Bayesian model - Two Source Trophic Position with $\alpha_r$ and carbon mixing model

Estimate trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. (2024) doi:10.1139/cjfas-2024-0028 using a Bayesian framework.

Usage

two_source_model_arc(bp = FALSE, lambda = NULL)

Arguments

bp: 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).
lambda: 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.

Value

returns model structure for two source model to be used in a brms() call.

Details

We will use the following equations derived from Post 2002 and Heuvel et al. (2024) doi:10.1139/cjfas-2024-0028 :

$$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$
$$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$
$$\delta^{13}C = c_1 \times \alpha_r + c_2 \times (1 - \alpha_r)$$
$$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$
$$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

For equation 1)

This equation is a carbon use 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) 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 $\alpha_r$ 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 $\lambda_2$).

Examples

two_source_model_arc()
#> alpha ~ ar * (max_alpha - min_alpha) + min_alpha 
#> ar ~ 1
#> d13c ~ (c1 * ar) + (c2 * (1 - ar)) 
#> ar ~ 1
#> d15n ~ dn * (tp - l1) + n1 * ar + n2 * (1 - ar) 
#> ar ~ 1
#> tp ~ 1
#> dn ~ 1

Bayesian model - Two Source Trophic Position with \(\alpha_r\) and carbon mixing model

Usage

Arguments

Value

Details

See also

Examples