Assessing ecological responses to environmental change using statistical models

Authors


Correspondence author. E-mail: claire@stats.gla.ac.uk

Summary

  • 1There is a clear need to improve our ability to assess the ecological consequences of environmental change. Because of the complexity of ecosystems and a need to disentangle the effects of multiple pressures, predictions are often reliant on models and expert opinion. These require validation with observed data; in this respect, long-term data sets are particularly valuable.
  • 2Innovative statistical methods (nonparametric regression and additive models) are presented for identifying nonparametric ecological trends and changes in seasonality in response to environmental change. These are illustrated through the example of Loch Leven, a shallow freshwater lake. Monitoring data for 35 years are examined, spanning periods of enrichment, ecological recovery and changing climate.
  • 3Models are developed for phosphorus and nitrogen; temperature and rainfall; Daphnia grazers; and chlorophyll a, with the ecological objectives of examining trends in water quality and the corresponding trends in nutrient availability, grazer abundance and climate.
  • 4The analysis highlighted a generally decreasing availability of P over the study period, and generally increasing nonparametric trends in nitrate concentration and rainfall. Increasing spring temperatures were also evident, as were significant nonparametric changes in density of summer grazers.
  • 5Significant reductions are highlighted in spring and summer chlorophyll a, related to the return of Daphnia to the loch. However, no response in chorophyll a to the later declining trends in P is apparent, but seasonality has changed.
  • 6Synthesis and applications. The analysis highlights the value of nonparametric statistical models for assessing complex ecological responses to environmental change. The models outlined can examine key ecological impacts of climate change, particularly effects on the timing of seasonal events and processes. The models are illustrated using long-term water-quality data from Loch Leven to explore patterns in key environmental drivers and ecological responses affecting freshwater ecosystems. Analysis of chlorophyll a, in particular, highlighted the value of examining the seasonal trends separately, with different trends evident for winter and spring and a changing seasonal pattern.

Introduction

There is a clear need to improve our ability to assess the ecological consequences of environmental change. Because of the complexity of ecosystems, predictions are often reliant on models and expert opinion (Sutherland 2006). These require validation with observed data; in this respect, long-term data sets are particularly valuable.

Assessing environmental change at an ecosystem level often requires assessing whether annual trends are significant and whether seasonality is changing. However, ecological time series are often very complex, with nonlinear and nonparametric trends over time and strong seasonality. More novel approaches to statistical analysis of ecological time series are needed to account for these issues. It is of particular interest to explore the average pattern over the years (annual trend), the average pattern over the years for each season separately (seasonal trend), and the average pattern within the year (seasonality) for responses.

This paper details innovative statistical methods for identifying trends and seasonality in ecological responses in complex, long-term ecological data sets. These are illustrated through the example of Loch Leven, a shallow freshwater lake in central Scotland. Over 35 years’ monitoring of water quality and plankton populations has been carried out, spanning periods of toxic pollution, nutrient enrichment, ecological recovery and changing climate. The data set is typical of long-term ecological time series in that there is a need to disentangle the effects of multiple pressures acting on the site. In particular, at Loch Leven it is of interest to understand how climate changes may affect ecological recovery from nutrient enrichment.

Climate change is recognized to affect the ecological quality of lakes in many ways, through changes to physical and hydrological parameters (water temperature and flushing rates), chemical regimes (catchment- and sediment-derived nutrient loading, water colour) and biological changes (Eisenreich 2005). Models, expert opinion and experimental mesocosm studies can all suggest how lake ecosystems may respond to change, but all these approaches require validation using observations from long-term monitoring studies.

The development and application of additive and nonparametric regression models are illustrated for this purpose. Models are developed for: (1) soluble reactive phosphorus (SRP) and nitrate (NO3-N), the main nutrients potentially limiting phytoplankton production in this system; (2) temperature and rainfall, important climatic variables; (3) Daphnia, the dominant phytoplankton grazer in the system; and (4) chlorophyll a, a measure of the phytoplankton standing crop and a key measure of the ecological status of freshwaters in the European Union Water Framework Directive (European Parliament 2000).

