| Quantity | Previous assessment (2025 value) | Updated assessment (2026 value) | Change (2026 minus 2025) |
|---|---|---|---|
| SSB | 490.867 | 466.951 | -23.916 |
| Depletion | 0.172 | 0.165 | -0.008 |
| B_Bmsy | 0.521 | 0.497 | -0.024 |
| Catch | 140.001 | 133.001 | -7.000 |
| RY | 104.865 | 98.766 | -6.099 |
| Catch_RY | 1.335 | 1.347 | 0.012 |
Namibian hake model update, 2026
Executive summary
Summary of changes in assessment inputs
The accepted 2025 model configuration was extended through 2026 without changing the base-model assumptions. The update adds the 2026 catch, CPUE, survey biomass, and composition rows provided in namhakdata26.dat. The species-specific survey series in CapeHakes.csv was extended through 2026 for Merluccius paradoxus and M. capensis.
Automated checks confirmed continuous annual ADMB matrices from 1964 through 2026, the expected dimensions of all weight, catch, composition, CPUE, and survey blocks, agreement among all terminal-year controls, and valid 54321 and 12345 end-of-file sentinels.
Summary of results
The 2026 base model converged with an objective function of -16.852, a maximum gradient of 1.41e-04, and a positive-definite Hessian. Estimated terminal spawning biomass was 467 and terminal spawning biomass relative to BMSY was 0.497.
All six sensitivity configurations met the same convergence criteria as the base model. After the selectivity slopes were estimated on the log scale with a first-difference smoothing penalty, the dome-shaped selectivity case converged with a maximum gradient below 0.001 and a positive-definite Hessian.
No TAC, ABC, or OFL recommendation is made in this document. Those quantities require the agreed Namibian management procedure and management inputs.
Introduction
This operational update applies the 2025 base-model structure to information available through 2026. Stable model methods are not repeated in full; the emphasis is on new inputs, bridge results, model diagnostics, and sensitivity to the principal structural assumptions considered in the previous assessment.
Fishery and management history
The assessment input contains annual catches from 1964 through 2026. Management-history text and the final 2026 management decision should be inserted following review by MFMR.
TODO: add the final 2026 TAC decision and any in-season management changes.
Data
| Check | Result |
|---|---|
| Annual model years | 1964–2026 |
| Weight-at-age dimensions | 63 × 10 |
| Fishery composition dimensions | 63 × 11 |
| CPUE dimensions | 63 × 7 |
| Survey dimensions | 63 × 5 |
| Control EOF sentinel | 54321 |
| Data EOF sentinel | 12345 |
Survey data
Other data
No additional external data series were introduced in this update. Missing survey observations remain represented as missing or zero according to the existing model input convention; no values were imputed to create artificial annual continuity.
Analytic approach
The age-structured ADMB model and 2025 base assumptions were retained. The control file specifies the terminal year in four locations: the model terminal year, final selectivity period, final recruitment-estimation year, and final Ksp period. All four are set to 2026.
The bridge comparison changes only the data and terminal year. Sensitivities fix survey catchability at 1.0 and 0.5, estimate natural mortality, allow dome-shaped fishery selectivity, and fix stock-recruit steepness at 0.5 and 0.9. Convergence requires a finite objective, maximum gradient no greater than 0.001, and a positive-definite ADMB Hessian.
Results
Bridge comparison
Figure 2 and Figure 3 show that adding the 2026 observations caused only modest revisions to the historical spawning-biomass trajectory. For the common terminal year 2025, the updated model estimated spawning biomass at 522, compared with 491 in the previous assessment—an increase of approximately 6.4%. Despite this upward revision to the 2025 estimate, the updated trajectory continued downward to 467 in 2026.
On the relative scale, estimated 2025 depletion increased from 0.172 in the previous model to 0.184 after the data update, before declining to 0.165 in 2026. The overlapping uncertainty intervals and closely aligned historical series indicate that the update changes the recent endpoint more than the overall interpretation of the stock trajectory.
