Brooks et a!.: Stock assessment of protogynous fish 
13 
if fishing preferentially removes males by targeting 
larger (older) individuals, for example through gear 
selectivity or management regulations. A disproportion- 
ate reduction of males could lower fertilization rates 
if not enough males are available to fertilize the eggs 
of mature females (i.e., the reduction could result in 
sperm limitation). 
The possibility of reduced fertilization rates raises 
the question of whether protogynous stocks are more 
susceptible than gonochoristic stocks to overexploita- 
tion. Several studies have concluded that protogynous 
stocks are more susceptible, based on hypothesized 
patterns of reproduction, sexual transition, and fish- 
ing (Huntsman and Schaaf, 1994; Alonzo and Mangel, 
2004, 2005). At least one study (Bannerot et al., 1987) 
indicates that, under some conditions, protogynous 
stocks are more resilient to exploitation. Either way, 
management of protogynous stocks merits the consid- 
eration of unconventional techniques (Shepherd and 
Idoine, 1993; Armsworth, 2001; Heppell et ah, 2006). 
In the United States, fishery management under the 
Magnuson-Stevens Fishery Conservation and Manage- 
ment Reauthorization Act of 2006 emphasizes the con- 
cept of maximum sustainable yield (MSY). Internation- 
ally, the use of MSY as a reference point for evaluating 
sustainable development is well established (FAO, 1999). 
Standard MSY-based biological reference points — the 
benchmarks used to gauge stock status — include fish- 
ing mortality rate at MSY (F MSY ), spawning biomass at 
MSY ( S MSY ), and MSY itself. All depend fundamentally 
on the spawner-recruit relationship, which is typically 
a function of spawning biomass (S). 
In conventional stock assessments, S is computed 
from females only (SO, and fertilization rate is im- 
plicitly assumed to be constant. Some assessments of 
protogynous stocks have emphasized the importance 
of males, by computing S from spawning biomass of 
males alone (S m ) or from the sum of both sexes ( S b ) 
(Punt et al., 1993; Vaughan et al., 1995). Early use of 
S b was in per-recruit analyses (Vaughan et al., 1992; 
Punt et al., 1993; Vaughan et al., 1995), and later, in 
spawner-recruit relationships (Vaughan and Prager, 
2001 ). 
The measure of spawning biomass — Sf, S m , or S b — 
used in an assessment plays a key role in estimates of 
biological reference points, and thus in subsequent man- 
agement advice. For example, in U.S. fishery manage- 
ment, a stock is considered to be overfished if the most 
recent estimate of S is sufficiently less than S MSY . (The 
level associated with “sufficiently” varies by stock, but 
the criterion to determine that level often takes natural 
mortality into account.) Declaring a stock overfished 
triggers development of a rebuilding plan to increase 
the stock to S MSY . In general, the choice of measure 
used to represent spawning biomass influences analyses 
on which management is based, including any esti- 
mate of stock status. Although various measures are 
used in assessments, the properties of reference points 
estimated from Sf, S m , or S b have not been examined 
comprehensively. 
We use simulations to evaluate the performance of 
each measure of spawning biomass. To begin, we simu- 
late a protogynous stock over an array of biological and 
fishery characteristics and calculate biological reference 
points for each case. Then we apply an assessment 
model to estimate those same reference points using 
each of the three S measures. The estimated reference 
points are compared to their simulated counterparts to 
quantify estimation error. These results are intended 
to help stock assessment biologists identify a robust 
measure of spawning biomass that is appropriate for 
the protogynous stock being modeled. 
Materials and methods 
Two deterministic models were constructed, both struc- 
tured by age and sex, to describe a protogynous stock. 
The first, referred to as the simulation model, was con- 
sidered a representation of the real world. It was used 
to compute true values of MSY-based biological refer- 
ence points (BRPs), which determine stock status. The 
second, the assessment model, was used to estimate 
those same reference points. Both models included age- 
specific values of maturity, mortality, sex ratio, and size. 
They differed only in computation of recruitment: the 
simulation model derived recruits directly from fertil- 
ized eggs, and the assessment model derived recruits 
indirectly from the spawning biomass of males, females, 
or both. Thus, with the assessment model the common 
assumption is that fertilization rates are static. Because 
that assumption creates the only structural difference 
between the simulation and assessment models, the 
source of any estimation error of computed quantities 
(BRPs) could be isolated and the most robust measure 
of spawning biomass could be identified. In this sense, 
estimation error refers to error caused by model mis- 
specification, rather than from fitting data. To quantify 
error systematically, BRPs were computed and estimated 
under many combinations of biological parameters and 
fishery conditions, as described below. 
Simulation model 
This study used an age-structured population model to 
compute the number of individuals at age (N a ), 
N 
a 
N„ 
-iM+F, 
a-1 ' 
[N a _ 1 e~ {M+Fa ~ l) / (1- e~ {M+Fa) ) 
2 < a < 50 
a = 50 
( 1 ) 
where N l represents the number of recruits (described 
below), and the maximum age (50) was treated as a 
plus group. The parameter M is natural mortality rate 
(constant across age), and F a is fishing mortality rate at 
age, equal to the product of total fishing mortality rate 
(F) and selectivity at age (s a ). Selectivity was assumed 
to be knife-edge, that is, s a = 0 for all ages younger 
than the first vulnerable age class (a s ) and s a = 1 
otherwise. 