Materials and Methods

study site

Loch Leven is situated in lowland Scotland, in the Perth and Kinross area. It is the largest shallow, eutrophic lake in Great Britain, with an area of 13·3 km2, mean depth 3·9 m and maximum depth 25·5 m. The water draining into the loch comes from direct rainfall and run-off from the agricultural catchment and is used by various industries downstream. The loch is an important trout fishery and is also a Ramsar site (http://www.ramsar.org) and National Nature Reserve. An action programme to improve the ecology and water quality of the loch, focused on reducing P loadings, began in the 1980s (Bailey-Watts & Kirika 1987, 1999). For further site information see Jupp & Spence (1977); Bailey-Watts (1978); Carvalho & Kirika (2003).

data

The Centre for Ecology & Hydrology has monitored approximately 150 variables at the loch since 1968. Samples are taken predominantly from Reed Bower, an area near the centre of the loch, and the sampling dates are a mixture of weekly, biweekly and monthly with long periods of missing data, especially in the 1980s (Ferguson et al. 2007).

There are six key variables in this study. For soluble reactive phosphorus (SRP), nitrate (NO3-N), Daphnia and chlorophyll a, the raw sampling dates have been aggregated to monthly means and a natural log transform has been applied to each variable. The data have also been aggregated to seasonal means to explore trends over the time period for each season, where winter is (Dec, Jan, Feb), spring is (Mar, Apr, May), summer is (Jun, Jul, Aug) and autumn is (Sep, Oct, Nov). For air temperature, average daily values have been calculated using (max + min)/2 and data have been aggregated to monthly and seasonal means: this is referred to as monthly/seasonal mean air temperature throughout this paper. However, for rainfall, monthly and seasonal cumulative rainfall values are used.

statistical methods

Classical approaches to modelling trends and seasonality in water quality data include Mann Kendall and seasonal Kendall tests (Hirsch, Slack & Smith 1982; Hirsch & Slack 1984). However, such tests assume monotonic trends. Here we highlight the extra information that can be gained from allowing greater flexibility in statistical models using additive and nonparametric regression models, with correlated errors incorporated where necessary.

The following three models can be used to explore trends and seasonality fully for each of the variables of interest:

y = µ + m1(year) + m2(month) + ɛ, ɛ = N(0,Vσ2)((eqn 1))
y = µ + m(years) + ɛ, ɛ = N(0,σ2)((eqn 2))
y = µ + m(year,month) + ɛ, ɛ = N(0,Vσ2)((eqn 3))

where m( ... ) is a univariate smooth function for year or month in models 1 and 2, or a bivariate smooth function for both in model 3. The errors, ɛ, are assumed to be normally distributed, with mean 0 and variance σ2 in model 2, and mean 0, correlation matrix V and variance σ2 in models 1 and 3.

Model 1 is used to consider the trend m1(year) and seasonality m2(month) over time for each response variable, for example log SRP, and model 2 is used to consider seasonal trends, m(years). In the latter case, the response contains the mean for a particular season for a variable of interest overtime, for example air temperature in spring. Model 3 is used to consider how the seasonal pattern across months, within each year, changes over the time period. In models 1 and 3, an AR(1) correlation structure is assumed for the errors. Therefore each model is fitted initially assuming independent errors to estimate the lag 1 correlation of the residuals, ρ. The matrix V is then constructed using inline image

model fitting

Model 2 is a nonparametric regression model with independent errors (see, for example, Bowman & Azzalini 1997). The local linear method of smoothing (Cleveland 1979) is used to fit the estimated smooth function, m̂(years), where m̂( ... ) denotes the estimated function, and a normal kernel density is used for the weights, with mean zero and standard deviation h. The smooth function m̂(years) can be expressed as Sy and the smoothing parameter, h, is determined by choosing the degrees of freedom (df) to be tr(S) = 5. Here the value of the degrees of freedom has been chosen to define the complexity of the model with this choice (df = 5) allowing a moderate degree of nonlinearity. However, results are not very sensitive to different values for degrees of freedom in this study of Loch Leven. This common value also allows consistency across models, with all smooth functions having the same degrees of freedom. While this choice of 5 df is appropriate for Loch Leven, for data sets that are sparser or have a different data aggregation this may not be appropriate. There are various other methods for selecting smoothing parameters, including cross-validation and generalized cross-validation (Hastie & Tibshirani 1990; Wood 2006). However, selection of the smoothing parameter becomes more difficult when correlation is involved, and automatic methods may not be appropriate.

