Siddeek et al.: Development of harvest control rules for hard-to-age crab stocks 
where S" = the total selectivity (a curve for selection of 
all susceptible sizes of retained and discarded 
crab to pot gear, fixed at the assessment model 
estimates); and 
Z' =the instantaneous total mortality during 
year t (including components for the directed 
pot fishery and bycatch in the groundfish 
fishery). 
A cap on the proportion of legal-sized male abundance 
that can be caught is included in the state HCRs for sev- 
eral Alaska crab stocks, such as the stock of red king crab 
(Paralithodes camtschaticus) in Bristol Bay (Zheng et al., 
1995). The HR, may therefore be further modified for con- 
servation purposes by constraining the predicted catch (in 
numbers) by the directed pot fishery to not exceed 25% of 
the legal-sized male abundance (crab size >136 mm CL; 
Siddeek et al., 2020), in other words, 
yeg@s js0.255" Ny, 
where L = the legal minimum size of golden king crab in 
the Aleutian Islands. 
The predicted catch (in weight) by the directed pot fishery 
cannot exceed the retained component of the ABC (cur- 
rently adopted precautionary state management policy is 
to avoid exceeding the federal fishery management limit, 
the OF L), in other words, 
ers A 
Yale < 0.75 x Deal t, jYj- 
Representing uncertainties 
Uncertainty about future recruitment Many types of stock— 
recruitment relationships can be fitted to the results 
from an assessment, but the Ricker model (Ricker, 1954) 
was chosen for this study given its previous use for king 
crab species (Lithodidae spp.) (e.g., Zheng et al., 1995; 
Bechtol and Kruse, 2009). Future recruitment is gener- 
ated with variation and temporal autocorrelation, with 
this equation: 
MMB, x -125n(5n) Ska o,-oR 
15 Fo wap, ° 
0 
€, =Pretatyl- Pre» and 
e~ N (0.0%), 
R, 
where R, = the number of recruits at unfished equilibrium; 
MMB, ,, = the MMB (in metric tons) in year t-k given 
a k-year lag between spawning and recruits 
entering the model (k=8; Daly et al., 2019); 
MMB, = the unfished MMB; 
h = the steepness parameter; 
Pp =the extent of autocorrelation in the recruit- 
ment deviations; 
Op = the standard deviation of recruitment; and 
Q; = a normalized gamma function that determines 
the distribution of recruits to each size class: 
1,+2.5 
ii gamma(x| o.,, B,.)dx 
1,-2.5 
Oe and (9) 
J n 1}+2.5 A 
Dealers gamma(s| a, B,)dx 
ax 
a,—1 By 
gamma(x |o,, B,) = x, 
B, NG.) 
where @, = a parameter of the gamma distribution; 
B,. = a parameter of the gamma distribution; 
1; = the midpoint of size class j; and 
n = the number of recruiting size classes, fixed to 5. 
Uncertainty about the state of the stock in 2018 For simplic- 
ity, most parameter values were fixed at their best esti- 
mates. However, uncertainty was introduced to the size 
composition for the first projection year (2018), according 
to the following equation: 
oe 8; ze 
Noo18,; =N2018,je ~ ? (10) 
hg Ed MN (0, YW). 
where N 2018, ; = the estimate of the number of males in 
size class j at the start of 2018; 
69018; = the standard error of the logarithm of the 
estimate of N 2018, ;; and 
V =the variance—covariance matrix for the 
numbers by size class at the start of 2018. 
Estimates of N201s,; and V were obtained from the assess- 
ment model. Stock status in each projection was deter- 
mined by using the best estimates from the assessment 
model rather than by sampling parameter vectors from a 
posterior or a bootstrap distribution, as would commonly 
be done in a full MSE. 
Uncertainty when applying the federal and state harvest 
control rules Uncertainty in the estimates of MMB and 
MMA are accounted for by replacing MMB and MMA in 
Equations 3 and 6 as follows: 
95 
uD 
: 5 
MMB¢™"e4 — MMB,e 2, (11) 
5: = Pp. t+ yl - Pee, and 
D, ~ N(0, on): and 
oN 
MMAS*™=ted — MMAe * 2 , (12) 
MO, =PyO + yl - Pn St, and 
DO, ~ N(0, ox), 
where MMBetimated = the estimate of MMB for year f; 
