Reference points

Introduction

Reference points for salmon typically use the Ricker stock function that predict the return from the number of spawners.

The Ricker function is

$R = a\times S\exp(-bS)$

where $R$ is the return and $S$ is the spawners. Parameters $a$ is in units of returns/spawner and $b$ is in units of reciprocal spawners.

Let $S_\textrm{MSY}$ be the spawner abundance that maximizes catch $R-S$ , which can be obtained by maximizing the function:

$R - S = a\times S\exp(-b S) - S$

The implicit solution, where $\alpha = \log(a)$ , satisfies

$(1 - bS_\textrm{MSY})\exp(\alpha - bS_\textrm{MSY}) = 1$

and can be solved numerically (Scheuerell 2016).

$S_\textrm{MSY}$ is also the spawner abundance that maximizes excess returns above the one-to-one line.

The harvest rate corresponding to $S_\textrm{MSY}$ is

$U_\textrm{MSY} = bS_\textrm{MSY}$

The quantity $S_\textrm{gen}$ is the spawner abundance that would reach the spawner abundance at MSY after one generation without fishing (Holt et al. 2009), and satisfies the equation:

$S_\textrm{MSY}=a\times S_\textrm{gen}\exp(-b\times S_\textrm{gen})$

There are several implicit assumptions associated with these reference points:

The stock-recruit relationship predicts the number of returns, not juveniles
All returns are the same age
All exploitation occurs on mature individuals
All returns are equally vulnerable to the fishery
All spawners are equally fecund

Reference points in salmonMSE

salmonMSE is an age-structured model that can relax the assumptions from the introduction:

The stock-recruit relationship predicts the number of outmigrating juveniles, not the return, from the egg production
The calendar-year return can comprise of multiple brood years
Exploitation can occur on both immature and mature individuals
Fish vulnerability can vary by age
Older, larger spawners can be more fecund than younger fish

Reference points are calculated with more structural complexity compared to the simple approach. However, comparisons below indicate the many situations where the salmonMSE reference points overlap with the typical salmon approach.

Calculations for MSY

Given exploitation in both the immature and mature component of the population, the juvenile survival of an outmigrating individual (“smolt”) to age $a$ is

$\ell_a = \prod_{j=1}^{a-1}\exp(-[M_j+v^\textrm{PT}_jF^\textrm{PT}])(1 - p_j)$ where $M$ is natural mortality, $v^\textrm{PT}$ is vulnerability at age in the preterminal fishery, $F^\textrm{PT}$ is fishing mortality in the preterminal fishery, and $p$ is maturity.

The corresponding return per juvenile $r_a = \ell_a \exp(-v^\textrm{PT}_aF^\textrm{PT}])(1 - p_a)$ .

The corresponding escapement per juvenile is $s_a = r_a \exp(-v^\textrm{T}_aF^\textrm{T}])$ , with $v^\textrm{T}$ is vulnerability at age in the fishery and $F^\textrm{T}$ is fishing mortality in the terminal fishery.

The corresponding egg production per juvenile $\phi = \sum_a s_a f_a p^\textrm{female}$ .

The equilibrium juvenile production $J$ is calculated from $\phi$ as:

$J = \log(a'\phi)/(b'\phi)$

where $a'$ is in units of juvenile per egg and $b'$ is in units of per egg.

The total return is $\sum_a J \times r_a$ and total spawners is $\sum_a J \times s_a$ .

The maximum sustainable yield state variables are values obtained from maximizing the total catch:

$C = \sum_a J\times\ell_a\times(1 - \exp(v^\textrm{PT}_aF^\textrm{PT})) + \sum_a J\times r_a\times(1 - \exp(v^\textrm{T}_aF^\textrm{T}))$

The optimization calculates the corresponding fishery effort $E_\textrm{MSY}$ . The user needs to specify the relative effort to the preterminal and terminal fisheries ( $e_1$ and $e_2$ ), with $F^\textrm{PT,MSY} = e_1 \times E_\textrm{MSY}$ and $F^\textrm{T,MSY} = e_2 \times E_\textrm{MSY}$ .

An alternative optimization scheme maximizes excess recruitment ( $R - S$ ):

$R - S = \sum_a J \times r_a - \sum_a J \times s_a$

Calculations for Sgen

Sgen is calculated by a subsequent optimization algorithm to search for the fishery effort $E_\textrm{gen}$ that generates an equilibrium distributions at age, then projects the population for $n_\textrm{gen}$ years with no fishing, such that the spawning abundance is $S_\textrm{MSY}$ at the end of the projection.

If there is partial maturity at several age classes, the number of years for the projection is unclear. The population can recover to $S_\textrm{MSY}$ with either increased survival, increased juvenile production, or both. Currently, salmonMSE will use the first age of maturity, e.g., typically two years for Chinook salmon, for the projection duration. Comparisons in the next section indicate that Sgen between the two methods frequently agree using this projection duration.

