.

Panel (a) shows fluctuations in 26 signals α(n, t) with a synchronous 20-year-period sine wave component obscured by independent local noise. Panel (d) shows 26 signals β(n, t) with a synchronous 20-year-period cosine component, also obscured by independent local noise of the same strength. Panels (b) and (e), which are wavelet mean field magnitudes (WMFM; Methods) of (a) and (d), respectively, display synchrony as a function of time and timescale and reveal synchrony at timescale 20 years. Panels (c) and (f) show WMFMs of populations influenced by α and β, respectively, and also by independent local noise time series of the same strength. β acts with a 5-year lag, 1/4 of the underlying 20-year periodicity. Synchrony is still visible, and is similar in strength, but is muted relative to (b) and (e) by the additional local noise acting on the populations. Panel (g) shows the WMFM of populations subject to a mixed influence of α and β, normalized so this influence had the same variance through time as the influences α and β had individually for panels (c) and (f). Independent local noise of the same strength was again applied. Synchrony is stronger in (g) than in (c) and (f), demonstrating interaction effects of the synchronizing agents α and β. Panel (h) shows the time-averaged square of the WMFM of (g), in green (called the mean squared WMFM or mean squared synchrony in Methods), which is greater than that of (c), in blue, or (f), in red. Panel (i) shows how interaction effects are a result of the phase relationships between the drivers. As the phase of the co-sinusoids underlying β are modified in further simulations, the interaction effect goes from positive to negative as the phase shift passes 0.5π. Significance contours on Fig.1b,c,e,f,g represent wavelet phasor mean field magnitude (Methods; WPMFM) significance thresholds at the 0.1, 0.05, 0.01, and 0.001 levels, respectively for the dot, dot-dash, dash and line contours. These significance thresholds are relative to a null hypothesis of no association between the phases of the 26 transforms. See text for mathematical details. WMFMs are not guaranteed to be ≤ 1 at all times, but ours were except for (e), which had maximum squared value 1.0111. For clearer plotting, we reassigned values > 1 in (e) to 1. Sync. = synchrony.

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For sampling locations indexed by n and sampling times indexed by t, define the (biotic or abiotic) environmental random variable (Fig 1a) , where is the ‘synchronous component’ of α(n, t) and α_L(n, t) is a ‘local noise’. This and other local noise terms will be considered to be standard-normally distributed for all n and t, and independent through time t, across space n, and of other local noises. We take w = 2π/20, so the timescale of oscillations we consider in this example is 20 years. We use n = 1, …, 26 and t = 0, …, 59 in this example because these values are similar to the analogous values in the empirical data we study below. It is straightforward to show that contributes 1/4 and contributes 3/4 of the variance through time of α, so that the ‘combination ratio’ of the synchronous and local-noise components of α is 1:3. Likewise define the environmental variable (Fig 1d), where we set for φ a constant, and β_L(n, t) is another local noise. Again, the combination ratio of the synchronous and local-noise components is 1:3. We initially consider the case φ = 0, later relaxing this assumption.

The wavelet mean field magnitude (WMFM; Methods) is a statistical technique that takes spatiotemporal data such as α(n, t) and provides a plot that shows how much synchrony exists between these time series for each time and timescale of analysis [7]. The WMFM approach picks out the synchronized fluctuations in α (Fig 1b) and in β (Fig 1e) as bright features at timescale 20 years spanning the duration of the time series, introducing the WMFM technique by way of demonstration.

Three alternative population-dynamical variables γ⁽ⁱ⁾(n, t) for i = 1, 2, 3 are now constructed, each influenced by different combinations of α and β. Each variable α and β is partially spatially synchronous, and can thus have a synchronizing effect on γ⁽ⁱ⁾. We will imagine α represents a top-down influence such as predation, and β represents a climatic influence. Depending on the causal mechanism, the two signals α and β may bear different phase relationships to the fluctuations they produce in a population variable γ. For example a low value of α may tend to produce high values of γ, yielding a negative correlation and a π phase difference between their fluctuations, whereas the effect of β may be subject to a time delay introducing a different phase shift. We define , where the are local noise, , , and , where is chosen so that the variance through time of is 1, the same as the variance through time of and (Appendix S1). Thus in all three cases the partially synchronous components are combined with the local noise components in combination ratio 1:3. The negative influence of α on γ⁽¹⁾ will produce a negative correlation between these variables. The 5-year lag of the effects of β on γ⁽²⁾ is a quarter of the 20-year cycle built into β (π/2 radians phase difference). The γ⁽³⁾ example models a combined influence of top-down (α) and climatic (β) Moran effects, though the strength of the combined effect is kept equal to the strength of the individual top-down and climatic effects acting, respectively, on γ⁽¹⁾ and γ⁽²⁾ through α and β. Thus effects can be fairly compared across our three population models.

