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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.

Usage

two_source_model_ar(bp = FALSE, lambda = NULL)

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 supplied data.frame needs values for both \(\delta^{15}\)N baseline (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:

  1. $$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$

  2. $$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$

  3. $$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

  4. $$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

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 \(\lambda_2\)).

See also

brms::brms()

Examples

two_source_model_ar()
#> alpha ~ ar * (max_alpha - min_alpha) + min_alpha 
#> ar ~ 1
#> d15n ~ dn * (tp - l1) + n1 * ar + n2 * (1 - ar) 
#> ar ~ 1
#> tp ~ 1
#> dn ~ 1