Brooks et al.: Stock assessment of protogynous fish 
15 
Table 1 
Model parameters. Values in braces are levels used for the primary analysis, where the assessment model did not account for 
dynamics of fertilization. 
Parameter Value(s) Description 
M 
10.1,0.2,0.31 
L„ 
1000 
K 
0.64M 
a g 
0.96/M 
c g 
10.75, 1.0, 1.251 
a g 
ii 
Pg 
10.2, 0.4, 0.8, 1.61 
a P 
2.3/M 
P p 
(0.2, 0.4, 0.8, 1.61 
C s 
10.75, 1.0, 1.251 
a s 
« s = c s « g 
V 1 
1x1 0 s 
v 2 
3.0 
£i 
1.0 
% 
3.0 
K 
10.2, 0.3, ..., 1.01 
h 
(0.4, 0.6, 0.8} 
Ro 
1x10 s 
Natural mortality rate 
Asymptotic maximum length 
Growth coefficient (Gardner et al., 2005) 
Mean age at 50% maturity (Gardner et al., 2005) 
Age at 50% maturity relative to the mean 
Age at 50% maturity 
Slope of logistic maturity function 
Age at 50:50 sex ratio (Gardner et al., 2005) 
Slope of logistic sex-transition function 
Age at selection relative to maturity 
Age at selection 
Weight-at-age coefficient 
Weight-at-age exponent 
Fecundity-at-age coefficient 
Fecundity-at-age exponent 
Steepness of fertilization function (f) 
Steepness of spawner-recruit function 
Unfished recruitment 
were used to describe K and a g , and to derive the age 
at 50:50 sex ratio by substituting K and L 50 into the 
von Bertalanffy model and solving for a ( a p =2.3/M ). 
Remaining parameters were set to values or ranges 
considered reasonable (Table 1). Note that results will be 
independent of E p v,, and L , because these parameters 
are merely scalars. 
Biological reference points (BRPs) 
This study focused on four BRPs: maximum sustainable 
yield ( MSY ) and the associated fishing mortality rate 
(F msy )> spawning biomass (S MSY ), and spawning poten- 
tial ratio ( SPR msy ), defined as fertilized eggs per recruit 
in relation to that at the unfished level. True values of 
BRPs were computed numerically from the simulation 
model by maximizing equilibrium yield computed over a 
range of F at intervals of 0.01. For each F, equilibrium 
yield (Y F ) was calculated from the Baranov catch equa- 
tion (Baranov, 1918) 
Y F^^N a w a (l-e- Za ). ( 8 ) 
a a 
The MSY was defined as maximum Y F , F MSY as the F 
resulting in MSY, and SPR msy as the corresponding 
spawning potential ratio (SPR). Unlike those three refer- 
ence points, the value of S MSY is specific to the measure 
of spawning biomass (f, m, and b) and was therefore 
computed as such, 
&MSY ~ 'y, Nq ( 1 P a )g a W a 
a 
= (9) 
a 
®MSY ~ 'y',Nqg a w a 
a 
where N a = the equilibrium number at age at MSY. 
Although fertilized eggs, rather than spawning bio- 
mass, determined recruitment in the simulation model, 
values of Sj^g Y were computed because of the key role 
that plays in determining whether a stock is over- 
fished. Equation 9 provided values in units comparable 
to estimates from the assessment model, where spawn- 
ing biomass did determine recruitment. 
Assessment model and estimation 
of biological reference points 
The assessment model was structurally identical to the 
simulation model with the single exception that recruits 
were computed from a measure of spawning biomass 
(mature females, males, or both), rather than from fer- 
tilized eggs. This difference represents a simplifying 
assumption common to almost all assessment models. Its 
inclusion allowed examination of how that assumption 
affects estimates of BRPs and identification of a robust 
measure of spawning biomass. 
In the assessment model, recruitment (R) was com- 
puted from spawning biomass (S=Sf, S' n , or S b ) by using 
the same functional form as Equation 7, 
