APPENDIX S3
Identifying coupled fast and slow resource channels
To identify the presence of coupled fast and slow resource channels in food webs,
we first isolated the fast channel as described in the main text. We then generated an
adjacency matrix, A, of slow fluxes by omitting the links involved in the fast channel and
substituting 1 for all of the remaining non-zero elements. For the adjacency matrix, A, of
a directed graph, An yields a new matrix with element [i, j] corresponding to the number
of directed walks of length n from node i to node j. Thus, the matrix 𝐏 = ∑𝐿𝑛=1 𝐴𝑛 , where
L equals the number of binary links in a food web (the maximum possible path length
between nodes, excluding loops), contains elements that represent the number of walks of
any length between each pair of nodes. To determine whether predators coupled fast and
slow channels, we first identified the top species in the fast channel by locating nodes
with Ti. = 0, where Ti. is the number of outgoing links (i.e. a species that contributes to the
fast channel that has no consumers). Likewise, basal species were identified by locating
nodes in the original food web matrix, T, with T.j = 0 (i.e. a species with no resources),
where T.j is the number of incoming links. A non-zero value of Pij between a top species,
j, in the fast channel and any basal species, i, from the original food web matrix indicates
the presence of fast and slow energy channels that are coupled by predators.
Analysis of fast and slow resource channels
To model the presence of both fast and slow resource channels as a function of
covariates, we used a generalized linear mixed model (GLMM), and assumed the
response variable (presence or absence of coupled fast and slow resource channels)
followed a binomial distribution. We used a logit function to link the linear predictor to
the data and random effects were assumed to be independent and normally distributed
with means equal to zero. This is a standard approach to modeling binary response data
with mixed random and fixed effects (Jiang 2007).
Many statistical tests that are available for making inferences from GLMMs (e.g.,
Wald tests and likelihood ratio tests) have been shown to be unreliable for testing
hypotheses about fixed effects (e.g. Pawitan 2000, Pinheiro and Bates 2000). As a more
robust alternative, we used a parametric bootstrap technique to estimate p-values for
model parameters (Rice 1995). On the linear predictor scale, we fitted a model of the
form Y = B1(time) + B2(isolation) + B3(time × isolation) + G + H + E, where Y is the
linear predictor of presence or absence of coupled fast and slow resource channels, B1-B3
are regression coefficients, G and H are random effects of mesocosm identity (to account
for the fact that multiple repeated measurements were made on each mesocosm) nested
within spatial block (the seven source lakes), and E is a vector of errors. We used the
lmer function in the lme4 package to fit the model (Bates & Sarkar 2007). We found that
statistical estimation algorithms implemented in lme4 were unstable when the time ×
isolation interaction was included so we omitted this interaction from the analysis. To
compute p-values, we used a Monte Carlo simulation to generate 1000 independent
datasets based on the estimated fixed effect coefficients, variances, and random effects
from the fitted model (Pinheiro & Bates 2000). We then re-fitted the model described
above to each simulated dataset and recorded coefficient estimates, yielding a distribution
consisting of 1000 estimates for each fixed effect coefficient of interest. We then used
this distribution to compute the probability that a particular coefficient differed from zero
(i.e. p-value for the coupled channels analysis in Table 1, main text) and to compute
confidence intervals for each coefficient.
Not inoculated
Inoculated
Analysis
Random Effect
Std. Dev.
Std. Dev.
C
Lake (block)
Mesocosm ID
(nested within lake)
1.2 × 10-6
1.8 × 10-2
2.9 × 10-2
2.3 × 10-2
Lake (block)
Mesocosm ID
(nested within lake)
3.8 × 10-2
3.2 × 10-2
2.3 × 10-2
5.4 × 10-2
Lake (block)
Mesocosm ID
(nested within lake)
7.2 × 10-2
6.8 × 10-2
1.7 × 10-2
5.5 × 10-2
Lake (block)
Mesocosm ID
(nested within lake)
4.3 × 10-1
1.1
< 1.0 × 10-6
1.8
Cdiff
F75
Coupled
channels
Table S3.1. Estimated random effects standard deviations from linear and generalized
linear mixed model (LMM and GLMM, respectively) analyses described in the main text
and Appendix S3. Random effects for the GLMM analysis of coupled resource channels
are reported on the scale of the linear predictor.
Literature cited
Bates, D. & Sarkar, D. (2007). lme4:linear mixed effects-models using S4 classes. R
package version 0.9975-11.
Jiang, J. (2007). Linear and generalized linear mixed models and their applications.
Springer, New York ; London.
Pawitan, Y. (2000). A reminder of the fallibility of the Wald statistic: Likelihood
explanation. The American Statistician 54:54-56.
Pinheiro, J. & Bates, D. (2000). Mixed-Effects Models in S and S-plus. Springer Verlag,
New York.
Rice, J. (1995). Mathematical Statistics and Data Analysis. 2nd edition. Wadsworth,
Belmont.