For more than one covariate, nonparametric regression can be extended to an additive model (Hastie & Tibshirani 1990) such as model 1. Correlated errors are incorporated for model 1, which is fitted using the backfitting algorithm. A normal kernel density is used to construct the weights for the year component. However, for cyclic terms such as month, a smooth function can be obtained using local constant regression with a circular smoother used for the weights; for full details see Giannitrapani et al. (2005); Ferguson et al. (2007). Smooth functions, m̂j(xj) = Sjy, can be obtained for each of the components j = 1,2 in the model, and for each component df = tr(Sj) – 1 = 4 to reflect the fact that each term, after backfitting, is constrained to have mean zero, thus eliminating 1 df.

Model 3 is an extension of model 2 to two dimensions. In this case, the formulation of the bivariate smooth component is similar to that used for the year component in the additive model, with a product of weight functions formed using a normal kernel density for year and a circular smoother for month. The function m̂(year,month) can be expressed as Sy and the smoothing parameters are determined by setting df = tr(S) = 10. Again, this value for degrees of freedom has been chosen to define the complexity of the model allowing a moderate degree of nonlinearity for the bivariate component.

For each variable, three time periods were considered to explore the system: 1968–2002, 1971–2002 (excludes the first 3 years when Daphnia were absent as a result of probable pesticide pollution), and 1988–2002 (recent period with continuous monitoring).

For each of the variables of interest, plots are produced against each of the covariates in the additive models. On each plot, the fitted values are displayed along with a shaded band indicating ±2 SE from these estimates. Details of the standard error (SE) calculations are provided by Giannitrapani et al. (2005); Ferguson et al. (2007). For nonparametric regression, seasonal figures are provided with a reference band for ‘no effect’ displayed on each plot; for details about the construction of the reference band see Bowman & Azzalini (1997). The reference band highlights where the mean would be expected to lie if the nonparametric effect is not significant. This is described further in following section.

model testing

An approximate F test (Hastie & Tibshirani 1990) is used to test hypotheses concerning model components. Construction of the residual sums of squares for models 1 and 3 is modified to incorporate correlation; see Giannitrapani et al. (2005) for details. For both models being compared, the correlation matrices and smoothing parameters are equal.

For the additive models, it is of interest to test whether components are significant in addition to one another (the hypotheses of no effect vs. effect, model 4), and whether the nonparametric effect is necessary or a linear component is adequate (the hypotheses of linear vs. nonparametric effect, model 5). The month term is not tested for a linear effect as it is a cyclic component.

Model 2 was tested in terms of the hypotheses no effect vs. effect, in order to investigate if seasonal trends are significant (model 6). The reference bands provided on the corresponding figures for such models indicate where the mean (the model specified in the null hypothesis, H0) would be expected to lie if H0 is true. The hypotheses in model 7, which include an additive and bivariate model respectively, are compared to investigate if the seasonal pattern across the year changes significantly over the time period considered. In order to allow comparison between the bivariate and additive model, the smoothing parameters have to be equal. As a compromise, the geometric mean of the univariate and bivariate smoothing parameters has been calculated for each model component, and these new smoothing parameters are used for both models in the testing procedure.

H0: E{y} = µ + m1(year)
H1: E{y} = µ + m1(year) + m2(month) ((eqn 4))
H0: E{y} = µ + β(year) + m2(month)
H1: E{y} = µ + m1(year) + m2(month) ((eqn 5))
H0: E{y} = µH1: E{y} = µ + m(years)((eqn 6))
H0: E{y} = µ + m1(year) + m2(month)H1: E{y} = µ + m(year,month)((eqn 7))

software

The statistical analysis was carried out using r, free software available at http://www.r-project.org. Figures and tests for seasonal trends were produced in r using the sm library (Bowman & Azzalini 1997) using, in particular, the sm.regression function. Examples of code for the other figures (for additive models) can be found in Appendices S1 and S2 in Supplementary Material.

Results

exploring trends and seasonality

Figure 1(a) illustrates generally decreasing availability of SRP. However, a slight increase is evident in the late 1980s and early 1990s. There are significant generally decreasing annual trends for all time periods considered (Table 1), and strong seasonality (Fig. 1b). Table 2 highlights that, while the whole period (1968–2002) is borderline nonparametric, trends can be considered linear over the later period.

Figure 1.

Trend and seasonality plots for log soluble reactive phosphorus (SRP) as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·36.

Table 1.  Additive model test results for no effect – approximate F test
VariableP for no effect vs. effect
SRP*Nitrate NO3-NAir temperatureRainDaphniaChlorophyll a
  • *

    SRP, soluble reactive phosphorus.