Comparison between methods

Currently, salmonMSE parameterizes the operating model with $a$ (returns/spawner), $\phi$ (juveniles/egg) and $Smax = 1/b'$ (eggs).

The following function facilitates comparison of the salmonMSE and Ricker SRR reference points in the various examples. Some conversion is also needed to convert between $b'$ (1/eggs) to $b$ (1/spawners).

In summary, if the age-structured reference point calculation only includes terminal exploitation, then the reference points are equivalent with values from the Ricker model unless both fecundity and vulnerability vary with age.

compare_ref <- function(a = 3, # Units of recruits/spawner
                        b = 1/1000, # Units of reciprocal spawners
                        maxage = 5,
                        rel_F = c(0, 1), # e_1 and e_2
                        p_female = 1,
                        vulPT = rep(0, maxage),
                        vulT = c(0, 0, 0.2, 0.5, 1),
                        M = c(1, 0.8, 0.6, 0.4, 0.2),
                        fec = c(0, 1500, 3000, 3200, 3500),
                        p_mature = c(0, 0.1, 0.2, 0.3, 1)) {
  
  require(salmonMSE)

  # Calculate eggs per smolt
  phi <- salmonMSE:::calc_phi(
    Mjuv = M,
    p_mature = p_mature,
    p_female = p_female,
    fec = fec,
    s_enroute = 1
  )

  # To convert Smax from units of spawners to eggs:
  # 1. Calculate unfished spawners from Ricker SRR, set R = S --> S0 = log(a)/b
  # 2. Calculate spawners per juvenile (spro)
  # 3. Calculate smolt per egg (phi)
  # 4. unfished eggs (eo) = unfished spawners * juvenile per spawner * egg per smolt
  # 5. Smax_eggs = log(a) / unfished eggs
  spro <- salmonMSE:::calc_phi(
    Mjuv = M,
    p_mature = p_mature,
    p_female = p_female,
    fec = fec,
    s_enroute = 1,
    output = "spawner"
  )
  
  Smax <- 1/b
  so <- Smax * log(a)
  eo <- so / spro * phi
  beta_eggs <- log(a)/eo
  Smax_eggs <- 1/beta_eggs

  SRRpars <- data.frame(
    kappa = a,
    Smax = Smax_eggs,
    phi = phi,
    SRrel = "Ricker"
  )

  # salmonMSE ref
  ref <- calc_MSY(
    M, fec, p_female, rel_F, vulPT, vulT, p_mature,
    s_enroute = 1, SRRpars = SRRpars
  )

  ref["Sgen"] <- calc_Sgen(
    M, fec, p_female, rel_F, vulPT, vulT, p_mature,
    s_enroute = 1, SRRpars = SRRpars, SMSY = ref["Spawners_MSY"]
  )
  ref["Catch/Return"] <- ref["KT_MSY"]/ref["Return_MSY"]

  ref_salmonMSE <- structure(
    ref[c("UPT_MSY", "UT_MSY", "Catch/Return", "Spawners_MSY", "Sgen")], 
    names = c("UMSY (preterminal)", "UMSY (terminal)", "Terminal Catch/Return", "SMSY", "Sgen")
  )
  
  # Ricker SRR calcs
  ref_Ricker <- local({
    umsy <- calc_Umsy_Ricker(log(a))
    smsy <- calc_Smsy_Ricker(log(a), b)
    sgen <- calc_Sgen_Ricker(log(a), b)
    structure(c(umsy, smsy, sgen), names = c("UMSY", "SMSY", "Sgen"))
  })
  
  output <- list(
    salmonMSE = ref_salmonMSE,
    Ricker = ref_Ricker
  )
  return(output)
}

Situation 1 - single age of maturity

Situation 1 specifies the salmonMSE (age-structured) model with the simple assumptions identified in the introduction. The salmonMSE and Ricker (stock-recruit only) reference points are identical.

compare_ref(
  a = 3,
  b = 1/1000,
  maxage = 5,
  rel_F = c(0, 1),
  p_female = 1,
  vulPT = rep(0, 5),
  vulT = c(0, 0, 0, 0, 1),
  M = c(-log(0.01), 0, 0, 0, 0), # SAR = 0.01
  fec = c(0, 0, 0, 0, 1),
  p_mature = c(0, 0, 0, 0, 1)
)
#> Loading required package: salmonMSE
#> $salmonMSE
#>    UMSY (preterminal)       UMSY (terminal) Terminal Catch/Return 
#>             0.0000000             0.4678228             0.4678228 
#>                  SMSY                  Sgen 
#>           467.8334508           188.2447444 
#> 
#> $Ricker
#>        UMSY        SMSY        Sgen 
#>   0.4678265 467.8265256 188.2417444

Situation 2 - partial age maturity, equal vulnerability

Situation 2 specifies the salmonMSE (age-structured) model with age-specific juvenile mortality and maturity. All spawners are identical with fecundity = 1 and fully vulnerable to the fishery.

