764 
Fishery Bulletin 95(4), 1997 
We have bracketed the e m m term to emphasize that 
Equation 8 is a simple algebraic equation for x= e r - . 
Note that in the limit p s = 0, we recover Equation 4 
from Equation 8. Equation 8 is very similar to Equa- 
tion 1 of Goodman (1984), amounting to a transla- 
tion into parameters available for fish populations. 
Equation 8 may also be obtained as the low density 
limit of the simplified age-structured model of Clark 
(1976), as modified by Mertz and Myers (1996). 
It is clear for a moderately large slope at the ori- 
gin ( a) that age of maturity (a) is the most impor- 
tant factor in determining r m (Fig. 1). The solid line 
in Figure 1 shows, for reference, the case p s = 0, for 
which r m may be calculated from Equation 4. The 
three broken lines in Figure 1 represent p s , = 0.7, 
0.8, 0.9, a range that should encompass all North 
Atlantic cod stocks (see next section). For this range, 
survival after reproduction (p s ) has only a minor ef- 
fect on r . 
m. 
Data sources and treatment 
The data we used are estimates obtained from as- 
sessments compiled by Myers et al. (1995b). Popula- 
tion numbers and fishing mortality were estimated 
by using sequential population analysis (SPA) of com- 
mercial catch-at-age data for most marine popula- 
tions. Sequential population analysis techniques in- 
clude virtual population analysis ( VPA), cohort analy- 
sis, and related methods that reconstruct population 
size from catch-at-age data (see Hilborn and Walters, 
1992) chapters 10 and 11, for description of the meth- 
ods used to reconstruct the population history). 
Briefly, the commercial catch-at-age is combined with 
estimates from research surveys and commercial 
catch rates to estimate numbers-at-age in the final 
year and to reconstruct previous numbers-at-age 
under the assumption that commercial catch-at-age 
is known without error and that natural mortality- 
at- age is known and constant. 
The population boundaries in the North Atlantic 
generally follow those of the Northwest Atlantic Fish- 
eries Organization (NAFO) or the International 
Council for the Exploration of the Sea (ICES) (Fig. 
2). Many populations cover more than one NAFO or 
ICES unit area, e.g. the cod population off Labrador 
and Northeast Newfoundland, known as “northern” 
cod, inhabits three NAFO divisions (2J, 3K, and 3L) 
and is designated as 2J3KL cod. There are three 
minor populations that are not included in the com- 
parative analysis: Flemish Cap, Gulf of Maine, and 
the English Channel. There are no reliable catch data 
for the Flemish Cap population (NAFO 3M) or the 
English Channel population (ICES Vlld), and the 
1.2 
1.0 
0.8 
<5 0.6 
>, 
Q 
Q. 
4 s 0.4 
0.2 
0.0 
Figure 1 
Rate of population growth (r m ) as a function of slope at the origin ( a ) at minimum population 
size, age of maturity (a), and adult survival rates p s = 0 (solid line), p s = 0.7 (dotted line),p s = 
0.8 (short-dashed line), and p s = 0.9 (long-dashed line) 
Slope at the origin (a) 