Year (68–02)<0·0010·0110·175<0·001<0·001<0·001
Month (68–02)<0·001<0·001<0·001<0·001<0·001<0·001
Year (88–02)0·026<0·0010·4580·1660·6980·838
Month (88–02)<0·001<0·001<0·001<0·001<0·001<0·001
Table 2.  Additive model test results for linearity – approximate F test
VariableP for linear vs. nonparametric effect
SRP*Nitrate NO3-NAir temperatureRainDaphniaChlorophyll a
  • *

    SRP, soluble reactive phosphorus.

Year (68–02)0·0570·0050·7540·169<0·001<0·001
Year (88–02)0·3190·0040·4010·2220·5620·725

For nitrate (Fig. 2a), there are significant annual trends for all time periods considered (Table 1) and these are generally increasing. However, the greatest increase appears to be from the mid-1990s onwards. There is also strong seasonality (Fig. 2b) and Table 2 highlights evidence of a nonparametric trend in each time period.

Figure 2.

Trend and seasonality plots for log NO3-N as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·52.

While a slight increasing trend and strong seasonality are evident in Fig. 3 for mean air temperature, the annual trends in all time periods are not significant. Table 2 highlights that it is reasonable to assume that trends are linear, and this is also true for cumulative rainfall. A general increase in rainfall (Fig. 4a) is evident over the whole period. However, the trend in the latter period (1988–2002) is not significant, Table 1.

Figure 3.

Trend and seasonality plots for mean air temperature as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·34.

Figure 4.

Trend and seasonality plots for cumulative monthly rainfall as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·03.

Tables 1 and 2 highlight that there is a significant nonparametric trend for Daphnia over the whole time period. However, following the reappearance of Daphnia grazers in the loch in 1971, there have been no significant trends in annual mean grazer densities, P = 0·529 (for 1971–2002, not quoted in the table). There is significant seasonality in Daphnia densities, with peaks in late spring and early summer (Table 1; Fig. 5b). Table 2 highlights that over the whole period (1968–2002) the trend appears to be nonparametric. However, discounting the early years, the trend could be considered linear (P = 0·728 for 1971–2002, not quoted in the table).

Figure 5.

Trend and seasonality plots for log Daphnia as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·68.

For chlorophyll a, additive models indicate significant nonparametric declining trends and strong seasonality (Tables 1 and 2; Fig. 6) for the whole period 1968–2002. However, for the period 1988–2002, while seasonality is still strong, the annual nonparametric trend is not significant and Table 2 highlights that it is highly likely to be linear.

Figure 6.

Trend and seasonality plots for log chlorophyll a as a response from January 1968 to December 2002. Separate component plots for year and month with shaded regions to identify ±2 SE from the estimate. The lag 1 correlation of the residuals is 0·49.

exploring seasonal trends

The main decrease in SRP was in summer, autumn and winter (Fig. 7) over the whole period. However, nonparametric trends in the latter period are not significant with the exception of a borderline P value for autumn (Table 3).

Figure 7.

Scatterplots with nonparametric regression for log SRP seasonally from January 1968 to December 2002. A reference band for ‘no effect’ is also provided.

Table 3.  Nonparametric regression test results for ‘no effect’– approximate F test
SeasonSRP*NO3-NAir temperatureRainDaphniaChlorophyll a
  • *

    SRP, soluble reactive phosphorus.

P for no effect vs. effect, 1968–2002
Spring0·0480·1360·0570·098<0·001<0·001
Summer0·0380·0510·9030·162<0·0010·003
Autumn0·0920·6500·1530·1560·0360·321
Winter0·0500·3690·1420·0220·0040·021
P for no effect vs. effect, 1988–2002
Spring0·6110·1380·3280·2150·3650·627
Summer0·1370·0710·4610·0350·5660·225
Autumn0·0500·5270·2690·5100·7940·366
Winter0·1040·0790·5620·8570·5950·539

However, for mean air temperatures, while annual nonparametric trends were not significant, a borderline significant, generally increasing trend is highlighted for spring temperatures over the whole period (1968–2002), but not for the later years (1988–2002) (Table 3; Fig. 8). There also appears to be a generally increasing trend in winter. However, this nonparametric trend is not significant. Results are more significant when water temperature (P = 0·021 for spring 1968–2002, not quoted in the table) is considered, as opposed to mean air temperature.

