138 
Fishery Bulletin 107(2) 
These relationships are depicted in Figures 7 and 10 
of Powell et al. (2009). The trajectory for surplus pro- 
duction under these constraints is compared in Figure 
3 and detailed in Figure 4. Recruitment rate rises as 
abundance declines (Fig. 4). This is anticipated from 
the compensation inherent in the relationship between 
broodstock and recruitment. The box-count mortal- 
ity rate shows a maximum somewhat above an abun- 
dance of 2xl0 9 (Fig. 4). These relationships define a 
trend between surplus production and abundance that 
is divergent from the normal Schaefer curve (Ricker, 
1975; Hilborn and Walters, 1992; Haddon, 2001; Zabel, 
2003), as expected. The single type-I reference point is 
at iV=9.3xl0 9 . This is an estimate of carrying capac- 
ity, K. Typically a single type-II reference point would 
exist, N msy , at about y, but in this case two maxima 
in surplus production exist, one higher, N^ nsy , than 
the other, N L msy . N^ lsy is at iV=4.86 x 10 9 . This is the 
abundance classically interpreted as N msy , and, indeed, 
surplus production is maximal at this point and the 
value is approximately -j- The second type-II reference 
point occurs at A=1.43 x 10 9 . Unlike the simple Schaefer 
curve depicted in Hilborn and Walters (1992), Haddon 
(2001), and Zabel (2003), a local minimum in surplus 
production exists between these two type-II surplus 
production maxima, at 1V=2.57 x 10 9 . In this case, sur- 
plus production remains above zero, S t > 0. An increase 
in abundance above this level and a decrease in abun- 
dance below this level both increase surplus produc- 
tion. This reference point, herein designated type III, 
always occurs between two maxima in surplus produc- 
tion and is characterized by - s -=0 and 0 (Table 2). 
dN dN 
The unusual nature of the surplus production curve 
in Figure 4, that yields the local minimum in surplus 
production and a secondary surplus production peak 
at a lower abundance, is produced by the depensatory 
and compensatory segments of the box-count mortality 
relationship established by the relationship between the 
occurrence of epizootics and abundance in the Delaware 
Bay oyster stock. 
Figures 5-7 show three alternative trajectories for 
the change in surplus production with abundance in 
the Delaware Bay oyster stock obtained by small modi- 
fications of the parameters governing recruitment and 
mortality. The first is obtained by using the 54-yr me- 
dian unrecorded mortality rate, rather than the 54-yr 
mean rate. The median is distinctly higher. Again, 
the surplus production trajectory includes one type-I, 
two type-II, and one type-III reference points (Figs. 3 
and 5). The abundance associated with the four refer- 
ence points remains unchanged, although the surplus 
production values associated with the type-II maxima 
and type-III minimum are lower than in the preceding 
case (Table 2). 
The second alternative is obtained after a perusal of 
Figure 10 in Powell et al. (2009) that shows that the 
mortality rate for stock abundances frequented by epi- 
zootics often falls below the curve provided by Equation 
12. This is a function of stock dispersion that modulates 
the likelihood of epizootic mortality rates (Powell et al., 
2009). In fact, on the average, box- 
count mortality rate reaches epizo- 
otic levels only half the time. Thus, 
Figures 3 and 6 show the trend in 
surplus production when epizoot- 
ics are assumed to occur only half 
the time, and box-count mortality 
rate is expressed as the average of 
a year with an epizootic and a year 
without one. The type-III reference 
point is nearer the N^ sy value in 
this surplus production trajectory, 
so that the valley between N L 
and is something more than a 
shoulder on the surplus production 
curve. Thus, the value of the sur- 
plus production maxima, averaged 
over a number of years, is strongly 
influenced by the frequency and in- 
tensity of epizootics (Table 2). 
The final alternative addresses 
the uncertainty that exists in the 
shape of the broodstock-recruit- 
ment curve at low abundance. 
Linearizing the curve at low abun- 
dance (Eq. 11) yields a surplus 
production trajectory depicted in 
Figure 8 of Powell et al. (2009). 
The relationship is unique in gen- 
erating a second type-I reference 
