Basics of Selectivity

selectivity
logistic
dome-shape

Definition of selectivity

Selectivity: relative vulnerability of a demographic group of the fished population to capture by a fishery or survey, with at least one demographic group being fully selected (Cadrin et al., 2015).

Selectivity is a combination of two processes:

  • the relative probability of capture for a demographic group (i.e., contact selectivity; Miller and Fryer, 1999)
  • proportion of the group that is available to the fishery in time and space (i.e., population selectivity; Miller and Fryer, 1999)
    • aka vulnerability

Assessments models typically require a particular form of selection curve and estimates a population selectivity curve. Selectivity is used to link observed composition data to model predictions about population abundance-at-age/-size. If multiple gear types operate in the fishery, or if the catch-at-age compositions from different segments of the fleet have distinct characteristics, then the catch-at-age data can be partitioned into separate matrices for each gear or fleet type with separate selectivity curves and parameters for each type.

Selectivity and fishing mortality - separability

Selectivity is the portion of a demographic group that is vulnerable to capture by fishing:

\[ F_{t,a} = F_t s_a \tag{1}\]

  • \(F_t\) is fishing mortality at time \(t\) for fully-vulnerable ages (i.e., a year-specific fishing mortality multiplier)
    • typically managed through limiting total fishing effort in a year
  • \(s_a\) is the selectivity of the age group
    • typically managed by regulating fishing technology and behavior

This definition is typically used for age-structure population models, but the same equation can be related to any other demographic groups (e.g., size intervals, life history stages). Eq 1 quantifies either constant or average selectivity during the time interval. The function expressed in eq 1 is termed separability because \(F_{t,a}\) can be separated into the components \(F_t\) and \(s_a\).

Recall that catchability (\(q\)) is the effect of of a unit of fishing effort (\(E\)) directed on the population, with the effect measured as the exponential rate of fishing mortality imposed on the population over a time interval \(t\):

\[ F_t = q E_t \tag{2}\]

Considering that catchability and selectivity have a relationship with fishing mortality, Eq 1 & 2 can be combined:

\[ F_{t,a} = s_a q E_t; \hspace{4mm} q_a = s_a q \tag{3}\]

Selectivity then plays an essential role as a component of the survival equation in abundance at age:

\[ N_{t,a} = N_{t-1,a-1}e^{-(F_{t-1} s_{a-1} + M_{t-1,a-1})} \tag{4}\]

  • \(N_{t,a}\) is the abundance of survivors of an age class at the end of year \(t\)

Eq 4 is the primary process equation for all age-structured stock assessment models.

Selectivity functions

There are several forms of fishing selectivity. Sampson and Scott (2012) defined four categories:

  1. Asymptotic: the oldest age classes are fully vulnerable 1a. Knife-edged: the simplest form of asymptotic form
  2. Increasing: selection increases with age
  3. Saddle: there is at least one local minimum selection in the intermediate range of age classes
  4. Domed: the age for maximum selectivity (\(s = 1\)) is imtermediate in the range of age classes

Logistic

Asymptotic selectivity is often modeled using a logistic function. There are many variations of the logistic selectivity function.

\[ s_a = \dfrac{1}{1 + exp(-\sigma_s (a - a_{50}))} \]

  • \(\sigma_s\) is the logistic slope parameter
  • \(a_{50}\) is the age of 50% vulnerability to the fishery (i.e., \(s_{a_{50}} = 0.5\))

Another parameterization of logistic selectvity uses parameters \(a_{50}\) and \(a_{95}\), which represent the ages where a fish has a probability 50% or 95% chance of being captured.

\[ s_a = \dfrac{1}{1+exp\left( -log(19)\left(\dfrac{a - a_{50}}{a_{95} - a_{50}}\right) \right)} \]

We need to ensure that the logistic function equals 0.95 when \(a = a_{95}\) and equals 0.5 when \(a = a_{50}\). Think about the odds of being captured:

  • The exponent of the log odds equals the odds ratio
  • The odds ratio is the probability of an event (i.e., getting captured) happening over the probablity of it not happening

Thus, the log-odds ratio for the age at which we have a 95% probability of getting captured is \(log(\dfrac{0.95}{0.05})\). This can be rewritten as \(log(\dfrac{19/20}{1/20})\), which then simplifies to \(log(19)\).

This is not necesssary for \(a_{50}\) because the odds ratio of two events with 50% probability is equal to zero.