Figure 8.

Scatterplots for mean air temperature from January 1968 to December 2002: (left) seasonal with nonparametric regression curves; (right) for spring with a reference band for ‘no effect’.

For cumulative rainfall, there is a significant, generally increasing trend for winter (1968–2002) (Fig. 9; Table 3). Spring and summer are generally wetter too, but nonparametric trends are not statistically significant over the whole period (Table 3).

Figure 9.

Scatterplots with nonparametric regression for cumulative seasonal rainfall from January 1968 to December 2002, with a reference band for ‘no effect’.

For Daphnia, in the period 1971–2002, summer densities are the only season to show significant changes (P = 0·025, not quoted in the table) with a decreasing trend evident until the early 1990s, followed by an increasing trend to 2002 (Fig. 10). For the later period, no seasonal nonparametric trends were apparent, and this is also true for chlorophyll a. However, there were significant reductions in spring and summer for the whole time period due to big reductions early in the first 3 years for chlorophyll a (Fig. 11; Table 3).

Figure 10.

Scatterplots for log Daphnia from January 1971 to December 2002: (left) seasonal with nonparametric regression curves; (right) for summer with a reference band for ‘no effect’.

Figure 11.

Scatterplots with nonparametric regression for log chlorophyll a seasonally from January 1968 to December 2002, with a reference band for ‘no effect’.

exploring changes in seasonality

Figure 12 highlights bivariate plots for log chlorophyll a and log SRP. In these plots, it is of interest to consider changes in the shape of the curved surface in the time (year) direction and changes in positions of peak/troughs in the monthly direction. Only log chlorophyll a shows any clear change in seasonality over the whole time period, with a significant P value of 0·009 (Table 4; Fig. 12, left). A much reduced and earlier spring peak (from April to February) is highlighted along with a more prominent ‘clear-water’ period in late spring/early summer (May/June). The seasonal patterns for the other variables under consideration have generally remained the same from 1968 to 2002 (Table 4). However, as illustrated in Fig. 12 (right), there is evidence of a slight change in the seasonality of SRP, although this is not significant.

Figure 12.

Bivariate plots for log chlorophyll a (left) and log soluble reactive phosphorus (SRP) (right) as responses with a bivariate term m(year,month). The lag 1 correlations of the residuals are 0·48 and 0·41, respectively.

Table 4.  Testing for changes in seasonality – approximate F test
VariableP for changes in seasonality
SRP*NO3-NAir temperatureRainDaphniaChlorophyll a
  • *

    SRP, soluble reactive phosphorus.

Year,month0·1130·2040·8700·3400·2240·009

Discussion

Freshwater communities, with their short generation times, allow exploration of impacts of environmental change on ecosystem structure and functioning.

The abundance of phytoplankton, in particular, is a key indicator of water quality and ecological status, recognized in recent European legislation (European Parliament 2000). Chlorophyll a concentration in the water column is a widely recognized simple measure of phytoplankton abundance, and so is of particular interest in any assessment of the impacts of environmental change in freshwaters. This study aimed to examine trends and seasonality in chlorophyll a, as well as the main potential drivers of change in the phytoplankton community, notably temperature and rainfall, nutrients (SRP and nitrate availability) and dominant grazer (Daphnia) densities.

Nonparametric regression and additive models were used to assess whether trends in these key ecological variables are significant, linear or nonparametric, and whether there have been any changes in seasonality.

annual trends

The clearest annual trend in the data set is the significant reduction in SRP concentrations, a key nutrient often limiting phytoplankton crops. In particular, trend analysis over the last 15 years studied indicated significant reductions, highlighting the success of more recent management to reduce point-source inputs from industry and sewage works in the catchment. There is also growing evidence that internal loads from the sediments have been decreasing since the late 1990s (Carvalho & Kirika 2002). Conversely, nitrate concentrations appear to be increasing in recent years, particularly since the mid-1990s. This nutrient is largely derived from diffuse agricultural sources in the catchment (Bailey-Watts & Kirika 1999) and may therefore be increasing partly in response to enhanced run-off associated with the increasing rainfall trend (Heathwaite & Johnes 1996), but may possibly also indicate increasing nitrate inputs in the catchment from agriculture, waste effluents and atmospheric sources. The nonparametric trends observed for both chlorophyll a and Daphnia were well described by the additive modelling approach. The models highlighted that significant reductions in chlorophyll a concentrations were apparent only early in the time series, largely following the reappearance of Daphnia populations. For the most recent period (1988–2002), no significant trends were apparent for either chlorophyll a or Daphnia. The lack of a chlorophyll a response to the more recent P reductions suggests that P may no longer be limiting phytoplankton over this period. In terms of an annual chlorophyll a response, increases in nitrate availability may have offset P reductions, or limitation by grazers may be more important.

