Brooks et al.: Stock assessment of protogynous fish 
21 
Table 5 
Sensitivity of relative errors in estimated biological reference points to each model factor in secondary analysis, where an incor- 
rect value of age at 50:50 sex ratio was assumed in the assessment model or where fecundity was incorrectly assumed to scale lin- 
early with weight. For each reference point (F MSY , S{^ Y , MSY, and SPR msy ), the measure of spawning biomass (female, male, or 
both) with the smallest total model error (total SS) demonstrated the least variability (values in italics). Table cells give propor- 
tion of total SS explained by each factor. Values >0.1 are indicated by bold font and values <0.01, by dashes. The term “Residual” 
is variation explained by all possible interaction terms. Factors (model parameters) are defined in Table 2. 
Factor 
f msy 
qf 
°MSY 
Female 
° MSY 
Male 
qb 
° MSY 
Both 
MSY 
SPRmsy 
Female 
Male 
Both 
Female 
Male 
Both 
Female 
Male 
Both 
M 
0.02 
— 
— 
— 
— 
— 
0.02 
— 




K 
0.38 
0.45 
0.41 
— 
0.19 
0.32 
0.37 
0.38 
0.45 
0.29 
0.26 
0.39 
h 
— 
0.05 
0.07 
0.02 
0.02 
0.12 
0.06 
0.21 
0.05 
0.16 
0.39 
0.08 
p P 
0.13 
0.02 
0.05 
— 
0.05 
0.05 
0.08 
0.03 
0.02 
0.05 
— 
— 
x p 
0.20 
0.24 
0.22 
0.89 
0.34 
0.14 
0.16 
0.17 
0.20 
0.13 
0.12 
0.18 
Xe 
— 
— 
— 
— 
0.02 
— 
— 
— 
— 
— 
— 
Residual 
0.25 
0.22 
0.24 
0.08 
0.39 
0.35 
0.30 
0.20 
0.27 
0.34 
0.22 
0.33 
Total SS 
1044 
148 
419 
226 
2869 
64 
248 
78 
116 
112 
55 
48 
Sensitivity of results to k may indicate that esti- 
mates of fertilization success, if obtainable, would be 
quite valuable. Although k itself may be difficult to es- 
timate directly, fertilization success could be assessed 
qualitatively if it shifts, for example from high to low, 
with a change in sex ratio. Such information would 
make it possible to infer a likely range for steepness 
of the fertilization function, and hence, to select the 
measure of spawning biomass most appropriate for 
that range. 
In addition to influencing assessment error, k in- 
fluences the values of BRPs themselves, and lower k 
results in higher S MSY and lower F MSY and MSY. Com- 
paring the BRPs of these simulated protogynous stocks 
with those of gonochoristic equivalents, we found that, 
on average, protogynous stocks could support higher 
F 'msy an d MSY when k> 0.5. This finding resulted from 
the condition that if age structures are equivalent, 
protogynous stocks are not inherently more vulner- 
able to exploitation than gonochoristic stocks, at least 
over moderate ranges of fishing mortality (Bannerot 
et ah, 1987). It indicates that protogynous stocks are 
not inherently more vulnerable to exploitation than 
gonochoristic stocks, at least over moderate ranges 
of fishing mortality. We caution, however, that higher 
^ msy does not imply more resilience to all levels of F. 
If fertilization rate depends on sex ratio, some level 
of F>F msy is still likely to be more detrimental to a 
protogynous stock, where that level would depend on 
characteristics of the stock in question. For example, 
if sex transition is rapid, and occurs across only a few 
ages, fishing could more readily deplete males, leading 
to fertilization failure and thus recruitment failure. 
Reproductive behavior could also affect a stock’s vul- 
nerability to exploitation. Spawning aggregations can 
make a species easier to target but probably better 
able to adapt to changes in sex ratios. Pair spawners, 
on the other hand, may be less easy to target, but more 
susceptible to effects of male depletion. 
Without any information on fertilization rates, a likely 
range of k could be postulated from evolutionary con- 
siderations. We expect that nature would select against 
values of k near its limits (0.2 and 1.0). At the lower 
bound of k = 0.2, any decline in the proportion of males 
would lead to a relatively steep reduction in fertilization 
success. Individual fitness could be increased by greater 
sperm production per male, thereby increasing fertiliza- 
tion success and driving k above 0.2. However, greater 
sperm production would likely be associated with an 
energetic cost. Thus, a tradeoff should exist between 
energy allocated toward sperm production versus other 
functions, such as somatic maintenance, foraging, or 
reproductive behavior (Alonzo and Warner, 2000; Scag- 
giante et al., 2005). The tradeoff may be worth the cost, 
but only to the extent that an increase in fertilization 
success improves fitness. At the upper range of k, the 
marginal gains in fertilization success are only real- 
ized if males are extremely depleted (i.e., as x F ^0 in 
Fig. 1). Furthermore, the value of k= 1.0 implies that 
a single male can fertilize the eggs of every female 
in the population, which is obviously not realistic. We 
therefore hypothesize that moderate values of k should 
be most prevalent. Theoretical predictions, and several 
field experiments, indicate that fertilization is less than 
100% and may decline as less sperm is released per 
spawning event (Petersen et al., 1992; Petersen and 
Warner, 2002). 
Moderate values of k correspond to the range where 
BRPs are best estimated from spawning biomass of 
both sexes, and we therefore recommend a default 
choice of S b when the degree of sperm limitation is 
unknown. The direction and degree of relative error 
indicate that S b would produce nearly perfect estimates 
of S MSY and risk-averse estimates of F MSY , and only a 