WMFMs of the γ⁽ⁱ⁾ reveal that synchrony is transmitted to all three population variables, but that, for φ = 0, the synchrony in γ⁽³⁾ was stronger than that in γ⁽¹⁾ and γ⁽²⁾, revealing interaction effects between Moran drivers. The WMFMs of γ⁽¹⁾ (Fig 1c) and γ⁽²⁾ (Fig 1f) show that the influence of α and β, respectively, have synchronized these population variables at the 20-year timescale. Synchrony in the populations is less than in the driver variables, as expected since the populations are also influenced by local noise. The WMFM of γ⁽³⁾ (Fig 1g) also shows synchrony at 20-year timescale, and this synchrony is stronger than that of γ⁽¹⁾ and γ⁽²⁾. Time-averaged squared WMFMs (Fig 1h) clarify the comparison of strengths of synchrony of the γ⁽ⁱ⁾. Because of the π/2 phase relationship between sine (in α_S) and cosine (in β_S) when φ = 0, and the π/2-phase-lagged effects of β on γ⁽³⁾, the synchronizing influences of α and β on γ⁽³⁾ reinforce one another. The asynchronous variations in α and β (local noise) tend to cancel out when these drivers are combined, leaving their synchronous components to have relatively larger influence.

We investigated interaction effects between Moran drivers further by considering φ ≠ 0. When φ ≠ 0, the effects of α_S and the lagged effects of β_S on γ⁽³⁾ no longer perfectly reinforce each other. For φ increasing from 0 to 0.5, reinforcement is partial, and positive interaction effects are seen; for φ increasing further from 0.5 to 1, negative interactions are seen (Fig 1i). Fig 1i shows typical (average over 50 realisations) time-averaged squared mean field magnitude values for γ⁽ⁱ⁾ at 20-year timescale for i = 1, 2, 3 in blue, red, and green, respectively. The crossover threshold of φ = 0.5 between the green and red or blue lines on Fig 1i corresponds to the fact that quarter-cycle-shifted sinusoids (π/2 phase shift) have correlation 0 with each other.

PCI fluctuations and synchrony have both top-down and climatic causes

At long timescales (> 4 years), the best wavelet model of PCI according to our model selection and testing procedure (Methods), which was based on wavelet regression and out-of-sample cross validation, included two predictor variables: C. finmarchicus abundance and sea surface temperature during the growing season. Models considered in the model selection procedure were constructed using combinations of the predictors listed in Tables 1 and 2. Other high-ranked models were similar, but not identical to the top model (S1 Table): all included C. finmarchicus and either growing season or annual average temperature; summer salinity and autumn cloud cover were included as predictors in some models. For simplicity and because all of the top-ranking models included C. finmarchicus and a temperature variable, and because other variables were only included in some of the top-ranking models, subsequent analyses focussed on the one top model. At short timescales (< 4 years) the best model included C. finmarchicus abundance, decapod larvae abundance, and echinoderm larvae abundance. Even the best model at short timescales performed poorly for our purpose of explaining synchrony (see below), so other short-timescale models were not listed.

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Table 1. The names of the plankton variables investigated.

Each time series was constructed by averaging monthly values over all twelve months, to produce one value per year per location.

https://doi.org/10.1371/journal.pcbi.1006744.t001

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Table 2. The names given to the environmental variables investigated.

Each time series was constructed by averaging monthly values over the relevant months, to produce one value per year per location.

