Experiment 2: grazing effects across the shore using a transect
The experimental design was based on the use of geostatistical tools (semivariograms and cross-semivariograms), which estimate how the variance of a measured variable depends on the distance intervals (or the lag) between samples. This allows the determination of spatial patterns in the data (Fortin & Dale 2005).
Positive values of LRR indicate that grazers promote the abundance of the algae through fertilization (Taylor & Rees 1998) or by releasing them from competitors (McQuaid & Froneman 1993), while negative values indicate reduction of the algal abundance by consumption. Zero values indicate no grazing effects.
The log-response ratios were analysed in three ways: (i) LRR was used as a dependent variable to study the spatial pattern of variability in grazing strength across the shore using semivariogram analyses; (ii) for descriptive purposes, LRR was regressed against MLWS to analyse the zonation of grazing strength across the shore; and (iii) the relationships between LRR and independent physical (aspect, elevation, water movement) and biological (densities of grazer size classes) predictor variables across the shore were studied using cross-semivariogram analyses. Average density of grazers in each block was estimated using two 1-m2 quadrats set next to each block on each of two dates (n = 4). We estimated the observation error of these quadrats (standard deviation) and calculated confidence intervals for each block. Then we simulated values 10 times within these confidence intervals and the fractal dimension was calculated to determine if the spatial patterns corresponded to those estimated using only the average value.
1 Spatial patterns in grazing strength were examined using semivariogram analysis. Here, semivariograms represent the relationship between the variability of the LRR and spatial scale or lag (Dale 2000; Turner, Gardner & O’Neill 2001), i.e. how variability changes with the lag. Semivariance was calculated for different lags across the shore using the formula:
- (eqn 2)
where N(h) is the number of pairs of data points separated by the lag h and (Zi+h−Zi) are the values of the studied variable at lags i and i+h. Twenty-seven lags were included in the transect, therefore the variation in the number of pairs of lags varied from 53 pairs at lag 1 m to 27 at lag 27 m. The spatial heterogeneity, defined as the change in the value of variance across lags, was estimated using the fractal dimension ‘D’ (Turner, Gardner & O’Neill 2001), calculated as:
- (eqn 3)
where D represents degree of partitioning into self-similar pieces (as spatial periodicity) of the variance of the variable of interest (Mandelbrot 1977). The value of D varies from 1 to 2, where values from 1 to 1.5 indicate ‘gradient behaviour’ of the variability in relation to the lag; values around 1.5 indicate a cut-off between a gradient pattern and the existence of large periodic patches across lags, which tend to become smaller and less periodic as D approaches 1.97. Finally, the range of values between 1.97 and 2.00 indicates the presence of a random pattern, i.e. the existence of small patches at small lags and lack of periodicity across lags, and consequently a lack of spatial heterogeneity, i.e. no relationship between variance and scale.
The term ‘m’ represents the slope in the regression between the natural logarithm (ln) of the semivariance and the ‘ln’ of the lags. This relationship can nest one or more subrelationships with different slopes, these are known as ‘scaling regions’. Scaling regions were detected using the three-step procedure described in Erlandsson & McQuaid (2004).
2 Zonation of grazing strength was analysed using polynomial regression analyses. The Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (BIC) were used to find the best adjustment for the relationship of LRR with distance from MLWS as the predictor variable. The AIC and BIC operate by rewarding goodness-of-fit, but include penalties for overfitting if the model has too many predictor parameters (Akaike 1974; Quinn & Keough 2003). The AICc formula involves a correction for deviations when the ratio n/K is lower than 40, as in our case:
- (eqn 4)
where ln = natural logarithm, n = number of observations, RSS = residual sum of squares and K is the number of parameters. To select the best-fitting model, we estimated ‘Wi’, which represents the probability of a model being the best among competing models. We accepted as the best fit, the model with the highest value of Wi. This involved first calculating (Δi), which is: AICi−AICc minimum. Wi was then calculated as:
- (eqn 5)
Additionally, in order to double-check our findings, we calculated the Schwarz Bayesian Criterion (BIC) (Quinn & Keough 2003), which it is more restrictive than the AICc. We used the following formula:
- (eqn 6)
The competing model with the lowest value for BIC represents the best adjustment (Quinn & Keough 2003).
3 The strength of grazing and a physical factor may exhibit positive spatial co-variance at some lags, while at others the relationship may be neutral or negative. The relationship between the strength of grazing effects, expressed as the LRR, and every predictor variable (abiotic and biotic) was analysed using cross-semivariance analysis to examine the spatial co-variability of both variables at specific lags. There was some collinearity among predictor variables, but this was never above r2 = 0.75, which is low (Sokal & Rohlf 1995), and the absence of collinearity is not an assumption for cross-semivariogram analyses (unlike multi-regression tests).
The cross-semivariance was calculated as:
- (eqn 7)
where N is the total number of data points; N(h) is the number of pairs of data points separated by the distance or lag h; Xi and Xi+h, and Zi and Zi+h are the values of LRR and the values of one of the physical factors, respectively at two different lags i and i+h. Significant differences of cross-semivariance values from zero were tested using the sampled randomized test (Sokal & Rohlf 1995).