Recruitment estimates are very similar over most of the shared time series (Figure 4), consistent with the limited historical revisions in the biomass bridge. The update produces more visible revisions in the most recent years: estimated 2025 recruitment decreases from 12.69 to 3.44, while the updated model estimates 2026 recruitment at 12.4. The apparent 2026 increase should be interpreted cautiously because it is the new terminal recruitment estimate and is consequently informed by less data than earlier cohorts.
The stock–recruit comparison extends the recruitment bridge by contrasting the 2026 base assumption with the two fixed-steepness sensitivities. The annual estimates, identified by the last two digits of their year and coloured by model, are overlaid with all three fitted curves in Figure 5.
The three models estimate broadly similar historical spawning biomass and recruitment, but impose different recruitment responses at low spawning biomass. The (h=0.5) curve has the weakest recruitment at low biomass, whereas (h=0.9) maintains comparatively high recruitment over the same range. As shown later in Table 4, the smaller recruitment-residual contribution for (h=0.5) means that its curve passes closer, in the model’s penalized residual metric, to the annual recruitment estimates. The wide scatter around all three curves nevertheless shows that the data provide limited information for choosing steepness solely from this comparison.
Time-varying fishery selectivity
The following figures use the year-normalized gradient ridge-plot presentation from the pollock model to compare time-varying fishery selectivity from the converged 2026 base model with the dome-shaped sensitivity. The same colour scale is used in both figures.
For the dome-shaped sensitivity, the descending-limb slope in selectivity period (p) was reparameterized as
\[ d_p = \exp(\eta_p), \]
where \(\eta_p\) is the estimated log_SelSlope and \(d_p\) is the positive descending slope used by the model. For ages at or above the specified onset age \(a_d\), the ascending logistic selectivity \(L_p(a)\) is modified as
\[ \widetilde S_p(a) = L_p(a)\exp\{-d_p(a-a_d)\}, \qquad a \geq a_d, \]
and the complete age-specific curve is subsequently normalized to have a maximum of one. Estimating \(\eta_p\), rather than directly estimating a bounded \(d_p\), guarantees a positive descending slope and removes numerical behavior associated with a slope estimate approaching a boundary.
Two penalties stabilize the period-specific estimates. The existing shrinkage penalty controls their overall magnitude,
\[ P_{\mathrm{level}} = 12.5\sum_{p=1}^{P}(d_p-0.5)^2, \]
while a first-difference penalty on the log slopes discourages abrupt changes between adjacent selectivity periods,
\[ P_{\mathrm{smooth}} = \frac{1}{2}\sum_{p=2}^{P} \left(\frac{\eta_p-\eta_{p-1}}{0.15}\right)^2. \]
These modifications retain time variation and permit dome-shaped curves, but reduce confounding among poorly informed period-specific slopes and the other parameters controlling the location and width of selectivity. In the final run, the regularized dome-shaped model had a maximum gradient of 3.39e-05 and a positive-definite Hessian.
