136 
Fishery Bulletin 107(2) 
The fraction dead determined from box counts is re- 
lated to the natural mortality rate m bc as 
m bc = - l0geil t ° 6c) , 3) 
where t = time. 
Boxes do not adequately measure the mortality of 
juvenile animals. The fraction dying not recorded by 
box counts, <J> 0 , is obtained by difference: 
O 0 = 
( N t - N t _, ) -(/?,_! - <& 6c AU - ® f N t _ x ) 
Nf~i + Rt-i 
(4) 
where <fy = the fraction taken by the fishery; 
R = the number of recruits into the population; 
and the first parenthetical term on the right- 
hand side represents the difference in abun- 
dance between two consecutive surveys. 
The two natural mortality rates, m bc (Eq. 3) and m 0 
(Eq. 5), are additive ( sensu Hassell et al., 1982; Holmes, 
1982), as the method for estimation includes the box 
counts as an input (Eq. 2) in contrast to fishing mortal- 
ity that can be compensatory under certain fishing sea- 
son scenarios (Klinck et al., 2001). <P 0 varied randomly 
over the time series with a 54-year mean of 0.274 and 
a 54-year median of 0.311 (Powell et al., 2008). The 
mortality rate can be obtained from <P 0 as 
log e (l-Q 0 ) 
(5) 
Fishing mortality was calculated as the fraction of the 
population present at the beginning of the year removed 
during that year by the fishery (catch): 
O 
f ~ 
catchf 
" 77 ' 
( 6 ) 
Additional mortality associated with the dredging pro- 
cess may occur; however, Powell et al. (2001, 2004) deter- 
mined that this source of mortality was inconsequential 
in comparison to the catch. Since the late 1950s, the 
fishery has rarely removed more than 7% of the stock 
annually, and normally much less, so that the yearly 
changes in stock abundance in Delaware Bay have been 
dominantly a product of natural processes over much of 
the time series (Powell et al., 2008). 
A crude estimate of age-frequency pattern was ob- 
tained by assuming equilibrium conditions. Yearlings, 
Y, were estimated from recruits (spat), R, based on 
observed one-year survivals of recruits between 1953 
and 1988 when yearlings were recorded as part of the 
survey. The yearling-to-spat ratio followed a weakly 
nonrandom pattern (Fig. 2) that provides a relationship 
between recruits and yearlings described by 
Y t +\ = 0.434e 
-3.659xl0 -11 N, 
tR f 
(7) 
Older age groups were modeled by assuming equivalent 
mortality across all ages. Thus, the number at age a is 
estimated as 
N a = Y e~ a(m ° +mbc \ (8) 
where m 0 and m bc are from Equa- 
tions 5 and 3, respectively. 
To model the relationship be- 
tween broodstock abundance and 
recruitment, we fit a relationship 
that produces declining recruit- 
ment at high abundance (overcom- 
pensation sensu Hancock, 1973; 
McCann et al., 2003), because 
shellfish can achieve densities suf- 
ficient to limit growth and repro- 
duction (e.g., Frechette and Bour- 
get, 1985; Frechette and Lefaivre, 
1990; Powell et al., 1995). Thus, 
from Hilborn and Walters (1992) 
R = N t _ ie 1 p (9) 
where R = the number of spat in 
millions; and 
N t-1 = oyster abundance in 
millions. 