seasonal trends

Annual trends do not, however, reflect changes in the seasonal processes occurring in a lake: it may be that P or N availability or grazer densities show more distinct seasonal trends. Trends in climate parameters such as temperature and rainfall also differ between seasons, with winter and spring temperatures generally increasing more than summer temperatures over recent decades.

As expected, trends were much more apparent when seasons were examined separately. In terms of climatic changes, the trend to warmer springs, as observed here, has also been observed in many other studies and potentially has direct physiological effects (e.g. enhanced growth rates) on plankton communities (Petchey et al. 1999; Anderson 2000) as well as effects on phenology (the timing of spring blooms and clear water phases), and therefore changing relationships between predators (Daphnia) and prey (phytoplankton) (Anneville et al. 2004; Winder & Schindler 2004).

The fact that chlorophyll a also showed significant declining trends, particularly in spring, suggests it could be an indirect response to the warmer spring temperatures (a direct response is generally assumed to result in an increase in chlorophyll a). The seasonal breakdown of chlorophyll a trends also appears to show a further recent recovery in spring and summer chlorophyll a since 2000 although, as for temperature, these trends are not significant if the last 15 years (1988–2002) are considered. However, it is not immediately obvious how warmer springs would result in reduced algal biomass. Other than temperature, potential drivers of the spring chlorophyll a reductions could be reduced SRP availability or increased Daphnia density. There is not much evidence for the former, as spring SRP concentrations generally increased from 1968 to 1975 (the period over which greatest reductions in chlorophyll a occurred) and have remained more-or-less unchanged since 1995 (the later period of further chlorophyll a reductions). Summer SRP concentrations have declined since 1995 and may therefore be responsible for some of the reductions observed in summer chlorophyll a since 1995. However, there is much more supporting evidence for the role of Daphnia in limiting spring phytoplankton, as the big decline in chlorophyll a concentrations in the early 1970s is consistent with the reappearance of Daphnia in the loch. This followed several years’ absence, thought to be due to regular pesticide pollution from industry in the catchment. There is also evidence for increased spring Daphnia densities since 2000, which could be responsible for the reductions in chlorophyll a observed over recent years.

The effects of pesticides were probably responsible for the changes in Daphnia density observed before 1971. Other studies have shown the positive effect on Daphnia abundance of increasing spring temperatures (Gerten & Adrian 2000), thought to be associated with enhanced reproductive rates in warmer waters (Peters & De Bernardi 1987). However, in the example of Loch Leven, there is no explicit evidence to suggest that temperature changes were responsible for determining spring Daphnia densities, although a previous analysis (Ferguson et al. 2007) showed significant positive relationships between late winter/early spring temperatures and spring Daphnia densities.

In contrast to changes in spring, winter chlorophyll a concentrations appear to show an increasing trend from the early 1980s. This may possibly be associated with enhanced growth rates associated with the slightly warmer winters observed, although neither trend was significant when the last 15 years (1988–2002) were considered. Rainfall also showed a generally increasing statistically significant trend in winter. This may have resulted in enhanced loading of nutrients, such as nitrate, that are predominantly from diffuse sources (Heathwaite & Johnes 1996), which could also have helped support increased phytoplankton productivity. However, increased rainfall also results in an increased flushing rate and potentially enhanced losses of phytoplankton from the lake (Bailey-Watts et al. 1990).

The differing chlorophyll a trends in winter and spring highlight the importance of examining seasonal trends. It is possible that reductions in spring phytoplankton are offset by increases in winter crops, resulting in no clear changes if examined as an annual measure.

seasonality

All variables showed significant seasonality, represented well by the smooth function of month in the additive models. The seasonality of SRP shows clear minima in spring, to levels close to the limit of detection, while nitrate remains high. In summer the opposite is true, with nitrate declining to undetectable levels while SRP concentrations increase. There is clearly a switch from a more P-limited system in spring to a more N-limited system in summer, which has implications for catchment management of both N and P sources, particularly diffuse sources of N in summer.