Model diagnostics and sensitivities
| Model | Objective | Maximum gradient | Status |
|---|---|---|---|
| 2025 baseline | -16.03433 | 0.00023 | Converged |
| 2026 update | -16.85233 | 0.00014 | Converged |
| q = 1.0 | 2.60241 | 0.00006 | Converged |
| q = 0.5 | 61.01969 | 0.00003 | Converged |
| Estimated M | -17.50260 | 0.00004 | Converged |
| Dome selectivity | -11.96047 | 0.00003 | Converged |
| h = 0.5 | -25.64017 | 0.00000 | Converged |
| h = 0.9 | -5.34565 | 0.00008 | Converged |
| Component | 2025 | 2026 | q = 1.0 | q = 0.5 | Est. M | Dome | h = 0.5 | h = 0.9 |
|---|---|---|---|---|---|---|---|---|
| CPUE | -47.59 | -48.59 | -50.14 | -47.64 | -49.14 | -46.03 | -48.54 | -48.81 |
| Survey B | -33.32 | -35.31 | -23.73 | -20.50 | -34.65 | -28.02 | -35.36 | -34.96 |
| Fishery CAA | -128.12 | -126.58 | -117.09 | -67.64 | -128.28 | -129.41 | -127.07 | -124.21 |
| Survey CAA | 90.83 | 89.88 | 95.77 | 115.45 | 87.54 | 125.50 | 89.82 | 90.30 |
| Recruit | 15.81 | 16.62 | 15.99 | 9.97 | 18.06 | 8.01 | 8.56 | 25.63 |
| Sel. penalty | 35.64 | 36.14 | 33.59 | 26.87 | 37.39 | 32.25 | 35.96 | 35.92 |
| Priors | 50.73 | 50.98 | 48.21 | 44.51 | 51.57 | 25.75 | 50.99 | 50.78 |
| Total | -16.03 | -16.85 | 2.60 | 61.02 | -17.50 | -11.96 | -25.64 | -5.35 |
The one-year-old biomass index is not shown as an objective component because its likelihood weight is zero in these runs. The negative component values arise from the likelihood formulations and their estimated variance terms; they do not indicate negative lack of fit. Totals are penalized objectives, so their absolute values are most directly comparable for configurations fitted to the same observations with the same penalty structure.
The notably lower total for the (h=0.5) model is almost entirely attributable to the recruitment-residual component. Its recruitment contribution is 8.56, compared with 16.62 for the 2026 base model: a reduction of 8.07 objective units. The full objective decreases by 8.79 units, while the CPUE, survey, age-composition, selectivity-penalty, and prior components change comparatively little. With steepness fixed at 0.5, the stock–recruit curve requires smaller estimated recruitment deviations under the model’s quadratic residual penalty. Thus the lower total primarily reflects better agreement with the assumed stock–recruit relationship, not a broad improvement in fit to every observed data type or, by itself, evidence that (h=0.5) should replace the base assumption.
Control rule application
Modeling Namibian hake survey data by species
Fisheries stock assessments require data that are reliably collected and compiled. Assessment models should also be configured to match the assumptions associated with the observed data. To account for survey trends between the two hake species, the estimated observation errors were applied in a state-space random-walk model. The observation model applies the observation-error variances \(\sigma_{j,t}^2\) for species \(j\) in year \(t\). The observed indices \(Z_{j,t}\) are fit to latent population trends \(\hat{Z}_{j,t}\):
\[ \log Z_{j,t} = \log \hat{Z}_{j,t} + \epsilon_{j,t}, \qquad \epsilon_{j,t} \sim N(0,\sigma_{j,t}^2). \]
The state equation and associated process-error variance \(\tau_j^2\) is:
\[ \log \hat{Z}_{j,t+1} = \log \hat{Z}_{j,t} + \eta_{j,t}, \qquad \eta_{j,t} \sim N(0,\tau_j^2). \]
The process-error variances may be shared or allowed to vary among indices, and the unobserved species population is estimated as a series of random effects. The model is fit by maximum likelihood in TMB using the REMA package. Species-specific survey observations through 2026 were fit using their supplied CVs as observation-error inputs.
Application to the Namibian hake stocks
The control rule was first applied using survey data and the relative proportions of the two hake species. A more robust formulation responds directly to the M. paradoxus biomass signal, independently of M. capensis. For this purpose, the survey smoother was fit to M. paradoxus alone. The long-term reference level is the geometric mean of predicted biomass from 1990 through 2026.
Figure 9 updates Figure 18 from the 2025 assessment with the 2026 M. paradoxus survey observation.
The historical adjustments for three values of the species reactivity parameter \(\gamma\) are shown in Figure 10. Values above the long-term mean are capped at one, so this component only reduces the combined stock advice.
