Get Started with ypr

Introduction

Fish are born, they grow, they reproduce and they die – whether from natural causes or from fishing. That’s it. Modelers just use complicated (or not so complicated) math to iron out the details. Cooper (2006)

Equilibrium based yield per recruit (YPR) methods (Walters and Martell 2004) estimate the capture rate that optimizes the yield under the assumption that there is no stochasticity and all density-dependence is captured by the stock-recruitment relationship. The remaining population processes of growth, reproduction and death are captured through a series of relatively straight-forward deterministic equations.

Growth

population <- ypr_population()
ypr_plot_schedule(population, "Age", "Length")

ypr_plot_schedule(ypr_population_update(population, L2 = 75, Linf2 = 200), "Age", "Length")
#> Warning: `ypr_population_update()` was deprecated in ypr 0.5.3.
#> ℹ Please use `ypr_update()` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

population <- ypr_population_update(population, Wa = 0.01, Wb = 3)
ypr_plot_schedule(population, "Length", "Weight")

Reproduction

population <- ypr_population_update(population, fa = 1, fb = 1)
ypr_plot_schedule(population, "Weight", "Fecundity")

population <- ypr_population_update(population, Ls = 50, Sp = 10, es = 0.8)
ypr_plot_schedule(population, "Length", "Spawning")

Death

ypr_plot_schedule(population, "Length", "NaturalMortality")

ypr_plot_schedule(ypr_population_update(population, nL = 0.15, Ln = 60), "Length", "NaturalMortality")

population <- ypr_population_update(population, Sm = 0.5)
ypr_plot_schedule(population, "Length", "NaturalMortality")

population <- ypr_population_update(population, Lv = 50, Vp = 50)
ypr_plot_schedule(population, "Length", "Vulnerability")

population <- ypr_population_update(population, rho = 0.5, Llo = 40, Lup = 70, Nc = 0.1)
ypr_plot_schedule(population, "Length", "Retention")

population <- ypr_population_update(population, pi = 0.3, Hm = 0.2)
ypr_plot_schedule(population, "Length", "FishingMortality")

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 = α ⋅ E ⋅ exp (−β ⋅ E)

population <- ypr_population_update(population, BH = 0L)
ypr_plot_sr(population, plot_values = FALSE)

The number of recruits at the carrying capacity (R_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_Rt = 1, lx_a = lx_a − 1 ⋅ (1 − N_a − 1) 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_Rt^F = 1, lx_a^F = lx_a − 1^F ⋅ (1 − (1 − N_a − 1) ⋅ (1 − U_a − 1)) for a > R_t

The lifetime number of eggs deposited per (unfished) recruit (ϕ) is then just

$$\phi = \sum_{a = R_t}^{t_\text{max}} lx_a \cdot F_a/2 \cdot S_a$$ where t_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 ϕ_F.

It important to realize that at the unfished equilibrium the annual number of recruits (R₀) is related to the annual egg deposition according to the following equation

E₀ = ϕ ⋅ R₀

By definition

$$\alpha = \frac{R_k \cdot R_0}{E_0} = \frac{R_k}{\phi}$$

The β term of the Beverton-Holt curve can then be found by rearranging the following formula

$$R_0 = \frac{\alpha \cdot \phi \cdot R_0}{\beta \cdot \phi \cdot R_0 + 1}$$

β ⋅ ϕ ⋅ R₀ + 1 = α ⋅ ϕ

$$\beta = \frac{\alpha \cdot \phi - 1}{\phi \cdot R_0}$$

The equivalent equation for the Ricker curve is arrived at as follows

R₀ = α ⋅ ϕ ⋅ R₀ ⋅ exp (−β ⋅ ϕ ⋅ R₀)

$$\frac{1}{\exp (-\beta \cdot \phi \cdot R_0)} = \alpha \cdot \phi$$

β ⋅ ϕ ⋅ R₀ = log (α ⋅ ϕ)

$$\beta = \frac{\log(\alpha \cdot \phi)}{\phi \cdot R_0}$$

The number of recruits at the fished equilibrium (R_0F) can then be found for the Beverton-Holt curve as follows

$$R_{0F} = \frac{\alpha \cdot \phi_F \cdot R_{0F}}{\beta \cdot \phi_F \cdot R_{0F} + 1}$$

β ⋅ ϕ_F ⋅ R_0F + 1 = α ⋅ ϕ_F $$R_{0F} = \frac{\alpha \cdot \phi_F - 1}{\beta \cdot \phi_F}$$

and for the Ricker

R_0F = α ⋅ ϕ_F ⋅ R_0F ⋅ exp (−β ⋅ ϕ_F ⋅ R_0F)

$$\frac{1}{\exp(-\beta \cdot \phi_F \cdot R_{0F})} = \alpha \cdot \phi_F$$

β ⋅ ϕ_F ⋅ R_0F = log (α ⋅ ϕ_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_max through the following transformations

β = β ⋅ κ/R_max R₀ = R₀/κ ⋅ R_max R_0F = R_0F/κ ⋅ R_max

where κ, which is the carrying capacity in the original scale, is α/β for the Beverton-Holt and α/(β ⋅ 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 × 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

Efficiency

The catchability q indicates the probability of capture for a unit of effort (E). It is assumed to be related to π according to the relationship π = 1 − exp (log (1 − q) ⋅ E) which can be rearranged to give $$E = \frac{\log(1-\pi)}{\log(1-q)}.$$

ypr_plot_yield(population, y = "Effort", harvest = TRUE, biomass = TRUE, Ly = 60)

ypr_plot_yield(population, y = "YPUE", harvest = TRUE, biomass = TRUE, Ly = 60)
#> Warning: Removed 1 row containing missing values or values outside the scale range
#> (`geom_line()`).