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Two timescale bands (<4 years and >4 years), as opposed to more bands, were used for simplicity and because wavelet methods have finite ability to resolve timescales of fluctuation in finite time series. The practical distinction between short (<4 year) and long (>4 year) timescale bands follows the same reasoning as [7]: considering lag-1 autocorrelation, a time series (or a spatially synchronous component thereof) dominated by >4-year components of variability will demonstrate slow changes in the sense that successive years will typically be positively correlated. If dominated by <4-year components, the correlation will be negative. The 4-year timescale is the boundary between persistence and anti-persistence in the ecological data.

Each of the variables included in the best models at short and long timescales, respectively, were tested for statistically significant spatial coherence (Methods) with PCI when controlling for other variables in the model. In other words, normalized predictor transforms included in a model were required to provide a significant improvement in model explanatory power over the model with that predictor removed (Methods). For long-timescales, the p-value associated with removing C. finmarchicus from the top-ranked model was 0.0023, and the p-value associated with removing growing season temperature was 0.0019. For short timescales, the p-values associated with separate removals of C. finmarchicus, decapod larvae, and echinoderm larvae abundance, respectively, from the top-ranked model were 0.0034, 0.0361, and 0.0017. Our best models were also tested and validated in several additional ways (Methods).

We extracted the phases of the β_k coefficients in our best fit models to determine the lag between the fluctuations in the kth predictor and the corresponding content in the PCI wavelet transforms. See Methods for details and S1 Fig for a plot of phase differences. A quarter-cycle phase shift existed between the long-timescale driver, temperature, and PCI, with high PCI values following low temperature values. Across the range of timescales in the long-timescale band, the phase of PCI leads the phase of temperature with coefficient phases between 0.5 and 2.1 radians. C. finmarchicus was found to be in antiphase with PCI at all frequencies, indicating an association between high C. finmarchicus abundance and low PCI. In the long-timescale band, the phase of PCI leads the phase of C. finmarchicus with coefficient phases between 2.8 and 4.2 radians; and in the short-timescale band between 3.0 and 3.5 radians. The two other short-timescale predictors, decapod larvae and echinoderm larvae, were in phase with the fluctuations in PCI. In the short-timescale band, decapod larvae had coefficient phases between 0 and 0.4 radians, and echinoderm larvae had coefficient phases between -0.4 and 0 radians.

The drivers and their phase relationships to PCI at low frequencies correspond to those illustrated in Fig 1. PCI maintains an antiphase relationship with C. finmarchicus abundance just as γ⁽¹⁾ does with driver α, and a quarter cycle phase shift with the temperature variable just as γ⁽²⁾ does with driver β. We can consider C. finmarchicus a top-down driver of PCI fluctuations, as the anti-phase relationship between the variables, which pertains even after including temperature in the model, implies variation in PCI driven by predation (see Discussion for further elaboration). As in Fig 1i, the resultant mean squared WMFM of PCI will depend on the interaction between the effects of these drivers, whether further synchronizing or desynchronizing, as investigated below.

Returning to short timescales, the in-phase relationships between PCI and echinoderm and decapod larvae may reflect environmental or other influences that impact these variables in similar ways. Reid and Beaugrand proposed that the larger phytoplankton production reflected in the growth in the PCI index that followed the 1980s North Sea regime shift increased sedimentation of planktonic detritus to the benthos leading to a cascade of ecological responses in the pelagos and benthos [19]. Kirby et al. and Lindley et al. showed that decapods and echinoderms responded to the climatic regime shift with a marked increase in abundance of both the benthic adult stage and their planktonic larvae [34, 35].

Fractions of synchrony explained

Having just demonstrated a dual climatic and top-down Moran effect on PCI, we now explore the proportion of PCI synchrony this effect can explain.

The WMFM plot for PCI (Fig 2a, colors) showed that synchrony (high WMFM) in PCI occurred at different time points within the time series for different frequencies. The wavelet phasor mean field magnitude plot (WPMFM; Methods) for PCI (S2 Fig; significance contours from that figure are superimposed on the WMFM plot of Fig 2a) showed that synchrony was significant for some times and timescales and not for others. Here the WMFM is being used to display synchrony and the WPMFM is being used to test for significance of phase synchrony. Significance testing only the phase associations highlights significantly synchronous features in the transforms without requiring a particular null hypothesis about the site-specific variability in the transforms (Methods). The significance contours of the WPMFM were constructed and are interpretable in the same way as for the simulated time series in Fig 1. Synchrony is especially apparent in the WMFM plot on long timescales, and in a feature in the mid 1980s. A ‘regime shift’ in the North Sea plankton community in the 1980s has been well documented (e.g., [26, 36]), and is likely to be associated with this feature.

