Computation of the Source Proportions
We constructed random samples from the posterior distribution of source proportions using
JAGS (MCMC JAGS Project 2012) and R (R Development Core Team 2012) software. Our
procedure incorporates all available data, including uncertainties. Very similar approaches for
sampling from the distributions of source proportions using stable isotope data were presented by
Batt et al. 2011 and Solomon et al. 2011). Here we summarize the procedure that we used.
Phytoplankton 2H
We estimated phytoplankton  2H distributions using the directly measured δ2H2O of water in
each lake and the isotopic discrimination factor εH for phytoplankton. The discrimination factor
was estimated using all 14 measurements of εH that have been made to date in the region. Ten of
the measurements of εH were made in this study, and four came from regrowth experiments
reported by Solomon et al. (2011). The mean and variance of these 14 measurements are -160.9
per mil and 17 per mil, respectively. Thus
𝜀𝐻 ~ 𝑁(−160.9, 0.0034)
[1]
which means that εH follows a normal distribution with mean -160.9 and precision (1/variance)
0.0034. The distribution of phytoplankton δ2H for lake i is also a normal distribution with mean
μ and precision τ:
𝛿 2𝐻𝑃ℎ𝑦𝑡𝑜𝑝𝑙𝑎𝑛𝑘𝑡𝑜𝑛 [𝑖]~ 𝑁(𝜇𝑃ℎ𝑦𝑡𝑜𝑝𝑙𝑎𝑛𝑘𝑡𝑜𝑛 [𝑖], 𝜏𝑃ℎ𝑦𝑡𝑜𝑝𝑙𝑎𝑛𝑘𝑡𝑜𝑛 [𝑖])
𝜇𝑃ℎ𝑦𝑡𝑜𝑝𝑙𝑎𝑛𝑘𝑡𝑜𝑛 = 𝛿 2𝐻2 𝑂 [𝑖] − 𝜀𝐻
[2]
Using equations [1] and [2] along with the measurements of δ2H2O, a random sample of δ2H can
be computed for each lake using JAGS. To carry out this calculation, it is necessary to assume a
prior distribution for τ. We used an uninformative prior distribution, specifically a uniform
distribution between 1 and 100 for the standard deviation (τ = 1/(standard deviation)2). This wide
range was chosen in order to have no discernible effect on the outcome. Guidelines for choice of
prior distributions are discussed in many textbooks of Bayesian statistics, such as Gelman et al.
(1996).
In summary, the distribution of phytoplankton δ2H for a lake is approximately normal with a
mean that depends on the directly measured δ2H2O in the lake and a regional distribution of εH.
The variance of the distribution for phytoplankton δ2H is largely determined by the variance of
εH.
For example, in Peter Lake the surface δ2H2O was -51.29‰. Combining this data with
the εH prior in the Bayesian model, the mean of the posterior distribution for Peter Lake surface
δ2HPhytoplankton is -212.27‰ with a standard deviation of 10.51‰. The metalimnion δ2H2O for
Peter Lake was -53.97‰ which is slightly more depleted in deuterium than the surface waters.
Therefore, the mean of the posterior distribution of metalimnion δ2HPhytoplankton in Peter Lake is
more negative at -214.87 (±10.32‰ SD).
Correcting Consumer δ2H for Dietary Water
A portion of consumers’ tissue H is assimilated from environmental water, and this dietary
contribution of δ2H must be accounted for in the estimation of resource use (Solomon et al.
2009). In an individual lake i, the proportion of Chaoborus δ2H that came from water (ω[i]) is
defined by the mixing equation
 2 HChaoborus [i]  (1  [i]) 2 H Zooplankton [i]   2 H 2O
[3]
Given direct measurements of δ2H in Chaoborus, their zooplankton prey, and lake water,
equation [3] is solved for ω[i]. We directly measured δ2H in Chaoborus, their zooplankton prey,
and lake water in each of the 40 lakes. We assumed that ω was constant for each trophic step, so
the ω[i] are samples of ω1, the value for one trophic step. We assumed that the trophic position of
Chaoborus was 2. Because dietary water is trophically compounded, the contribution of dietary
water relative to phytoplankton ω2[i] is
2 [i ]  1  (1  [i]) 2
[4]
(Solomon et al. 2009). Using equations [3] and [4] and a non-informative distribution for the
precision, we used JAGS to draw random samples from the posterior distribution of ω1 for
herbivorous zooplankton (cladocerans and all copepods except Mesocyclops) and ω2 for
Chaoborus and Mesocyclops. The posterior distribution of ω1 over all 40 lakes was normal with
mean 0.07 and precision 111.92. The posterior distribution of ω2 over all 40 lakes was normal
with mean 0.1325 and precision 33.33.
The δ2H of herbivorous zooplankton relative to phytoplankton, corrected for dietary water, is
 [i] 2 H 2O[i]
 2 H Corrected herbivores [i]   2 H herbivores [i]  1
[5]
1  1[i]
The δ2H of Chaoborus relative to phytoplankton, corrected for dietary water, is
2 [i] 2 H 2O[i]
2
2
 H Corrected Chaoborus [i]   H Chaoborus [i] 
[6]
1  2 [i]
Equation [6] was also used for Mesocyclops. Equations [3]-[6] were used to estimate δ2H of
consumers relative to phytoplankton, corrected for dietary water, in each of the 40 lakes. In the
remainder of this Appendix, we discuss the procedure for Chaoborus. The procedure for other
consumers was analogous.
