Estimate trophic position using a two source model with \(\alpha_r\) derived from Post 2002 and Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028") using a Bayesian framework.
two_source_model_arc(bp = FALSE, lambda = NULL)returns model structure for two source model to be used in a
brms() call.
logical value that controls whether informed priors are
supplied to the model for both \(\delta^{15}\)N and
\(\delta^{15}\)C baselines. Default is FALSE meaning the model will
use uninformed priors, however, the supplied data.frame needs values
for both \(\delta^{15}\)N and \(\delta^{15}\)C baseline
(c1, c2, n1, and n2).
numerical value, 1 or 2, that controls whether one or
two lambdas are used. See details for equations and when to use 1 or 2.
Defaults to 1.
We will use the following equations derived from Post 2002 and Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028"):
$$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$
$$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$
$$\delta^{13}C = c_1 \times \alpha_c + c_2 \times (1 - \alpha_c)$$
$$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_c + n_2 \times (1 - \alpha_c)$$
$$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_c + \lambda_2 \times (1 - \alpha_c))) + n_1 \times \alpha_c + n_2 \times (1 - \alpha_c)$$
For equation 1)
This equation is a carbon source mixing model with \(\delta^{13}C_c\) is the isotopic value for consumer, \(\delta^{13}C_1\) is the mean isotopic value for baseline 1 and \(\delta^{13}C_2\) is the mean isotopic value for baseline 2.
For equation 2)
\(\alpha\) is being corrected using equations in
Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028").
with \(\alpha_r\) being the corrected value (bound by 0 and 1),
\(\alpha_{min}\) is the minimum \(\alpha\) value calculated
using add_alpha() and \(\alpha_{max}\) being the maximum \(\alpha\)
value calculated using add_alpha().
For equation 3)
This equation is a carbon source mixing model with \(\delta^{13}\)C being
estimated using c_1, c_2 and \(\alpha_c\) calculated in equation 1.
For equation 4) and 5)
\(\delta^{15}\)N are values from the consumer,
\(n_1\) is \(\delta^{15}\)N values of baseline 1, \(n_2\) is
\(\delta^{15}\)N values of baseline 2,
\(\Delta\)N is the trophic discrimination factor for N (i.e., mean of 3.4),
tp is trophic position, and \(\lambda_1\) and/or
\(\lambda_2\) are the trophic levels of
baselines which are often a primary consumer (e.g., 2 or 2.5).
The data supplied to brms() when using baselines at the same trophic level
(lambda argument set to 1) needs to have the following variables, d15n,
n1, n2, l1 (\(\lambda_1\)) which is usually 2. If using baselines at
different trophic levels (lambda argument set to 2) the data frame needs
to have l1 and l2 with a numerical value for each trophic level (e.g.,
2 and 2.5; \(\lambda_1\) and \(\lambda_2\)).
brms::brms()