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Fig 2. Our best long-timescale model explained synchrony on long timescales, but our short-timescale model did not explain synchrony on short timescales.

The PCI squared WMFM plot (a, colors) showed that spatially synchronous fluctuations in PCI occurred at different time points within the time series for different frequencies. Contours indicate statistically significant phase synchrony (at the 0.1, 0.05, 0.01, and 0.001 levels, respectively for the different contours) and are taken from the WPMFM plot (Methods; S2 Fig). Synchrony predicted (Methods) by the best long-timescale wavelet model (b, left of white line) resembled real patterns of synchrony, but synchrony predicted by the best short-timescale model (b, right of white line) did not. Mean squared WMFM (c, green line; Methods) was moderately well approximated by the mean squared value of predicted synchrony for long timescales (c, blue line) but not for short timescales (c, red line). Sync. = synchrony; pred. = predicted.

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Our wavelet Moran theorem (Methods) led to model-predicted synchrony plots and fractions of synchrony explained that showed that our best models explained PCI synchrony effectively on long timescales, but poorly on short timescales. Model-predicted synchrony plots have the same format as WMFM plots but represent the synchrony that would have occurred if the only synchronizing agents were those predictor variables included in the model (Methods; this is in the notation used there). The model-predicted synchrony plot based on our best long-timescale model (Fig 2b, left of the white line) resembled the PCI WMFM plot (Fig 2a) for long timescales. In particular, a feature in the model-predicted synchrony plot (Fig 2b) representing high synchrony on the longest timescales examined (> 20 years) corresponds to a similar peak in the observed PCI WMFM plot; note the different color bars on the two panels when making this comparison. Synchrony is also high on both plots at timescales of around 8 years in the 1980s. In contrast, the model-predicted synchrony plot based on our best short-timescale model (Fig 2b, right of white line) bore no resemblance to the PCI WMFM plot (Fig 2a) for short timescales. Correspondingly, the mean squared WMFM or mean squared synchrony (Fig 2c, green line; this is the mean through time of the square of Fig 2a) only moderately exceeded the mean squared value of synchrony predicted by our best model on long timescales (Fig 2c, blue line; this is the mean through time of the square of Fig 2b for long timescales), but greatly exceeded it on short timescales (Fig 2c, red line). The fraction of long-timescale mean squared synchrony in PCI explained by our best long-timescale model was 61.1%. The fraction for short-timescales was only 3.1% (Methods). Thus growing season temperature and C. finmarchicus abundance were probably the dominant Moran effects producing synchrony (high WMFM) in PCI on long timescales (with temperature likely acting indirectly, see Discussion); and we have not explained short-timescale synchrony.

The significant fit on short timescales of the model with predictors C. finmarchicus, echinoderm, and decapod larvae abundances does not conflict with the result that the same model explains very little of the synchrony in PCI abundance; the two results illustrate that explaining dynamics and explaining synchrony are related but distinct concepts. The significant fit reflects the fact that local associations between the drivers and PCI are greater, on short timescales, than would be expected by chance. But spatial synchrony can only result when there is a strong association between drivers and PCI and, crucially, when the drivers themselves are substantially synchronous. The small fraction of synchrony accounted for at high frequencies indicates that most of the spatial synchrony at high frequencies results from factors other than the drivers we were able to identify. See the material on our wavelet Moran theorem in Methods for additional details.

Synchrony is more than its top-down and climatic components

Having just shown that top-down and climatic Moran effects produced most of the synchrony (mean squared WMFM) in PCI on long timescales, we now explore whether these effects interacted.

Randomizations showed that interactions between the Moran effects of growing season temperature and C. finmarchicus were significant and substantial in their influence on long-timescale PCI synchrony. In one randomization test, mean squared model synchrony ( in Methods) for our best long-timescale model was compared to the same quantity computed after C. finmarchicus data were replaced by surrogate datasets randomized so their patterns of synchrony were preserved but relationships with growing season temperature were eliminated (called spatially synchronous surrogates, Methods). Mean squared model synchrony is distinct from mean squared model-predicted synchrony, which was pictured in Fig 2c, blue line (Methods). The latter depends on model synchrony as well as on the strength of relationship between the model and PCI.

