21  Reference Points

Reference points are benchmark values that describe the state of a stock or fishery under specified conditions. They provide the yardstick against which current stock status is measured and against which management strategy performance is evaluated. In openMSE, reference points are calculated from the operating model after the historical dynamics have been run, and are stored in Hist@Reference and can be accessed with Reference(Hist).

Two broad categories of reference points are used in openMSE:

Additional reference points include the mean generation time (MGT), the reference yield under a fixed-\(F\) policy, and \(B_\text{Low}\) (a rebuilding-based limit reference point used in DFO Canada’s Precautionary Approach framework).

21.1 MSY Reference Points

MSY reference points are calculated automatically by Simulate() and stored in Hist@Reference@MSY. They can also be recalculated at any time using CalcMSY(), which takes a Hist or OM object and returns a refpointsMSY object.

Hist <- Simulate(myOM)

# Full reference object
Reference(Hist)

# Full refpointsMSY object
MSYRefs(Hist)

# Individual quantities
FMSY(Hist)
BMSY(Hist)
SBMSY(Hist)
MSYLandings(Hist)

By default, reference points are evaluated using parameters from the last historical year. Alternative years can be specified via the Years argument:

# Reference points at a specific year
CalcMSY(Hist, Years = 2010)

# Reference points at multiple years (evaluated independently)
CalcMSY(Hist, Years = c(2005, 2010, 2020))

Evaluating reference points at multiple years is useful for examining how \(F_\text{MSY}\) or \(B_\text{MSY}\) shift as biological or fishery parameters change over time.

CalcMSY() returns a refpointsMSY object with the following slots, all with dimensions Sim × Stock × Year:

Slot Description
FMSY Apical fishing mortality at MSY
BMSY Total biomass at MSY
SBMSY Spawning biomass at MSY
SPMSY Spawning production at MSY
SPRMSY Spawning potential ratio at MSY
MSYLandings Landed catch at MSY
MSYDiscards Dead discards at MSY

21.1.1 Removals vs Landings

CalcMSY() can maximise either total removals (landings plus dead discards) or landings alone, controlled by the type argument:

CalcMSY(Hist, type = "Removals")  # default
CalcMSY(Hist, type = "Landings")

Removals-based MSY finds the \(F\) that maximises total fish removed from the population; landings-based MSY finds the \(F\) that maximises retained catch. The two diverge when discards are substantial: the \(F\) that maximises landings may be lower than the \(F\) that maximises total removals, because at high \(F\) a greater proportion of removals are discarded fish that contribute to mortality but not to yield.

21.1.2 Multi-Stock Complexes

When a stock complex contains more than one stock, CalcMSY() finds the single \(F_\text{MSY}\) maximising total yield across all stocks. Stock-level quantities (BMSY, SBMSY, etc.) are then evaluated at this complex-level \(F_\text{MSY}\). See Chapter 29 for details.

21.2 Yield-Per-Recruit Reference Points

Note

\(F_\text{Max}\) and \(F_{0.1}\) are not yet calculated automatically. Code to compute and store these reference points is under development.

Two \(F\)-based reference points are derived from the yield-per-recruit (YPR) curve and do not require a stock-recruitment relationship:

  • \(F_\text{Max}\): the apical fishing mortality that maximises YPR.
  • \(F_{0.1}\): the apical fishing mortality at which the slope of the YPR curve is 10% of its initial slope.

21.3 Spawning Potential Ratio Reference Points

Note

\(F_\text{crash}\) and \(\text{SPR}_\text{crash}\) are not yet calculated automatically. Code to compute and store these reference points is under development.

The spawning potential ratio \(\text{SPR}(F) = \Phi_F / \Phi_0\) measures the fraction of unfished spawning production per recruit remaining at fishing mortality \(F\). Two SPR-based reference points are computed:

  • \(F_\text{crash}\): the fishing mortality at which equilibrium recruitment falls to zero.
  • \(\text{SPR}_\text{crash}\): the spawning potential ratio at \(F_\text{crash}\), i.e. the SPR below which equilibrium recruitment is zero.