The salmonMSE and Ricker-only reference points are identical.

compare_ref(
  a = 3,
  b = 1/1000,
  maxage = 5,
  rel_F = c(0, 1),
  p_female = 1,
  vulPT = rep(0, 5),
  vulT = rep(1, 5),
  M = c(1, 0.3, 0.2, 0.1, 0.1),
  fec = rep(1, 5),
  p_mature = c(0, 0.1, 0.2, 0.3, 1)
)
#> $salmonMSE
#>    UMSY (preterminal)       UMSY (terminal) Terminal Catch/Return 
#>             0.0000000             0.4678228             0.4678228 
#>                  SMSY                  Sgen 
#>           467.8334508           188.2447444 
#> 
#> $Ricker
#>        UMSY        SMSY        Sgen 
#>   0.4678265 467.8265256 188.2417444

Situation 3 - partial age maturity, varying vulnerability by age

Situation 3 is the same as 2 except with age-varying fishery vulnerability.

The salmonMSE and Ricker-only reference points are identical.

One nuance is how to report the exploitation rate, since it varies by age due to differential vulnerability. UMSY reported in salmonMSE is the maximum value experienced by an age class (typically the oldest), but the ratio of total catch to total return may be more appropriate metric.

compare_ref(
  a = 3,
  b = 1/1000,
  maxage = 5,
  rel_F = c(0, 1),
  p_female = 1,
  vulPT = rep(0, 5),
  vulT = c(0, 0.1, 0.2, 0.4, 1),
  M = c(1, 0.3, 0.2, 0.1, 0.1),
  fec = rep(1, 5),
  p_mature = c(0, 0.1, 0.2, 0.3, 1)
)
#> $salmonMSE
#>    UMSY (preterminal)       UMSY (terminal) Terminal Catch/Return 
#>             0.0000000             0.7330604             0.4678273 
#>                  SMSY                  Sgen 
#>           467.8250871           188.2409734 
#> 
#> $Ricker
#>        UMSY        SMSY        Sgen 
#>   0.4678265 467.8265256 188.2417444

Situation 4 - partial age maturity, varying fecundity and vulnerability by age

Situation 4 is the same as 3 except with age-varying fecundity.

The salmonMSE and Ricker-only reference points are not identical.

Preferential vulnerability by the fishery for return which are not identical in fecundity are additional complexities not implemented in Ricker-only reference points.

compare_ref(
  a = 3,
  b = 1/1000,
  maxage = 5,
  rel_F = c(0, 1),
  p_female = 1,
  vulPT = rep(0, 5),
  vulT = c(0, 0.1, 0.2, 0.4, 1),
  M = c(1, 0.3, 0.2, 0.1, 0.1),
  fec = c(0, 1000, 2000, 3000, 3500),
  p_mature = c(0, 0.1, 0.2, 0.3, 1)
)
#> $salmonMSE
#>    UMSY (preterminal)       UMSY (terminal) Terminal Catch/Return 
#>             0.0000000             0.6424027             0.4000761 
#>                  SMSY                  Sgen 
#>           523.7657546           260.2641560 
#> 
#> $Ricker
#>        UMSY        SMSY        Sgen 
#>   0.4678265 467.8265256 188.2417444

Situation 5 - partial age maturity, with immature exploitation

Reference points can also be calculated by considering preterminal exploitation, e.g., Hankin and Healey 1986, although this is not common practice.

Preterminal mortality is not explicitly modeled in Ricker-only reference points.

Thus, salmonMSE and Ricker-only reference points are not identical.

compare_ref(
  a = 3,
  b = 1/1000,
  maxage = 5,
  rel_F = c(1, 1),
  p_female = 1,
  vulPT = c(0, 0.1, 0.2, 0.3, 1),
  vulT = c(0, 0.1, 0.2, 0.4, 1),
  M = c(1, 0.3, 0.2, 0.1, 0.1),
  fec = c(0, 1000, 2000, 3000, 3500),
  p_mature = c(0, 0.1, 0.2, 0.3, 1)
)
#> $salmonMSE
#>    UMSY (preterminal)       UMSY (terminal) Terminal Catch/Return 
#>             0.4670406             0.4670406             0.2553111 
#>                  SMSY                  Sgen 
#>           535.7260998           315.6413128 
#> 
#> $Ricker
#>        UMSY        SMSY        Sgen 
#>   0.4678265 467.8265256 188.2417444

2025-07-31