The reduction of synchrony from the randomization, attributable to the elimination of interactions between C. finmarchicus and temperature Moran effects, was substantial (Fig 3, cyan line versus black line). Thus these interactions contributed substantially to the synchrony of PCI. When C. finmarchicus data were instead replaced by asynchronous surrogates (Fig 3, magenta line), eliminating both interactions with temperature and C. finmarchicus synchrony itself, the further decrease was modest compared to the decrease attributable to interactions. This highlights the importance of interaction effects. The average of across long timescales using spatially synchronous surrogates of C. finmarchicus was only 84% of the average using the actual, unmanipulated data. The actual data yielded higher average than did 97.7% of spatially synchronous surrogates. Thus PCI synchrony is significantly higher than it would be if its two main Moran drivers were independent. Furthermore, we applied our synchrony attribution theorem (Methods) to calculate the fractions of long-timescale synchrony (mean squared WMFM) in PCI explained by, respectively, growing season temperature, C. finmarchicus abundance, and interactions between these: 41.4%, 5.4%, and 14.3%. Interaction effects were more important than the direct effects of C. finmarchicus abundance for PCI synchrony.

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Fig 3. Randomizations revealed that interactions between Moran effects were important for long-timescale PCI synchrony.

Mean squared model synchrony (Methods) at long timescales for our best long-timescale model (cyan line), compared to using synchrony-preserving surrogates (black line) and asynchronous surrogates (magenta line) of C. finmarchicus data. The best model had predictors growing season temperature and C. finmarchicus abundance, and synchrony-preserving surrogates randomized away interactions between these climatic and top-down Moran effects while retaining the effects themselves. Black and magenta lines show average results across 1000 surrogates. C. fin. = C. finmarchicus; unsync. refers to surrogates of the C. finmarchicus data for which synchrony, as well as relationships with temperature data, has been randomized away.

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Data

The CPR survey has consistently monitored plankton in the seas around the British Isles, on a monthly basis since January 1946 [53]. The survey, operated by SAHFOS from 1999 until recently when it was taken over by the MBA, uses voluntary merchant ships of opportunity to tow the CPR sampling device. At any given time several devices are in use, towed along different shipping routes. The device traps plankton on a continuously moving ribbon of silk, which is later cut into sections for microscope analysis. Each section represents a 10 nautical mile transect made at a particular time and place. The number of organisms of various types is recorded, together with the PCI, which corresponds to the amount of chlorophyll retained on the ribbon. Detailed descriptions of the survey are published elsewhere [22, 54].

The PCI has four values corresponding to observations of ‘No Green’, ‘Very Pale Green’, ‘Pale Green’, and ‘Green’. Colebrook gives the dilution factors needed to render samples with different colors the same color: ‘Very Pale Green’ = 1, ‘Pale Green’ = 2, ‘Green’ = 6.5 [53]. We used Colebrook’s original dilution factors to generate our time series (Appendix S3). Other authors indicate that the PCI is well correlated with satellite ocean chlorophyll measurements during the satellite period [21, 22, 55].

The ecological and physical variables to be compared with PCI and examined as potential Moran drivers or covariates are shown in Tables 1 and 2, respectively. We examined densities of 13 major zooplankton components from the CPR, including the important phytoplankton consumers C. finmarchicus and C. helgolandicus, calanoid copepod species. Data about physical oceanographic variables were downloaded from the International Comprehensive Ocean-Atmosphere Data Set (ICOADS) Release 2.5. We investigate measurements of temperature and salinity made in the top ten meters of the water column, and measurements of wind speed and cloud cover made at the sea surface.

Measurements of each of the biological and physical variables were compiled into 26 time series representing 2 by 2 degree areas of sea (Appendix S3, S4 Fig). Our annualized data reflect changes in the growing conditions of the annual phytoplankton bloom. Many generations of organisms contribute to the phytoplankton bloom each year, and their overall abundance is driven by climatic and other factors that vary on long timescales, and in the case of some climatic phenomena, periodically. This results in corresponding inter-annual variation in the PCI, which can be identified with timescale-specific methods. Annual time series for several variables are in S5–S9 Figs. For each variable the annualized time series were subjected to Box-Cox normalization (see Appendix S4, S2 Table, and [56]).