Double logistic

The six parameter version of the double normal selectivity function can capture both asymptotic and dome-shaped shapes (commonly used in Stock Synthesis; Methot and Wetzel, 2013). The double normal has three components connected by steep logistic “joiners” to provide overall differentiability:

\[ s_a = {asc}_a (1-j_{1,a}) + j_{1,a}((1-j_{2,a}) + j_{2,a} {dsc}_a) \]

  • \({asc}_a\) is the ascending limb

\[ {asc}_a = p_5 + (1-p_5)\left(e^{-(a - p_1)^2 / e^{p_3}} - e^{{p_1}^2 / e^{p_3}} \right) / \left(1 - e^{{p_1}^2 / e^{p_3}} \right) \]

  • \({dsc}_a\) is the descending limb

\[ {dsc}_a = 1 + (p_6 - 1)\left(e^{-(a - \gamma)^2 / e^{p_4}} - 1 \right) / \left(e^{-(A-\gamma)^2 / e^{p_4}} - 1 \right) \]

  • \(j_{1,a}\) is the first joiner function

\[ j_{1,a} = \dfrac{1}{\left(1+e^{-20(a-p_1)/(1+|a-p_1|)}\right)} \]

  • \(j_{2,a}\) is the second joiner function

\[ j_{2,a} = \dfrac{1}{\left(1+e^{-20(a-\gamma)/(1+|a-\gamma|)}\right)} \]

  • \(p_1\) is the age at which selectivity=1 starts
  • \(\gamma\) is the age at which selecitivity=1 ends and is defined as:

\[ \gamma = p_1 + 1 + \left ( \dfrac{0.99 A - p_1 - 1}{1 + e^{-p_2}} \right) \]

  • \(p_2\) determines the age at which selectivity=1 ends (the width of the “top”; \(\gamma\) is the end point)
  • \(p_3\) determines the slope of the ascending section
  • \(p_4\) determines the slope of the descending section
  • \(p_5\) is the selectivity at age 0 (in logit-space)
  • \(p_6\) is the selectivity at age \(A\) (in logit-space)

The double normal can mimic the asymptotic selectivity function by setting \(p_6 = 1\).

Gamma

Dome-shaped selectivity can be represented using a gamma function: \[ s_a = \dfrac{a}{a_{max}}^{a_{max}/p} e^{(a_{max}-a)/p}; \hspace{6mm} p = 0.5 \left[\sqrt{{a_{max}}^2 + 4\gamma^2}- a_{max}\right] \]

  • \(a_{max}\) is the full vulnerability to the fishery (i.e., \(s_{a_{max}}\) = 1)
  • \(\gamma\) is the gamma slope parameter

Sensitivities of selectivity

The form of selectivity is assumed in various stock assessment models. \(F_{t,a}\) can either be directly estimated or derived from estimated paramters.

  • Catch curve
    • Estimates total mortatliy as the negative of log-linear slope in eq 4 by assuming full selectiivty for a range of ages, usually from a middle age to oldest age (i.e., knife-edge selectivity)
    • Chapman and Robson, 1960
  • Virtual population analysis (VPA)
    • Assumes the selectivity of oldest age relative to a younger age to estimate \(N_{t,a}\) to derive \(F_{t,a}\)
    • Selectivity in the most recent year is similar to previous years
    • Age range of full vulnerability is often assumed to derive \(F_t\) (i.e., knife-edge selectivity)
    • Shepherd and Pope, 2002
  • Statistical catch at age (SCAA)
    • Separability (eq 1) - estimate a time series of \(F_t\) and selectivity (\(s_a\)), either as estimated parameters or derived from functional selectivity functions
    • Maunder and Punt, 2013

References

Cadrin, S.X., DeCelles, G.R. and Reid, D. (2016). Informing fishery assessment and management with field observations of selectivity and efficiency. Fisheries Research, 184, 9-17.

Chapman, D., & Robson, D. S. (1960). The analysis of a catch curve. Biometrics, 354-368.

Maunder, M. N., & Punt, A. E. (2013). A review of integrated analysis in fisheries stock assessment. Fisheries research, 142, 61-74.

Methot Jr, R. D., & Wetzel, C. R. (2013). Stock synthesis: a biological and statistical framework for fish stock assessment and fishery management. Fisheries Research, 142, 86-99.

Millar, R. B., & Fryer, R. J. (1999). Estimating the size-selection curves of towed gears, traps, nets and hooks. Reviews in Fish Biology and Fisheries, 9, 89-116.

Sampson, D. B., & Scott, R. D. (2012). An exploration of the shapes and stability of population–selection curves. Fish and Fisheries, 13(1), 89-104.

Shepherd, J. G., & Pope, J. G. (2002). Dynamic pool models I: Interpreting the past using Virtual Population Analysis. Handbook of Fish Biology and Fisheries: Fisheries, 2, 127-163.