21.4 Reference Yield

The reference yield provides a practical benchmark for comparing MP performance. It is the highest yield (landings or removals) achievable under a fixed-\(F\) policy over the projection period, accounting for the actual simulated population trajectory and recruitment variability. It is computed separately for landings and removals, stored in Hist@Reference@Landings and Hist@Reference@Removals, and calculated automatically during Simulate() (landings by default; removals optionally).

For seasonal models, the effort pattern from the last complete historical year (all \(S\) seasons) is extracted as the baseline. A single scalar multiplier is then optimised to scale all fleets and all seasons simultaneously, preserving the relative seasonal effort distribution while searching over the overall level of fishing pressure.

21.5 Mean Generation Time

The mean generation time (MGT) is the mean age of a mature female in the population under natural mortality only at unfished equilibrium:

\[\text{MGT} = \frac{\sum_a a \cdot l_a \cdot m_a}{\sum_a l_a \cdot m_a}\]

where \(l_a\) is survivorship to age \(a\) under natural mortality only and \(m_a\) is maturity-at-age. MGT is used as the time unit for the \(B_\text{Low}\) calculation below and is computed by CalcMGT():

CalcMGT(Hist)

# At a specific year
CalcMGT(Hist, Years = 2010)

MGT is stored in Hist@Reference@MGT and has dimensions Sim × Stock × Year. In seasonal models, age classes are in decimal-year units, so MGT is returned in years regardless of seasonal resolution.

21.6 BLow

Note

\(B_\text{Low}\) is not calculated by default due to its computational cost. It must be enabled as an argument to Simulate(). Code for this is under development.

\(B_\text{Low}\) is a rebuilding-based limit reference point used in DFO Canada’s Precautionary Approach framework. It is defined as the current spawning biomass below which a population, if it were to recover with zero fishing, could not rebuild to \(B_\text{frac} \times SB_\text{MSY}\) within \(\text{HZN} \times \text{MGT}\) years. The default values are \(B_\text{frac} = 0.5\) and \(\text{HZN} = 2\).

Conceptually, if current spawning biomass falls below \(B_\text{Low}\), then even with immediate cessation of fishing the stock would require more than two mean generation times to recover to half of \(SB_\text{MSY}\).

\(B_\text{Low}\) is stored in Hist@Reference@BLow. Because it requires a forward simulation of the population trajectory, it is computationally expensive and is not calculated by default; it is enabled via an argument to Simulate().

21.7 Per-Recruit Curves

The underlying per-recruit calculations used to derive \(F_\text{MSY}\), \(F_{0.1}\), \(F_\text{Max}\), and \(F_\text{crash}\) can be examined directly using CalcPerRecruit() and CalcEquilibrium(). These are useful for understanding yield curves, plotting SPR against \(F\), and diagnosing model behaviour.

PR <- CalcPerRecruit(Hist, apicalF = seq(0, 1, by = 0.05))
Eq <- CalcEquilibrium(Hist)

Both functions accept a Years argument and a Complex argument.

21.8 Depletion and Stock Status

Current stock status is typically expressed as depletion, the ratio of current biomass to a reference level:

  • \(B/B_0\): current total biomass relative to unfished biomass
  • \(SB/SB_0\): current spawning biomass relative to unfished spawning biomass
  • \(SP/SP_0\): current spawning production relative to unfished spawning production
  • \(B/B_\text{MSY}\): current biomass relative to \(B_\text{MSY}\)
  • \(F/F_\text{MSY}\): current fishing mortality relative to \(F_\text{MSY}\)

Spawning biomass (\(SB\)) and spawning production (\(SP\)) measure related but distinct quantities. \(SB\) is the biomass of reproductively active fish (maturity-at-age \(\times\) weight-at-age \(\times\) numbers). \(SP\) is fecundity-weighted abundance, the quantity that directly drives the stock-recruitment relationship. When fecundity is proportional to weight, \(SB\) and \(SP\) are proportional and give the same depletion ratio; they diverge when fecundity is specified independently of weight (e.g. via age-specific egg production).

