Recruitment
With growth, reproduction and death defined, the final task is to estimate the recruitment (birth) rate.
This requires the lifetime number of spawners per spawner at low density (\(R_k\)) and the recruitment age (\(R_t\); by default 1) to be defined.
If recruitment follows a Beverton-Holt (\(BH = 1\)) curve then
\[R = \frac{\alpha \cdot E}{(\beta \cdot E + 1)}\]
population <- ypr_population_update(population, Rk = 3)
ypr_plot_sr(population, plot_values = FALSE)
where \(E\) is the annual eggs (stock) and \(R\) is the annual recruits at age \(R_t\).
With a Ricker curve (\(BH = 0\)) the relationship is as follows
\[R = \alpha \cdot E \cdot \exp (-\beta \cdot E)\]
population <- ypr_population_update(population, BH = 0L)
ypr_plot_sr(population, plot_values = FALSE)
The number of recruits at the carrying capacity (\(R_\text{max}\)) is a scaling constant that only affects the estimate of the yield.
Before calculating the recruitment it is important to introduce the concept of the (unfished) survivorship (\(lx_a\)) which is the probability of a recruit surviving to age \(a\) in the absence of fish mortality.
ypr_plot_schedule(population, "Age", "Survivorship")
The unfished survivorship (\(lx_a\)) is defined recursively by
\[lx_{R_t} = 1, lx_a = lx_{a-1} \cdot (1-N_{a-1}) \;\text{for}\; a > R_t\] where \(N_a\) is the annual interval natural mortality at age \(a\).
And the fished survivorship (\(lx_a^F\)) is
\[lx_{R_t}^F = 1, lx_a^F = lx_{a-1}^F \cdot (1 - (1 - N_{a-1}) \cdot (1 - U_{a-1})) \;\text{for}\; a > R_t\]
The lifetime number of eggs deposited per (unfished) recruit (\(\phi\)) is then just
\[\phi = \sum_{a = R_t}^{t_\text{max}} lx_a \cdot F_a/2 \cdot S_a\] where \(t_\text{max}\) is the maximum age considered (by default 20) and \(F_a\) the fecundity at age \(a\) is divided by two as the sex ratio is assumed to 1:1 and \(S_a\) is the probability of spawning.
The fished equivalent is denoted \(\phi_F\).
It important to realize that at the unfished equilibrium the annual number of recruits (\(R_0\)) is related to the annual egg deposition according to the following equation
\[E_0 = \phi \cdot R_0\]
By definition
\[\alpha = \frac{R_k \cdot R_0}{E_0} = \frac{R_k}{\phi}\]
The \(\beta\) term of the Beverton-Holt curve can then be found by rearranging the following formula
\[R_0 = \frac{\alpha \cdot \phi \cdot R_0}{\beta \cdot \phi \cdot R_0 + 1}\]
\[\beta \cdot \phi \cdot R_0 + 1 = \alpha \cdot \phi\]
\[\beta = \frac{\alpha \cdot \phi - 1}{\phi \cdot R_0}\]
The equivalent equation for the Ricker curve is arrived at as follows
\[R_0 = \alpha \cdot \phi \cdot R_0 \cdot \exp (-\beta \cdot \phi \cdot R_0)\]
\[\frac{1}{\exp (-\beta \cdot \phi \cdot R_0)} = \alpha \cdot \phi\]
\[\beta \cdot \phi \cdot R_0 = \log(\alpha \cdot \phi)\]
\[\beta = \frac{\log(\alpha \cdot \phi)}{\phi \cdot R_0}\]
The number of recruits at the fished equilibrium (\(R_{0F}\)) can then be found for the Beverton-Holt curve as follows
\[R_{0F} = \frac{\alpha \cdot \phi_F \cdot R_{0F}}{\beta \cdot \phi_F \cdot R_{0F} + 1}\]
\[\beta \cdot \phi_F \cdot R_{0F} + 1 = \alpha \cdot \phi_F\] \[R_{0F} = \frac{\alpha \cdot \phi_F - 1}{\beta \cdot \phi_F}\]
and for the Ricker
\[R_{0F} = \alpha \cdot \phi_F \cdot R_{0F} \cdot \exp (-\beta \cdot \phi_F \cdot R_{0F})\]
\[\frac{1}{\exp(-\beta \cdot \phi_F \cdot R_{0F})} = \alpha \cdot \phi_F\]
\[\beta \cdot \phi_F \cdot R_{0F} = \log(\alpha \cdot \phi_F)\]
\[R_{0F} = \frac{\log(\alpha \cdot \phi_F)}{\beta \cdot \phi_F}\]
Finally the estimates are rescaled so that the carrying capacity is identical to \(R_\text{max}\) through the following transformations
\[ \beta = \beta \cdot \kappa / R_\text{max} \] \[ R_0 = R_0 / \kappa \cdot R_\text{max} \] \[ R_{0F} = R_{0F} / \kappa \cdot R_\text{max} \]
where \(\kappa\), which is the carrying capacity in the original scale, is \(\alpha/\beta\) for the Beverton-Holt and \(\alpha/(\beta \cdot e)\) for the Ricker curve.
Yield
When the yield is simply the number of fish caught (irrespective of the weight or whether or not its harvested) then it is given by
\[Y = \sum_{a = R_t}^{t_\text{max}} R_{0F} \cdot lx_a^F \cdot \pi \cdot V_a\]
if only harvested fish are considered it becomes
\[Y = \sum_{a = R_t}^{t_\text{max}} R_{0F} \cdot lx_a^F \cdot \pi \cdot V_a \cdot R\]
and if the total weight (in kg) is important then its
\[Y = \sum_{a = R_t}^{t_\text{max}} R_{0F} \cdot lx_a^F \cdot \pi \cdot V_a \cdot R \cdot W_a/1000\]
and if only trophy fish are to be considered then its
\[Y = \sum_{a = R_t}^{t_\text{max}} \text{if}(La < L_y)\ 0\ \text{else}\ R_{0F} \cdot lx_a^F \cdot \pi \cdot V_a \cdot R \cdot W_a/1000\] where \(L_y\) is the minimum length of a trophy fish.
ypr_plot_yield(population, harvest = TRUE, biomass = TRUE, Ly = 60)
ypr_tabulate_yield(population, harvest = TRUE, biomass = TRUE, Ly = 60)
#> # A tibble: 2 x 8
#> Type pi u Yield Age Length Weight Effort
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 actual 0.3 0.18 0.0505 7.39 66.8 3006. 3.39
#> 2 optimal 0.441 0.265 0.0553 7.35 66.7 2978. 5.53
