Powell et al.: Multiple stable reference points in oyster populations 
137 
The recruitment rate F ; (A^_ 1 ) is calculated as 
log. 
1 + e 
*t-l 
r,w ( _ 1 )= 
( 10 ) 
We compared the results of Equation 10 to that obtained 
for a best-fit linear regression with zero intercept. The 
linear relationship is 
Surplus production S is calculated as the difference 
between additions to the population through recruit- 
ment and debits through mortality. The two processes 
are structurally uncoupled in time, however. First, mor- 
tality occurs differentially in time in relation to recruit- 
ment. Second, the method of data collection results in 
a time-integrated value of mortality, but a year ending 
value for recruitment, inasmuch as the death of recruits 
between settlement and the time of observation is not 
recognized as a component of the mortality term (see 
Keough and Downes, 1982; Powell et al., 1984; Caffey, 
1985). Consequently, in the absence of fishing, 
R t = 0A93N t _ v (11) 
Note that the linear fit travels through the recruitment 
values at low abundance slightly below that traversed by 
the Ricker curve (Fig. 8 in Powell et al., 2009). Powell et 
al. (2009) provide caveats concerning the use of a single 
broodstock-recruitment curve for the population over the 
entire 54-yr time series. The dispersion of the stock over 
the four bay regions exerts limitations on the ambit of 
stock performance at any specific time. 
Powell et al. (2009) develop an admittedly ad hoc em- 
pirical relationship to describe the relationship between 
box-count mortality and abundance: 
% = ® + * log, (#,_! + PI- 
'S, =tf,_ 1 (e r *f-l)- 
N, 
t - i 
\mbc t +m O t } 
(13) 
which reduces to the familiar equation 
+ (14) 
where t = the time increment between observations of 
recruitment. 
Note that the subscript t — 1 is used for the stock abun- 
dance value N because the stock survey occurs at the 
end of the year preceding the year for which surplus 
production is forecast and for which recruitment is mea- 
sured. 
(pN^ + xN^d 
where co=0.055, k= 0.03, p=1.0, (p=0.0025, x=0.1, ip=2.2, 
and o = 0.8, with N expressed as billions of 
animals. 
The specific mortality rate, m bc (N), is calculated with 
Equation 3. Equation 12 has the unique property of 
eliciting both depensatory and compensatory trends 
at low abundance. Powell et al. (2009) provide caveats 
concerning the use of the broodstock-mortality curve. 
The dispersion of the stock over the four bay regions 
exerts limitations on the ambit of stock performance at 
any specific time. At abundances greater than 4 x 10 9 , 
mortality was low. The fraction dying each year aver- 
aged 9.6 % for these nonepizootic years, a nonepizootic 
year being defined for convenience as a year in which 
the fraction dying is less than 20%. However, of the 32 
years with abundances less than 3xl0 9 , of which 14 
were epizootic years, only one had a fractional mortal- 
ity between 0.15 and 0.20. Accordingly, two divergent 
outcomes exist over a range of low abundances. In some 
years, the fraction dying approximates the long-term 
mean for high-abundance years, about 9.6%. In other 
years, epizootic mortalities occur. The likelihood of 
these two divergent outcomes is substantively affected 
by the dispersion of the stock (Powell et al., 2009). 
Modeling of population dynamics— results 
of simulations and discussion 
In the absence of fishing, the population increases when 
surplus production S t is positive (Eq. 14). The popula- 
tion decreases when S t is negative. Abundances where 
S t is zero offer potential biological reference points, 
as do cases where S t is maximal. Carrying capacity 
is an example of the former. In this case, mortality 
and recruitment balance and S t = 0. Surplus produc- 
tion declines as abundance nears carrying capacity 
and, therefore, the rate of change should be negative, 
but relatively constant; thus, J|<0 and ~^|~0. We 
will refer to reference points characterized by S t = 0, 
§<0 and H~0 as type-I reference points (Fig. 3). B msy 
is defined to be a maximum in surplus production. Sur- 
plus production declines as abundance declines below or 
rises above this point. Hence, S.> 0, and ^4<0. We 
will refer to maxima in surplus production as type-II 
reference points (Fig. 3). Because the time series under 
analysis is configured in terms of abundance rather than 
biomass, the designation N msy , rather than B msy , will be 
used hereafter. 
We present hereafter a series of simulations of the 
Delaware Bay oyster stock designed to examine the 
change in surplus production with abundance. We first 
consider a population for which recruitment rate fol- 
lows Equation 9, a compensatory curve, with a 54-yr 
average unrecorded mortality rate (Eq. 5), and with 
the box-count mortality rate described by Equation 12. 