Helper functions for extracting these quantities from a Hist or MSE object:

# Relative to unfished
B_B0(Hist)       # total biomass / B0
SB_SB0(Hist)     # spawning biomass / SB0
SP_SP0(Hist)     # spawning production / SP0

# Relative to MSY
B_BMSY(Hist)     # total biomass / BMSY
SB_SBMSY(Hist)   # spawning biomass / SBMSY
SP_SPMSY(Hist)   # spawning production / SPMSY
F_FMSY(Hist)     # total apical F (summed over fleets) / FMSY

All functions accept a df argument (TRUE (default) returns a data frame; FALSE returns an array).

21.9 Seasonal Models

Seasonal operating models subdivide each calendar year into \(S\) seasons with season-specific biology, mortality, and recruitment. Reference points for seasonal models are expressed on an annual scale and follow the same logic as annual models. The Years argument to CalcMSY(), CalcPerRecruit(), and CalcEquilibrium() should specify complete calendar years (integers); all seasons within each requested year are used automatically.

21.9.1 Annual reference points from seasonal dynamics

All reference point quantities are reported as annual values. \(B_\text{MSY}\), \(SB_\text{MSY}\), and \(SP_\text{MSY}\) represent equilibrium population state at \(F_\text{MSY}\); MSY is total annual catch across all seasons; SPR is the ratio of fished to unfished annual spawning production per recruit.

In a seasonal model, biomass fluctuates within the year as seasonal mortality and recruitment pulses occur. The per-recruit framework accounts for this by computing each quantity as a recruitment-weighted average over birth-season cohorts (see Section 29.11 for the full equations).

21.9.2 Cohorts and birth seasons

The seasonal recruitment pattern is controlled by SRR@R0: the proportion of annual recruitment entering in season \(s_0\) is \(\pi_{s_0} = R_{0,s_0} / R_{0,\text{annual}}\). Per-recruit quantities (spawning production, biomass, yield) are computed for each birth season separately and combined using these \(\pi_{s_0}\) weights. Full equations are given in Section 29.11.

Cohorts born in different seasons experience different lifetime mortality and fecundity trajectories, because they encounter each calendar season at different ages. The \(\pi_{s_0}\) weighting correctly averages over the full composition of annual recruitment across birth seasons.

21.9.3 Fecundity and spawning production

AdjustSeasonalFecundity() (called automatically during Simulate()) scales fecundity-at-age by season in proportion to the seasonal recruitment pattern. This means \(\Phi_0\) (unfished spawning production per recruit, the sum of fecundity-weighted survival over all ages and seasons) reflects the actual timing and magnitude of spawning across the year, consistent with whatever recruitment seasonality is specified. SPR \(= \Phi_F / \Phi_0\) retains its standard interpretation: the fraction of unfished spawning production per recruit remaining at fishing mortality \(F\), regardless of whether spawning is concentrated in one season or distributed across several.

21.9.4 Interpretation of \(F_\text{MSY}\)

\(F_\text{MSY}\) is the annual apical fishing mortality, the sum of instantaneous rates experienced by the most heavily fished age class across all seasons in a year. This is directly comparable to \(F_\text{MSY}\) from an annual model. The optimisation scales a peak-season effort scalar, with all other seasons proportional to it via the historical effort pattern; \(F_\text{MSY}\) is reported as the corresponding annual total.

21.10 Spatial Models

For spatially structured models, reference points are currently calculated using spatially-aggregated parameters and should be interpreted as spatial averages. A spatially-explicit equilibrium approach is planned as a future option.

For full technical details of all reference point calculations, see Chapter 29.