gaussian_mixture_pred_kernel: Gaussian-mixture predation kernel

Description

A predation kernel for which the log predator/prey mass ratio follows a mixture of Gaussian distributions.

Usage

gaussian_mixture_pred_kernel(ppmr, kernel_p, kernel_mean, kernel_sd)

Value

A vector giving the value of the predation kernel at each of the predator/prey mass ratios in the ppmr argument.

Arguments

ppmr: A vector of predator/prey mass ratios.
kernel_p: A numeric vector of relative component proportions.
kernel_mean: A numeric vector of component means on the log predator/prey mass-ratio scale.
kernel_sd: A numeric vector of positive component standard deviations.

Details

Writing the predator mass as $w$, the prey mass as $w_p$, and $x = \ln(w / w_p)$, the feeding kernel is $$ \phi_i(w, w_p) = \sum_j a_{ij} \exp\left[-\frac{(x - \mu_{ij})^2}{2\sigma_{ij}^2}\right], \qquad a_{ij} = \frac{p_{ij}/\sigma_{ij}} {\sum_k p_{ik}/\sigma_{ik}}. $$ for predator/prey mass ratios greater than or equal to one, and zero for smaller ratios.

This is proportional to the Gaussian-mixture probability density with mixing proportions $p_{ij}$, means $\mu_{ij}$, and standard deviations $\sigma_{ij}$. The scaling makes the sum of the component peak heights equal to one. Consequently the kernel is at most one, and a one-component mixture is identical to lognormal_pred_kernel() with beta = exp(kernel_mean) and sigma = kernel_sd.

The three component parameters are vectors of equal length. When this function is selected in a species parameter data frame, they should be held in the list-columns kernel_p, kernel_mean, and kernel_sd. The values in kernel_p must be non-negative with at least one positive value, but they do not need to sum to one because they are normalised by the function.

Examples

Run this code

ppmr <- exp(seq(0, 12, length.out = 200))
phi <- gaussian_mixture_pred_kernel(
    ppmr,
    kernel_p = c(0.3, 0.7),
    kernel_mean = c(4, 8),
    kernel_sd = c(0.8, 1.5)
)
plot(ppmr, phi, type = "l", log = "x")