mizer (version 2.3.0)

setBevertonHolt: Set Beverton-Holt density dependence

Description

Takes a MizerParams object params with arbitrary density dependence and returns a MizerParams object with Beverton-Holt density-dependence in such a way that the energy invested into reproduction by the mature individuals leads to the reproduction rate that is required to maintain the given egg abundance. Hence if you have tuned your params object to describe a particular steady state, then setting the Beverton-Holt density dependence with this function will leave you with the exact same steady state. By specifying one of the parameters erepro, R_max or reproduction_level you pick the desired reproduction curve. More details of these parameters are provided below.

Usage

setBevertonHolt(
  params,
  R_factor = deprecated(),
  erepro,
  R_max,
  reproduction_level
)

Arguments

params

A MizerParams object

R_factor

Use reproduction_level = 1 / R_factor instead.

erepro

Reproductive efficiency for each species. See details.

R_max

Maximum reproduction rate. See details.

reproduction_level

Sets R_max so that the reproduction rate at the initial state is R_max * reproduction_level.

Value

A MizerParams object

Details

With Beverton-Holt density dependence the relation between the energy invested into reproduction and the number of eggs hatched is determined by two parameters: the reproductive efficiency erepro and the maximum reproduction rate R_max.

If no maximum is imposed on the reproduction rate ($R_{max}=\mathrm{\infty }$) then the resulting density-independent reproduction rate $R_{di}$ is proportional to the total rate $E_R$ at which energy is invested into reproduction, $$R_{di}=\frac{\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}}{2w_{min}}{E}_R,$$ where the proportionality factor is given by the reproductive efficiency erepro divided by the egg size w_min to convert energy to egg number and divided by 2 to account for the two sexes.

Imposing a finite maximum reproduction rate $R_{max}$ leads to a non-linear relationship between energy invested and eggs hatched. This density-dependent reproduction rate $R_{dd}$ is given as $$R_{dd}=R_{di}\frac{R_{max}}{R_{di}+R_{max}}.$$

(All quantities in the above equations are species-specific but we dropped the species index for simplicity.)

The following plot illustrates the Beverton-Holt density dependence in the reproduction rate for two different choices of parameters.

This plot shows that a given energy $E_R$ invested into reproduction can lead to the same reproduction rate $R_{dd}$ with different choices of the parameters R_max and erepro. R_max determines the asymptote of the curve and erepro its initial slope. A higher R_max coupled with a lower erepro (black curves) can give the same value as a lower R_max coupled with a higher erepro (blue curves).

For the given initial state in the MizerParams object params one can calculate the energy $E_R$ that is invested into reproduction by the mature individuals and the reproduction rate $R_{dd}$ that is required to keep the egg abundance constant. These two values determine the location of the black dot in the above graph. You then only need one parameter to select one curve from the family of Beverton-Holt curves going through that point. This parameter can be erepro or R_max. Instead of R_max you can alternatively specify the reproduction_level which is the ratio between the density-dependent reproduction rate $R_{dd}$ and the maximal reproduction rate $R_{max}$.

If you do not provide a value for any of the reproduction parameter arguments, then erepro will be set to the value it has in the current species parameter data frame. If you do provide one of the reproduction parameters, this can be either a vector with one value for each species, or a named vector where the names determine which species are affected, or a single unnamed value that is then used for all species. Any species for which the given value is NA will remain unaffected.

The values for R_max must be larger than $R_{dd}$ and can range up to Inf. If a smaller value is requested a warning is issued and the value is increased to the value required for a reproduction level of 0.99.

The values for the reproduction_level must be positive and less than 1. The values for erepro must be large enough to allow the required reproduction rate. If a smaller value is requested a warning is issued and the value is increased to the smallest possible value. The values for erepro should also be smaller than 1 to be physiologically sensible, but this is not enforced by the function.

As can be seen in the graph above, choosing a lower value for R_max or a higher value for erepro means that near the steady state the reproduction will be less sensitive to a change in the energy invested into reproduction and hence less sensitive to changes in the spawning stock biomass or its energy income. As a result the species will also be less sensitive to fishing, leading to a higher F_MSY.

Examples

# NOT RUN {
params <- NS_params
species_params(params)$erepro
# Attempting to set the same erepro for all species
params <- setBevertonHolt(params, erepro = 0.1)
t(species_params(params)[, c("erepro", "R_max")])
# Setting erepro for some species
params <- setBevertonHolt(params, erepro = c("Gurnard" = 0.6, "Plaice" = 0.95))
t(species_params(params)[, c("erepro", "R_max")])
# Setting R_max
R_max <- 1e17 * species_params(params)$w_inf^-1
params <- setBevertonHolt(NS_params, R_max = R_max)
t(species_params(params)[, c("erepro", "R_max")])
# Setting reproduction_level
params <- setBevertonHolt(params, reproduction_level = 0.3)
t(species_params(params)[, c("erepro", "R_max")])
# }