
Bayesian model - Two Source Trophic Position with \(\alpha_r\)
Source:R/two_source_model_ar.R
two_source_model_ar.Rd
Estimate trophic position using a two source model with \(\alpha_r\) derived from Post 2002 and Heuvel et al. 2024 using a Bayesian framework.
Arguments
- bp
logical value that controls whether informed priors are supplied to the model for both \(\delta^{15}\)N baselines. Default is
FALSE
meaning the model will use uninformed priors, however, the supplieddata.frame
needs values for both \(\delta^{15}\)N baseline (n1
andn2
).- lambda
numerical value,
1
or2
, that controls whether one or two lambdas are used. See details for equations and when to use1
or2
. Defaults to1
.
Details
We will use the following equations derived from Post 2002 and Heuvel et al. 2024:
$$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$
$$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$
$$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$
$$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$
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. This
equation is added to the data frame using add_alpha()
.
For equation 2)
\(\alpha\) is being corrected using equations in
Heuvel et al. 2024
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) and 4)
\(\delta^{15}\)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 \(\lambda_2\)).