One major innovation of the statistical models outlined is that they allow analysis of changes in seasonality. This is only apparent for chlorophyll a with earlier (but much reduced) peaks in late winter/early spring (February/March) and more obvious minima in late spring/early summer (May/June). The earlier peak may be a response to slightly warmer winter and spring months, and the clearer minima could also be an indirect response to temperature via increased grazers over these later months. The chlorophyll a minima certainly occur concurrently with the Daphnia maxima over these 2 months. A previous analysis of relationships between ecological responses and environmental drivers (Ferguson et al. 2007) highlighted a significant positive relationship between spring water temperatures and spring Daphnia, providing supporting evidence for a climatic effect on chlorophyll a seasonality.

assessing ecological responses to environmental change

Understanding how ecosystems respond to global change is of vital importance for predicting impacts of anticipated climate warming. In particular, understanding how climate change may influence the ecological quality of freshwaters is particularly relevant, given the objective of the EC Water Framework Directive to achieve good ecological status in European waters by 2015 (European Parliament 2000).

The statistical modelling of the Loch Leven data sets illustrates a number of advantages for assessing environmental changes in ecological data sets. First, although many physical or chemical drivers of change may show more-or-less linear trends (e.g. temperature and P changes at Loch Leven), biological responses (e.g. chlorophyll a and Daphnia) often show more complex, nonlinear trends. For this reason, nonparametric models are required to assess patterns over time and throughout the year. Autocorrelation is also common in ecological data sets, and these models present methods for incorporating correlated errors where necessary. Cyclic components, such as month of the year, can also be included using a circular smoother, to enable investigation of seasonal patterns across the months of the year, in addition to trends.

Because of strong seasonality, environmental changes in freshwater ecosystems are often assessed using annual measures of nutrients or chlorophyll a (OECD 1982). However, this may mask important seasonal trends. The analysis at Loch Leven highlights the greater scope for identifying, or at least implicating, the drivers or processes responsible for changes without constraining trends to be linear. Clearly cause and effect cannot be identified, but at least strong and significant relationships between variables can be used to infer possible hypotheses for further investigation. With respect to this study, the role of spring Daphnia populations in the improvements in spring water quality merits further study.

While these models provide an extremely valuable exploratory view of the patterns within variables over time, relationships between variables and the effect of covariates on responses in the system are not considered. Ferguson et al. (2007) use extensions of these models with covariates incorporated in the modelling, along with terms for trend and seasonality, as well as considering both contemporaneous and lagged relationships between system responses and covariates. Ferguson et al. (2006, 2007) also highlight that univariate and multivariate varying-coefficient models are an effective way to illustrate how relationships between variables change throughout the year. Both modelling approaches are an aid to disentangling the effects of nutrient and climate variables on water quality and grazers, and help to provide insight into the different effects of climate change and eutrophication.

The analysis also highlighted the great value of long-term ecological research monitoring sites. Ecosystems are rarely affected by only a single pressure; it is much more likely that sites are affected by multiple pressures with synergistic or opposing effects, such as eutrophication and climate change. To be able to disentangle the effects of these pressures requires many decades of data. The trend analysis of the shorter 15-year period illustrates this well, with very few variables showing statistically significant trends that are distinguishable from natural ecosystem variability.

With climate change being a major political issue, it is likely to become increasingly important to demonstrate convincing, statistically supported evidence of ecological impacts. This study illustrates the application of the models for assessing changing patterns in seasonality through the use of bivariate nonparametric regression. Such models can be used to explore significant shifts in the phenology of seasonal events (e.g. spring clear-water phase, flowering) and the changing seasonality of processes (e.g. predator–prey relationships). They are likely to prove extremely valuable in highlighting the ecological impacts of climate change.

Acknowledgements

C.F. gratefully acknowledges research student funding from the Department of Statistics, University of Glasgow. Laurence Carvalho was, in part, funded for this work by the Eurolimpacs Project. Eurolimpacs is funded by the European Union under Thematic Sub-Priority 1.1.6.3 ‘Global Change and Ecosystems’ of the Sixth Framework Programme. The Centre for Ecology & Hydrology gratefully acknowledges Loch Leven Estates for providing access to the loch and assistance with field work over the years, and Loch Leven Estates data providers, particularly Glen George, Iain Gunn and Gavin Thomas for Daphnia data.

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