Statistical methods

We use a variety of methods, some new. Here we summarize the techniques and how to interpret their outputs, with details in the Supporting Information. Please see www.github.com/reumandc/wsyn for a software package for the R statistical programming language containing code for the techniques of this study and a vignette explaining how to use the code. We start with wavelet transforms, on which other methods are based. Time series can be regarded as composed of fluctuations on different timescales (i.e., of different characteristic oscillatory periods). The wavelet transform provides time- and timescale-specific information on these fluctuations. We use a complex Morlet transform, as demonstrated in S10 Fig and described mathematically in Appendix S5.

The spatial coherence [7] of two spatiotemporal variables and measured at locations n = 1, …, N and times t = 1, …, T quantifies, in a timescale-specific way, the strength of the association between the variables. The spatial coherence, which is based on the ‘power-normalized’ (Appendix S5) wavelet transforms and of and , is a function of timescale, σ. It takes values between 0 and 1, with higher values meaning a stronger association (Appendix S6).

Unlike commonly used correlation measures, the spatial coherence has the benefit of finding associations even if fluctuations are strongly related only on particular timescales, and even if there are phase delays between the corresponding fluctuations. A demonstration of the spatial coherence technique is in S11 Fig.

We approached question 1 from the Introduction via a multivariate linear modeling approach for wavelet transforms (Appendix S7) that made it possible to quantify determinants of synchrony in PCI and their interactions. If for k = 0, …, K are spatiotemporal variables (n = 1, …, N, t = 1, …, T) and are their power-normalized wavelet transforms (Appendix S5), then the linear models statistically explain variation in the response transforms in terms of the predictor transforms : (2)

We take to be PCI, and for k > 0 to be potential influences on PCI (different variables selected from Tables 1 and 2 in different models). Such models can be fitted with data; for a given model, the fitted coefficients of the model, which are functions of timescale, maximize (Appendix S7) the spatial coherence between the left and right sides of Eq 2 at every timescale. Two nested models can be compared statistically using surrogate resampling approaches (Appendix S8) which are standard in wavelet statistics; conceptually, these are analogous to standard F-tests between nested linear models in the sense that a more complex model is tested against a simpler alternative. But F-distributions are not used in the wavelet case. Conventional significance tests assuming serially independent data are inappropriate for spatially synchronous wavelet transform data in which correlations are present between the values. The Fourier surrogates we use (Appendix S8) are a standard alternative.

A model selection and testing procedure (Appendix S9) determines which predictors k = 1, …, K should optimally be included for a given range of timescales. These predictors are the good candidates for statistically explaining PCI dynamics and synchrony on those timescales. For a model to be ranked, each predictor transform (k > 0) included in the model was required to provide a significant improvement over the model with that predictor removed. Among those models for which all predictors were significant, we performed leave-one-out cross validation to select the best model and avoid over fitting. Models were constructed separately for short (< 4 years) and long (> 4 years) timescales. The choice of these two timescale ranges was justified in the main text and in Appendix S9.

Best models selected by the model selection procedure were tested in additional ways (Appendix S10-Appendix S12) beyond the testing that was built into the selection procedure (Appendix S9). First, the phases of the model coefficients β_k(σ) are supposed to represent the phase lag of effects (or associations) at timescale σ of on (Eq (2) for notation). Strong associations which are consistent over time and space should therefore produce a correspondence between the phase of β_k(σ) and the average phase, over space and/or time, of , where the overbar denotes complex conjugation; the phase of this quantity is the phase difference between and at timescale σ, time t, and location n. We tested for this phase correspondence and found it to be good (Appendix S10). The phases of the β_k for our best-fitting models were extracted for interpretation of model output (Results), to determine the lag between the fluctuations in the kth predictor and the corresponding content in the PCI wavelet transforms. Details of this process are also in Appendix S10.

Another test of the potential usefulness of our top-ranked models as tools to reveal Moran effects had to do with whether the models represented ‘local associations’. An alternative possibility is that PCI is statistically associated with these variables on average across the the British seas sampled, but that seawater mixing and dispersal prevent the dynamics of PCI from being attributable to local drivers. This alternative possibility would suggest a dominant role of dispersal as a synchronizing agent rather than Moran effects. We tested for local associations between PCI and the four predictor variables appearing in top models at long or short timescales through a spatial permutation test. Results indicated that associations were local in all cases, with details in Appendix S11.

