Fitzhugh et al.: Size- and age-dependence in batch spawning 
417 
the highest harvest rates can be achieved by removing 
first those ages with the lowest reproductive value. For 
our purposes, we use reproductive value to demonstrate 
the importance of older fish toward population growth 
and to quantify how that importance depends on the 
pattern of annual number of batches. 
We computed reproductive value of a stationary popu- 
lation as the expected number of offspring produced by 
an individual from age a until the end of the lifespan 
(3) 
x=a l a 
the probability that an individual survives 
to age x; and 
the mean number of offspring produced by an 
individual of age x (Stearns, 1992), here the 
product of annual fecundity and maturity. 
Because this analysis was conducted on a per-recruit 
basis, l x was equivalent to N a of Table 1. The term ljl a 
was the probability of surviving to age x conditional on 
having survived to age a. To examine implications for 
optimal management strategies, we computed V a across 
a range of fishing rates. For each of the 4 qualitative 
patterns in annual spawning frequency, we report V a in 
the absence of fishing in addition to the age for which V" a 
was maximized as a function of fishing rate. 
Results 
Review of the literature — spawning frequency 
and duration 
In our review of spawning frequency, we found 208 
articles that reported multiple batches and indicated 
some information about the frequency or number of 
batches occurring within a specified time period. Only 
20% (41 articles, 34 species) presented findings about 
spawning frequency related to age or size (Table 3). 
Of this subset, most (28 articles, 21 species) reported 
increases of spawning frequency with age or size. Some 
(9 articles, 9 species) reported no effect, but several of 
these articles cautioned about lack of power. Only 4 
articles (4 species) indicated a decrease in spawning 
frequency with age or size. 
We found that if we tried to restrict our review of 
spawning frequency to stocks or species that exhibit 
indeterminate fecundity, we would have had a chal- 
lenging task. Of the articles (ra = 208) reporting some 
information about spawning frequency, only 52 were 
explicit about fecundity type or oocyte development pat- 
tern, and most (38) of them indicated an indeterminate 
type. Of the 41 articles reporting a spawning frequency 
trend by size or age, 10 indicated a fecundity type and 
7 reported an indeterminate type. 
In our review of spawning duration, we found 33 ar- 
ticles (28 species) that reported results related to age 
or size (Table 4). Of these articles, most (28 articles, 23 
species) of them reported increased spawning duration 
with age or size. Several articles noted no change (5 
articles, 5 species). None of them mentioned a decreased 
duration with age or size. 
Implications for stock assessment and management 
Across all levels of fishing, spawning potential ratio 
was highest when batch production decreased with 
age and lowest when it increased with age (Fig. IB). 
Therefore, an incorrect assumption about annual batch 
production could lead to substantial bias in biological 
reference points and consequently in resulting manage- 
ment advice. For example, in this model, F 40% equals 
0.24 for age-invariant batch production, F 40% equals 0.11 
for increasing production, F 40% equals 0.53 for decreas- 
ing production, and F 40% equals 0.26 for dome-shaped 
production. If the actual pattern was that of increas- 
ing batch production but was incorrectly assumed to 
be age invariant, the estimated F 40% would be biased 
high by 118%. 
As with spawning potential ratio, reproductive value 
was sensitive to the spawning pattern. In the absence 
of fishing, older females were considerably more im- 
portant to population growth in cases of constant or 
increasing batch production than in cases of decreas- 
ing or dome-shaped batch production (Fig. 2A). For 
example, with the increasing pattern, reproductive 
value exceeded a value of 5 and was maximized near 
age 14, indicating that a fish of that age can produce 
more than 5 times the number of offspring throughout 
its remaining lifespan than can a fish of age 1. With 
the decreasing pattern, the maximum reproductive 
value was less than twice the value of fish at age 1 and 
occurred near age 6. As the rate of fishing increased, 
the age of maximum reproductive value shifted toward 
older fishes in each case but did so more quickly and 
to much older ages in cases of constant or increasing 
batch production (Fig. 2B). This result highlights the 
importance of older individuals for population growth. 
Discussion 
Our review of literature on batch spawning revealed that 
most studies have not examined, or at least have not 
reported, size- or age-dependent effects. In those studies 
where spawning frequency was examined, the majority 
of species (62%) were found to spawn more often with 
increasing age or size. Similarly, in those studies where 
spawning duration was examined, the majority of spe- 
cies (82%) were found to spawn over a longer duration 
with increasing age or size. These patterns of increasing 
spawning frequency and duration were common but not 
universal. Several studies found no discernible change 
with age or size, and patterns of decreases with age or 
size appeared to be rare. Although 1 of 2 articles on 
Atlantic cod ( Gadus morhua) found spawning frequency 
to decrease with age (Kjesbu et ah, 1996), that article 
( A max) : 
where l x = 
= 
