26  Unfished Conditions

Unfished conditions provide the baseline reference against which stock status and biological reference points are measured. Two unfished states are computed:

• Equilibrium (Hist@Unfished@Equilibrium): the steady-state population produced by R0, current natural mortality, and (for seasonal models) the seasonal pattern of recruitment, with no fishing. Computed analytically or by iteration.
• Dynamic (Hist@Unfished@Dynamic): the population trajectory that would have occurred over the historical period had fishing never taken place (i.e. catchability set to zero). Accounts for time-varying biology.

Both states are stored in a popdynamics object and are computed by CalcUnfished_Equilibrium() and CalcUnfished_Dynamic() respectively. Number is a list of length nStock, each element an array with dimensions Sim × Age × Year × Area. Biomass, SBiomass, and SProduction are arrays with dimensions Sim × Stock × Year (summed over ages and areas).

26.1 Survival-at-Age

Cumulative survival from recruitment to age \(a\) is computed by CalcSurvival(). Two variants are produced: survival to age \(a\) (used for numbers-at-age) and survival to the spawning event at fraction \(\omega\) through the age-\(a\) time step (SpawnTimeFrac; used for spawning biomass and production). When \(\omega = 0\) (the default) the two are identical.

For an annual model, end-of-step survival is:

\[S_{a,t} = \begin{cases} \exp\!\bigl(-M_{1,t}\,\omega\bigr) & a = 1 \\[6pt] S_{a-1,t}\;\exp\!\bigl(-(M_{a-1,t}(1-\omega) + M_{a,t}\,\omega)\bigr)\;(1 - \xi_{a-1,t}) & a > 1 \end{cases}\]

where \(M_{a,t}\) is natural mortality-at-age and \(\xi_{a,t}\) is semelparous post-spawning mortality (zero for iteroparous species; see Section 24.5).

For the plus group (when Ages@PlusGroup = TRUE), the plus-group age class accumulates fish that survive beyond the maximum discrete age. The survival is adjusted to account for the ongoing geometric series of survivors:

\[S_{n_\text{age},t} \;\leftarrow\; \frac{S_{n_\text{age},t}}{1 - \exp(-M_{n_\text{age},t})}\]

When SpawnTimeFrac = 0 (spawning at the start of each step, the default) the formula reduces to the familiar discrete Leslie-matrix survival \(S_{a,t} = \exp(-\sum_{i=1}^{a-1} M_{i,t})\) for iteroparous species.

26.2 Unfished Number-at-Age

Unfished numbers-at-age are computed under equilibrium conditions: the stable age structure that would arise if the scheduled biology persisted indefinitely with no fishing and no recruitment deviations. For the dynamic unfished state, numbers are obtained by running the full historical simulation with catchability set to zero (see Section 26.5).

26.2.1 Annual model

For a non-seasonal operating model, equilibrium unfished number-at-age is the product of unfished recruitment \(R_0\) and survival:

\[N^0_{a,t} = R_{0,t} \cdot S_{a,t}\]

For multi-area models, numbers are then apportioned across areas by the unfished spatial distribution \(\boldsymbol{\pi}_{a,t}\) (see Section 26.4):

\[N^0_{k,a,t} = N^0_{a,t} \cdot \pi_{k,a,t}\]

Unfished number-at-age is computed by CalcUnfishedNumber().

26.2.2 Seasonal model

When OM@Seasons > 1, each season has its own recruitment and natural mortality values, so equilibrium numbers-at-age cannot be expressed in closed form. Instead, CalcUnfishedNumber() iterates the seasonal cycle (applying recruitment, mortality, and ageing each season) until the numbers-at-age converge to a stable seasonal pattern. Within each season \(s\), recruitment enters age class 1 and survivors age up one class:

\[N^{(s)}_1 = R_{0,s}\]

\[N^{(s)}_{a+1} = N^{(s^-)}_a \exp(-M_{a,s})\,(1-\xi_{a,s}), \quad a = 1, \ldots, n_\text{age}-1\]

where \(N^{(s^-)}_a\) denotes numbers from the previous season, mortality is split around the spawning event as in Section 26.1, and the plus group accumulates survivors beyond the maximum age. The converged \(N^{(s)}_a\) are then apportioned across areas as above.

26.3 Biomass and Spawning Quantities

All four aggregate quantities are summed over ages and areas from the unfished numbers-at-age. Spawning quantities use \(N^{0,\text{SP}}_{a,t}\), the numbers at the point of spawning within the time step; these are computed using the spawning survival variant (see Section 26.1) with SP = TRUE in CalcUnfishedSurvival().

\[B^0_t = \sum_a N^0_{a,t}\, \bar{W}_{a,t}\]

\[\mathit{SB}^0_t = \sum_a N^{0,\text{SP}}_{a,t}\, \bar{W}_{a,t}\, \text{mat}_{a,t}\]

\[\mathit{SP}^0_t = \sum_a N^{0,\text{SP}}_{a,t}\, f_{a,t}\]

where \(\bar{W}_{a,t}\) is mean weight-at-age, \(\text{mat}_{a,t}\) is maturity-at-age, and \(f_{a,t}\) is fecundity-at-age (see Section 24.6). Spawning production per recruit is:

\[\phi^0_t = \frac{\mathit{SP}^0_t}{R_{0,t}}\]

\(\phi^0\) appears in the stock–recruitment parameterisations (see Section 24.7) and in the per-recruit reference point calculations (see ?sec-tech-reference-points).

26.3.1 Multi-stock spawning production

In multi-stock operating models, a stock’s recruitment may depend on the spawning production of a different stock (e.g. a mixed-sex model where recruitment is driven by female spawning production but the stock being modelled represents males). The SRR@SPFrom slot names the source stock; if set, CalcUnfished_Equilibrium() replaces that stock’s SProduction with the named stock’s value after the standard computation.

26.4 Spatial Distribution

When the operating model has more than one area, CalcUnfishedDist() computes the stationary (asymptotic) spatial distribution under unfished conditions from the Spatial@Movement array. The movement array has dimensions Sim × FromArea × ToArea × Age × Year; for each Sim × Age × Year combination the \(n_\text{area} \times n_\text{area}\) movement sub-matrix \(\mathbf{M}_{a,t}\) is extracted and its stationary distribution \(\boldsymbol{\pi}_{a,t}\) is found by CalcAsymDist():

\[\boldsymbol{\pi}_{a,t}\,\mathbf{M}_{a,t} = \boldsymbol{\pi}_{a,t}, \quad \sum_k \pi_{k,a,t} = 1\]

The result is stored in Spatial@UnfishedDist with dimensions Sim × Area × Age × Year. For a single-area model all values are 1.

Spatial dynamics during the historical and projection periods are developed in Chapter 27 and Chapter 30.

26.5 Equilibrium vs Dynamic Unfished

The equilibrium unfished state uses the scheduled values of \(R_0\), \(M\), and growth for each year but does not propagate cohorts forward through time; it computes the steady-state age structure that would arise if those year-specific schedules persisted indefinitely. It is the appropriate reference for biological reference points such as \(B_0\) and \(\phi^0\).

The dynamic unfished state runs the full historical simulation with catchability set to zero, so cohorts propagate forward time step by time step (season by season for seasonal models) with actual recruitment deviations and time-varying biology. It tracks what the stock would have been in each historical time step without fishing.