getDiffusion: Get diffusion rate from predation

Description

Calculates the diffusion rate $D_i(w)$ (grams^2/year) for each species. This diffusion rate has two components:

The diffusion due due to the variability in prey sizes. This is the diffusion term from the jump-growth equation.
Any externally specified diffusion, which is added via setExtDiffusion()

Usage

getDiffusion(object, ...)

Value

MizerParams: An ArraySpeciesBySize object (predator species x predator size) with the diffusion rates.
MizerSim: An ArrayTimeBySpeciesBySize object (time step x predator species x predator size) with the diffusion rates at every time step. If drop = TRUE then dimensions of length 1 will be removed.

Arguments

object

A MizerParams or MizerSim object.

...

Additional arguments that depend on the class of object.

For a MizerParams object:

n: A matrix of species abundances (species x size). Defaults to the initial abundances stored in object.

n_pp

A vector of the resource abundance by size. Defaults to the initial resource abundance stored in object.

n_other

A named list of the abundances of other dynamical components. Defaults to the initial values stored in object.

t

The time for which to do the calculation. Defaults to 0.

For a MizerSim object:

time_range: The time range over which to return the rates. Either a vector of values, a vector of min and max time, or a single value. Defaults to the whole time range of the simulation.
drop: If TRUE then any dimension of length 1 is removed from the returned array.

Details

The diffusion due due to the variability in prey sizes is determined by summing over all prey species and the resource spectrum and then integrating over all prey sizes $w_p$, weighted by predation kernel $\phi(w,w_p)$: $$ d_i(w) = (1-f_i(w))(\alpha_i(1-\psi_i(w)))^2\gamma_i(w) \int \left( \theta_{ip} N_R(w_p) + \sum_{j} \theta_{ij} N_j(w_p) \right) \phi_i(w,w_p) w_p^2 \, dw_p. $$ Here $N_j(w)$ is the abundance density of species $j$ and $N_R(w)$ is the abundance density of resource. The overall prefactor $\gamma_i(w)$ determines the predation power of the predator. It could be interpreted as a search volume and is set with the setSearchVolume() function. The predation kernel $\phi(w,w_p)$ is set with the setPredKernel() function. The species interaction matrix $\theta_{ij}$ is set with setInteraction() and the resource interaction vector $\theta_{ip}$ is taken from the interaction_resource column in species_params(). $f(w)$ is the feeding level calculated with getFeedingLevel(). $\psi(w)$ is the proportion of the available energy that is invested in reproduction instead of growth, obtained with psi().

The diffusion integral is normally evaluated efficiently with a fast Fourier transform, which assumes that the predation kernel depends only on the ratio of predator to prey size. If a custom predation kernel that depends on predator and prey size separately has been set with setPredKernel(), the integral is instead evaluated by direct summation over the full predation kernel, as in getEncounter().

References