Finally, spatial coherences between nine individual phytoplankton species abundances from the CPR data set and C. finmarchicus were tested in earlier work [38], and those results were also consistent with our findings that PCI and C. finmarchicus were significantly spatially coherent and in anti-phase (Appendix S12).

Given a spatiotemporal dataset x_n(t), we used two tools for describing the synchrony among these time series, and its time and timescale dependence. The wavelet mean field magnitude (WMFM) plot, here denoted |r_σ(t)|, is the magnitude of . The WMFM plot is applied in Fig 1b, 1c and 1e–1g, from which one can get a sense of the nature of the output of the method and how to interpret that output. See Appendix S13 for details. We refer to the quantity as the mean squared synchrony of the x_n(t), where {⋅}_t here and henceforth denotes the time average of the square of the quantity in braces. The wavelet phasor mean field magnitude (WPMFM; Appendix S13) provides a plot in the same format as the WMFM, but quantifies only the phase synchrony of the x_n(t) as a function of time and timescale. Phase synchrony is always between 0 and 1 and equals 1 at the t and σ for which time series have identical phases of oscillation. Phase synchrony can be straightforwardly tested for significance at t and σ. Details on these techniques are in Appendix S13 and a demonstration is in S12 Fig. WMFM and WPMFM plots for PCI are in S2 Fig, and for several other variables in S13–S16 Figs.

Having determined robust wavelet models (Eq 2) of PCI transforms for long and short timescales as described above, we applied a new ‘wavelet Moran theorem’ (Appendix S14) to quantify to what extent the synchrony in PCI can be explained by the predictors included in our best models. The theorem is a generalization of an earlier result [7]. A wavelet model can be a good model, in the sense that it is validated by our testing procedures and explains a significant amount of variation in , without contributing to synchrony. Our Moran theorem allows us to quantify how well the predictors explain synchrony. In words, the theorem states that at timescale σ, if the only synchronizing influences on a biological variable are the biotic or abiotic variables for k = 1, …, K, then the strength of synchrony of the biological variable is the strength of synchrony in the best-fitting wavelet model times a spatial coherence of that model with the biological variable. In formulas, , where is the WMFM of , is a WMFM for the best-fitting wavelet model, and is a spatial coherence of the model with the biological variable. The left side measures synchrony of the biological variable and the first multiplicand measures model synchrony for a model that accounts for the variables , k = 1, …, K. The theorem formalizes the reasoning that synchrony of a driven variable should depend on synchrony of the combined effects of the drivers and on the strength of the influence of the drivers. Additional independent drivers may act to increase synchrony of the , turning the approximate equality, ≈, into an inequality, >. The right side of the Moran formula is then the amount of synchrony attributable to the drivers for k = 1, …, K.

The theorem also provides a ‘predicted synchrony’ plot for a model, which has format like the WMFM plot and can be compared to it, but shows the pattern of synchrony that would pertain if the only synchronizing influences on were the variables (Appendix S15). Observed synchrony of (i.e., ) is expected to be strictly greater than predicted synchrony, the difference corresponding to synchronizing influences not included in the model. Using the Moran theorem, it is straightforward to quantify the fraction of synchrony explained by the model for a timescale band (Appendix S15). With being PCI, we calculated predicted synchrony and fractions of synchrony explained for long and short timescales using our best long- and short-timescale models, respectively.

To investigate whether interaction effects of Moran drivers were important for the synchrony of PCI, we considered mean squared model synchrony of the best models chosen by our methods, and we analyzed how it was affected by randomizations of predictor variables that eliminated any interaction effects that may have occurred (Appendix S16).

We also proved an extension of our wavelet Moran theorem (theorem 1 Appendix S14), dubbed the ‘synchrony attribution theorem’ (theorem 2 Appendix S17), that makes it possible to partition synchrony explained by a wavelet model into components explained by individual predictors, and a component explained by interaction effects between predictors. We applied the theorem with again being PCI. We tested our methods on the numerical example of the Theoretical illustration section of Results (Appendix S18).