Using Peter Lake again as an example, the uncorrected Chaoborus δ2H was -157.5‰.
Correcting the Chaoborus δ2H for dietary water using a Bayesian framework yields a δ2HCorrected
Chaoborus of -180.2±10.0‰ (SD).
Contributions of Terrestrial and Phytoplankton 2H to Chaoborus
The final step combines the distributions of end member and corrected Chaoborus δ2H to
compute the distributions of the terrestrial and phytoplankton contributions to Chaoborus δ2H.
These distributions fit the mixing model
𝛿 2𝐻𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 𝐶ℎ𝑎𝑜𝑏𝑜𝑟𝑢𝑠 [𝑖] = 𝜙𝑇𝑒𝑟𝑟 [𝑖]𝛿 2𝐻𝑇𝑒𝑟𝑟 + 𝜙𝑃ℎ𝑦𝑡𝑜 [𝑖]𝛿 2𝐻𝑃ℎ𝑦𝑡𝑜 [𝑖] [7]
subject to the constraint 𝜙𝑇𝑒𝑟𝑟 [𝑖] + 𝜙𝑃ℎ𝑦𝑡𝑜 [𝑖] = 1 . The distributions of δ2H in Chaoborus
corrected for dietary water (eq [6]) and phytoplankton (eq [2]) were obtained from the steps
above. The distribution of δ2H in terrestrial sources computed from 81 samples was normal with
mean -129.5 and precision 0.0043. The prior distributions of the terrestrial and phytoplankton
contributions to Chaoborus δ2H, 𝜙𝑇𝑒𝑟𝑟 [𝑖] and 𝜙𝑃ℎ𝑦𝑡𝑜 [𝑖] respectively, were center-log
transformed uniform distributions (Semmens et al. 2009, Solomon et al. 2011).
Random samples of 2500 points from the posterior distributions of 𝜙𝑇𝑒𝑟𝑟 [𝑖] and 𝜙𝑃ℎ𝑦𝑡𝑜 [𝑖]were
computed for each lake using JAGS. In the main text we report the quartiles of 𝜙𝑇𝑒𝑟𝑟 [𝑖].
Medians of the 𝜙𝑇𝑒𝑟𝑟 [𝑖] posterior distributions were used in the regression analyses reported in
the main text.
Contributions of Terrestrial, Epilimnetic Phytoplankton and Metalimnetic Phytoplankton
In addition to the two potential sources considered in the model above, a three-source model
including terrestrial material, epilimnetic phytoplankton and metalimnetic phytoplankton was
used for Chaoborus in the 15 lakes with a metalimnetic O2 maximum. Two isotopes (δ2H and
δ13C) were used in the three-source model in order to solve for three unknown source fractions
(ϕTerr, ϕEpi, and ϕMeta). This model combined the δ2H and δ13C end member distributions for
epilimnion phytoplankton (δ2HEpi and δ13CEpi), metalimnion phytoplankton (δ2HMeta and δ13CMeta)
obtained from the steps described above, and terrestrial material with δ2HCorrected Chaoborus and
δ13CChaoborus.
Epilimnion and metalimnion phytoplankton δ13C was computed from measurements of aqueous
dissolved inorganic δ13C and photosynthetic fractionation of carbon isotopes, εC, using the
analog of equation [2] for carbon. We assumed a non-informative prior distribution for the
precision of εC. Observations for computing the posterior distribution of εC were the same 14
samples used to compute εH.
Posterior distributions for the source proportions in the three-source mixing model conform to
the equations
𝛿 2𝐻𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 𝐶ℎ𝑎𝑜𝑏𝑜𝑟𝑢𝑠 [𝑖] = 𝜙 𝑇𝑒𝑟𝑟 [𝑖] 𝛿 2𝐻𝑇𝑒𝑟𝑟 + 𝜙𝐸𝑝𝑖 [𝑖] 𝛿 2𝐻𝐸𝑝𝑖 [𝑖] + 𝜙𝑀𝑒𝑡𝑎 [𝑖]𝛿 2𝐻𝑀𝑒𝑡𝑎 [𝑖]
𝛿 13𝐶𝐶ℎ𝑎𝑜𝑏𝑜𝑟𝑢𝑠 [𝑖] = 𝜙 𝑇𝑒𝑟𝑟 [𝑖]𝛿 13𝐶𝑇𝑒𝑟𝑟 + 𝜙𝐸𝑝𝑖 [𝑖] 𝛿 13𝐶𝐸𝑝𝑖 [𝑖] + 𝜙𝑀𝑒𝑡𝑎 [𝑖] 𝛿 13𝐶𝑀𝑒𝑡𝑎 [𝑖]
[8]
subject to the constraint ϕTerr [i] + ϕEpi [i] + ϕMeta [i] =1. The mean and precision of δ13CTerrestrial
(-29.2 and 0.82 respectively) was obtained from the same samples from Solomon et al. (2011) as
described in the main text. The prior distributions of terrestrial, epilimnion and metalimnion
phytoplankton contributions to Chaoborus isotope values were center-log transformed uniform
distributions (Semmens et al. 2009; Solomon et al. 2011). Random samples of 2500 points from
the posterior distributions of ϕTerr[i], ϕEpi[i], and ϕMeta [i] were computed for each lake using
JAGS.
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Semmens B.X., Ward E.J., Moore J.W. & Darimont C.T. (2009) Quantifying inter- and intrapopulation niche variability using hierarchical Bayesian stable isotope mixing models,
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