Control rule developments
The two-species control rule combines the overall biomass trend from the assessment with the species-specific survey signal. Its purpose is to reduce the exploitation rate when either combined spawning biomass or M. paradoxus biomass is low. The specification evaluated in the 2025 report is retained:
\[ TAC_y = \begin{cases} 0, & p_y < 0.3\bar{p}, \\ \delta \left(\frac{1}{5}\sum_{i=0}^{4}RY_{y-i}\right) f(B_y,\lambda) \min\left(1,\frac{\tilde{p}_y}{\bar{p}}\right)^\gamma, & p_y \geq 0.3\bar{p}, \end{cases} \]
where:
\[ f(B_y,\lambda) = \begin{cases} \left(\dfrac{B_y/B_{MSY}}{0.5}\right)^\lambda, & B_y/B_{MSY} < 0.5, \\ 1, & B_y/B_{MSY} \geq 0.5. \end{cases} \]
The rebuilding factor \(\delta\) is 0.8 when spawning biomass is below \(B_{MSY}\) and 1.0 otherwise. The \(\lambda\) term responds when spawning biomass falls below \(0.5B_{MSY}\), while \(\gamma\) controls the response to the M. paradoxus biomass ratio.
Table 5 applies the same four \(\lambda\)-\(\gamma\) options shown in the 2025 report to the unchanged 2025 model and the 2026 update.
| Statistic | 2025 baseline | 2026 update |
|---|---|---|
| Recent 5-year average replacement yield | 150.754 | 149.262 |
| Current spawning biomass / BMSY | 0.521 | 0.497 |
| 2026 M. paradoxus biomass / long-term mean | 0.868 | 0.868 |
| Option 1: lambda = 1.5, gamma = 0.25 | 116.404 | 114.254 |
| Option 2: lambda = 2, gamma = 0.25 | 116.404 | 113.923 |
| Option 3: lambda = 2, gamma = 0.1 | 118.906 | 116.371 |
| Option 4: lambda = 2, gamma = 1 | 104.663 | 102.432 |
The response curves retained from the 2025 report are shown in Figure 11 and Figure 12. For the 2026 update, M. paradoxus predicted biomass is 0.868 of its 1990–2026 geometric mean, and the base-model spawning biomass is 0.497 of \(B_{MSY}\).
Projections
A set of simple 40-year projections was produced from the 2026 base-case model. The scenarios hold catch at 50%, 100%, or 110% of the 2026 catch; increase or decrease catch by 10% for 10 years before returning to the 2026 catch; or set catch to the model-estimated replacement yield. Their effects on catch and depletion are compared in Figure 13.
Harvest recommendations
The ADMB model produced replacement-yield and long-term reference-point outputs for each converged configuration. A management recommendation is not presented because the final choice among control-rule options and other management inputs was not supplied as part of this update. The calculations in Table 5 reproduce the options evaluated previously and should not be interpreted as adopted TAC advice.
Risk table and recommendation
The 2026 base model passed the declared numerical diagnostics. Structural uncertainty remains material: estimated natural mortality and fixed survey catchability change the inferred scale of the stock. The dome-shaped selectivity case now passes the numerical criteria, but it remains a structural sensitivity rather than the selected base configuration.
Status determination
The base-model estimate of spawning biomass relative to BMSY is reported in Table 1. Formal legal or management status terminology should be supplied by MFMR using the applicable Namibian framework.
Ecosystem considerations
No new ecosystem covariates were incorporated in the assessment likelihood. The species-specific survey series is retained for evaluating the relative signals for M. paradoxus and M. capensis.
Data gaps and research priorities
- Evaluate the biological plausibility and stability of the regularized dome-shaped selectivity alternative before using it for management inference.
- Document the provenance and preprocessing of each annual input block.
- Add retrospective analyses and likelihood profiles to the operational workflow.
- Archive the final management-procedure inputs alongside the accepted assessment.
Acknowledgements
The assessment update was prepared by John Kathena and Jim Ianelli with data provided by the Namibian Ministry of Fisheries and Marine Resources.
Auxiliary files
The assessment source, annual controls, automated validation tests, ADMB template, and model outputs are maintained in the NamibianHake repository.