Fitzhugh et a!.: Size- and age-dependence in batch spawning 
421 
that have been shown or must be assumed to be indeter- 
minate spawners. When information is lacking to make 
a distinction about fecundity type, the recommendation 
is to follow an indeterminate approach that, in theory, 
will give accurate estimates of annual fecundity regard- 
less of fecundity type (Lowerre-Barbieri et al., 2011b). 
Applications of our model showed substantial sensitiv- 
ity to age dependence in the annual number of batches. 
If the annual number of batches increased with age but 
was incorrectly assumed constant, stock assessment 
models tended to overestimate the biological reference 
points used for setting harvest rates. We found this 
result to be true for rates based on spawning potential 
ratio (Fig. IB), as well as for those rates based on maxi- 
mum sustainable yield. 
This source of error in the determination of harvest 
rate may be of particular concern when management is 
based on output controls, such as annual catch limits, 
where the estimation of absolute scale of a population 
matters. However, it is not straightforward to predict 
a priori the consequences of all the interactions within 
an assessment model. For example, age dependence 
of spawning may not affect the estimated number of 
recruits (Morgan et al., 2007) but could affect the esti- 
mated spawning biomass and, therefore, the estimated 
spawner-recruit curve. As a result, the erroneous as- 
sumption of age invariance may be a common source of 
the error-in-variables problem identified in spawner-re- 
cruit estimation (Walters and Ludwig, 1981). Whether 
through error-in-variables or other latent assumptions, 
inadequately accounting for the effects of age structure 
on reproductive potential can result in poor prediction 
of stock status (Witthames and Marshall, 2008; Murua 
et al., 2010). 
Our study adds to a growing list of age-dependent 
characteristics shown to influence reproductive success, 
including skipped spawning (Jprgensen et al., 2006; 
Secor, 2008; Rideout and Tomkiewicz, 2011), egg or 
larval quality (O’Farrell and Botsford, 2006; Spencer 
et al., 2007), and senescence (Woodhead, 1979; Porch 
et al., 2007). The first 2 characteristics likely would 
(although not necessarily) increase the value of older 
fishes beyond that considered by our models, and senes- 
cence would likely decrease the value of older fishes for 
population growth. Another consideration more difficult 
to ascertain is the influence of males on reproductive 
success (Van Doornik et al., 2008). Although our mod- 
els predicted recruitment through the conventional use 
of only females, some other assessments model stock 
productivity as a function of both sexes (Brooks et al., 
2008). Sperm limitation may be important in popula- 
tion dynamics if sex ratio fluctuates with changes in 
population structure, as it does, for example, in her- 
maphroditic fishes, particularly if the sperm quality, 
production, or allocation depends on age or size (Brooks 
et al., 2008; Uusi-Heikkila et al., 2012). 
In many assessments, spawning biomass is used in 
place of total egg production, either for simplicity or 
because no reliable information on batch fecundity ex- 
ists. Because batch fecundity typically scales with body 
weight (Hunter et al., 1985; Porch et al., 2007; but see 
Dick, 2009), the use of spawning biomass may serve as 
a first-order approximation, particularly when no other 
measure of reproductive potential is available. To ad- 
ditionally account for increased number of batches, the 
age-based vector of body weight can be multiplied by an 
age-based vector (6) of annual number of batches. In 
essence, this technique generalizes the usual approach, 
where 6 = 1 is assumed. An increasing vector would 
seem more credible and, in data-limited cases, could 
be obtained by borrowing data from similar species or 
from meta-analysis. 
Indeed, we initially considered a meta-analysis of 
the studies from our literature review. However, after 
examining the data as reported, we did not think we 
could provide meaningful, quantitative analyses. In ag- 
gregate, those studies were more useful in the qualita- 
tive sense of identifying patterns of increase, decrease, 
or no change in the number of batch spawnings. We 
anticipate that quantitative meta-analysis will become 
possible as more detailed data on batch spawning are 
collected. Lowerre-Barbieri et al. (2011a) provide guid- 
ance on the standardized information and approaches 
needed for estimating spawning frequency and duration. 
We also desired to categorize the results of our litera- 
ture review by fecundity type, but we could identify only 
a trend for indeterminate type among studies reporting 
multiple spawning. Most studies did not explicitly in- 
dicate either fecundity type or oocyte recruitment pat- 
tern. For warm-water species, indeterminate fecundity 
type is often assumed but not proven, and, as mentioned 
earlier, this assumption can be a robust one for estimat- 
ing annual fecundity (Lowerre-Barbieri et al., 2011b). 
Careful review within the literature by species may 
yield more insight, but fecundity type often may defy 
easy categorization because the synchrony of oogenesis 
and resulting fecundity type may vary even within a 
species. Different stocks, especially those separated by 
latitudinal zones, may exhibit different fecundity types 
(Abaunza et al., 2003). Therefore, multiple criteria are 
recommended for defining fecundity type (Murua and 
Saborido-Rey, 2003; Lowerre-Barbieri et al., 2011b) and, 
accordingly, some corrections have been made to previ- 
ous classifications (Arocha, 2002). 
Conclusions 
The 2 principal findings of this study are that, for fishes 
with indeterminate or uncertain fecundity type, age 
(size) dependence in the annual number of batches is 
more common than age invariance and that this depen- 
dence merits consideration in population models. For 
many species, stock assessments could be improved 
by collecting age-specific information on spawning 
frequency and duration. Without such information, or 
unless data indicate otherwise, the assumption that 
annual spawning increases with age may be more plau- 
sible than the currently common assumption of age 
invariance. 
