Myers et at: Population growth rate of Gadus morhua 
763 
recently collapsed Atlantic cod populations (Hutchings 
and Myers, 1994). 
Methods 
Model 
For fish populations, reproduction is generally ex- 
pressed as recruitment, the number of juvenile fish 
reaching, in a given year, the age of vulnerability to 
fishing gear. Thus, the reproduction curve (Royama, 
1993) for fish is displayed as a spawner-recruitment 
curve (Ricker, 1954), and r m must be derived from 
the slope of this curve near the origin (low popula- 
tion). This derivation will be presented immediately 
following a brief discussion of the standard popula- 
tion-recruitment curves. 
Juvenile fish become vulnerable to fishing gear, 
that is, they recruit at an age designated as j' . We 
consider the Ricker spawner-recruitment model 
which describes the number of recruits at age j' in 
year t+j\ N t+j\ j\ resulting from a spawning stock 
biomass (SSB) of S f . We follow the usual convention 
in fisheries science of assuming that the number of 
eggs produced is proportional to the biomass of 
spawners. 
The Ricker model has the form 
E(N t+r r ) = aS t e 
-ps, 
( 1 ) 
where a = the slope at the origin (measured perhaps 
in recruits per kilogram of spawners). Density-de- 
pendent mortality is assumed to be the product of P 
multiplied by the spawning biomass (S t ). 
For the forthcoming calculations the slope at ori- 
gin, a, must be standardized. First consider 
a -a SPR 
F = 0 ’ 
where SPR F=Q is the spawning biomass resulting 
from each recruit (perhaps in units of kg of spawn- 
ing fish per recruit) in the limit of no fishing mortal- 
ity (F= 0). This quantity, a , represents, on a lifetime 
basis, the number of recruits per recruit at very low 
spawner abundance or, equivalently, the number of 
spawners produced per spawner (assuming that 
there is constant survival from recruit to spawner). 
The quantity, a , required for our calculations is the 
number of spawners produced by each spawner per year 
(after a lag of a years, where a is age-at-maturity). 
If adult survival is p then a = 'L-pJa, or sum- 
mmg the geometric series 
If the annual survival fraction for spawners was 
zero, the population of spawners, N t , would obey the 
following equation: 
N =aN 
t+a 
(3) 
Equation 3 has the solution N t = na = a n N 0 , where N 0 
is the number of spawners at t = 0. It follows that 
the natural growth rate, per annum, of the popula- 
tion is 
r m - (1/ n)loga, 
(4) 
for the limit of small p g . The analogous result for the 
case of overlapping generations is derived below. 
When adult survival in not zero (p s ^ 0), one has 
an age-structured spawning population, and r m can- 
not be derived in the simple manner presented above. 
Rather, one must solve the Euler-Lotka equation 
(Charlesworth, 1994) to obtain r m in this situation. 
The Euler-Lotka equation is 
l,rn,e 
(5) 
where Z = the fraction of animals surviving to age 
j ; and 
m- = the number of offspring per animal pro- 
duced at age j. 
We now assume that ra • - m Q for fish of age a and 
older, and also, for j > a, L = l a pj ~ a , where l a is the 
fraction of juveniles that survive from age zero to 
age a, and, again, p s is the annual survival fraction 
of spawners. It follows from Equation 5 that 
( 6 ) 
A little manipulation, and the summing of a geomet- 
ric series, allows Equation 6 to be written as 
/ — r a 
a m 0 e =1 
1 ~ Ps e ^ 
(7) 
Since m 0 is the number of age-zero fish produced by 
each spawner, and since l a is the fraction of age-zero 
fish surviving through the juvenile stage to matu- 
rity, it follows that m 0 / Q = a , and thus Equation 7 
can be expressed as 
( 2 ) 
(e r " )° - p s (e r " ) a_1 - a = 0. 
a = a(l-p s ) = a-SPR f=0 (l-p s ). 
( 8 ) 
