As a negative linear Δ15N vs. dietary δ15N relationship implies a limit in δ15N values, we developed a dietary δ15N value-dependent enrichment model based on an alternative form of the von Bertalanffy growth equation:
with δ15NTP being the consumer isotope value at a given TP, δ15Nlim the saturating isotope limit as TP increases, δ15Nbase the isotope value for a known baseline consumer in the food web, and k the rate at which δ15NTP approaches δ15Nlim per TP step. This model is value-dependent in that δ15Nlim is reached when the rates of 15N and 14N uptake balance those of 15N and 14N elimination (i.e. Δ15N = 0 at δ15Nlim), the rates of which are assumed constant among consumers and diets (see Supplementary Material S4).
Solved for TP this equation becomes
Calculating TP from this model requires estimates of both δ15Nlim and k which are given from the meta-analysis as
The estimated scaled TP (TPscaled) for each consumer is then estimable from the posterior distribution of the meta-analysis, given its δ15N value (δ15NTP) and a δ15Nbase value for a given food web. Baseline TL2 consumers used to estimate δ15Nbase were zooplankton, including copepod, Euphausia frigida and mysid, Undinula vulgaris for the South African food web and copepod, Calanus hyperboreus for the Canadian Arctic food web. The full meta-analytical model was implemented in a Bayesian framework, using the PyMC package (Patil et al. 2010 ) for the Python programming language. The model was run for 100,000 iterations, with an 80,000 iteration burn-in and thinned by a factor of 10; convergence was assessed through visual inspection of chains, plots of the model fit with the data, and Bayesian P-values (Gelman et al. 2006 ). Model code is included in Supplementary Materials S5.
with δ15NTP being the consumer isotope value at a given TP, δ15Nlim the saturating isotope limit as TP increases, δ15Nbase the isotope value for a known baseline consumer in the food web, and k the rate at which δ15NTP approaches δ15Nlim per TP step. This model is value-dependent in that δ15Nlim is reached when the rates of 15N and 14N uptake balance those of 15N and 14N elimination (i.e. Δ15N = 0 at δ15Nlim), the rates of which are assumed constant among consumers and diets (see Supplementary Material S4).
Solved for TP this equation becomes
Calculating TP from this model requires estimates of both δ15Nlim and k which are given from the meta-analysis as
The estimated scaled TP (TPscaled) for each consumer is then estimable from the posterior distribution of the meta-analysis, given its δ15N value (δ15NTP) and a δ15Nbase value for a given food web. Baseline TL2 consumers used to estimate δ15Nbase were zooplankton, including copepod, Euphausia frigida and mysid, Undinula vulgaris for the South African food web and copepod, Calanus hyperboreus for the Canadian Arctic food web. The full meta-analytical model was implemented in a Bayesian framework, using the PyMC package (Patil et al. 2010 ) for the Python programming language. The model was run for 100,000 iterations, with an 80,000 iteration burn-in and thinned by a factor of 10; convergence was assessed through visual inspection of chains, plots of the model fit with the data, and Bayesian P-values (Gelman et al. 2006 ). Model code is included in Supplementary Materials S5.
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