Powell et al.: Multiple stable reference points in oyster populations: Crassostrea virginica in Delaware Bay 
119 
Table 2 
One-year transition probabilities and the frequency 
of occurrences for the eastern oyster ( Crassostrea 
virginica) population in each quadrant over the 
54-yr time series were calculated from the Dela- 
ware Bay oyster broodstock-recruitment distribu- 
tion (Fig. 7). Median abundance over 54 years was 
2.64xl0 9 and median recruitment was 1.53xl0 9 . 
Arrows indicate trajectories between quadrants. 
Quadrants are defined in Figure 10. 
Quadrant 
1 
2 
3 
4 
1 
0.500 
0.125 
0.188 
0.188 
2 — > 
0.444 
0.222 
0.000 
0.333 
3 
0.222 
0.333 
0.333 
0.111 
4 — * 
0.111 
0.111 
0.167 
0.611 
Frequency of 
occurrence 
0.320 
0.170 
0.170 
0.340 
Number of years 
17 
9 
9 
18 
15-yr period than at any other time before 2000. 
Thus, high broodstock abundance was not reward- 
ed by equivalently high recruitment. Three mech- 
anisms seem viable. The first is that fecundity 
declines at high abundance as availability of food 
becomes limited. Food limitation by high densities 
of filter feeders is well described (e.g., Peterson 
and Black, 1987; Rheault and Rice, 1996; Wilson- 
Ormond et al., 1997). The second is that cannibalism of 
larvae occurs, but this cause of mortality is of unlikely 
importance (Andre et al., 1993; Tamburri et al., 2007). 
The third is that predation rates on juveniles increase 
at high abundance. Although little evidence of this ef- 
fect exists (e.g., Whitlatch and Osman, 1994; Powell 
et al., 1995), a proportional increase in predation on 
juveniles at high abundance is consistent with optimal 
foraging theory (Hughes, 1980), under the assumption 
that oyster predators are optimal foragers (Powell et al., 
1995; see also Pyke, 1984; Pierce and Ollason, 1987). 
All are standard explanations for compensation in the 
broodstock-recruitment relationship (e.g., Myers and 
Barrowman, 1996). 
The broodstock-recruitment diagram (Fig. 7) indi- 
cates that low abundance limited total recruitment in 
some way. This relationship is clear despite the exclu- 
sion from this data series of an unknown number of 
adults and recruits in State of Delaware waters, along 
the fringes of the bay, particularly in the river mouths, 
and on the leased grounds downbay of the high-mortal- 
ity beds. Moreover, the leased grounds likely retained 
substantial numbers of adult animals before the mid- 
1980s, although estimates of abundance are not avail- 
able. Many fewer were present thereafter because of 
the demise of the bay-season fishery. 6 Interestingly, the 
>- 
.Q 
CO 
Quad 
L 
rant 2 
Quadrant 4 
Quad 
rant 1 
Quadrant 3 
Variable X 
Figure 10 
Mean first passage times for eastern oyster ( Crassostrea vir- 
ginica) were calculated by employing an arbitrary quadrant 
numbering convention. One-year transition probabilities were 
obtained by examining the position of consecutive x-y data 
pairs in quadrant space. Four transitions are possible for 
each starting position, the possibilities for quadrant 1 being 
depicted. Sixteen total trajectories are possible. 
Table 3 
Mean first passage times, as well as the distribution of 
occurrences of the eastern oyster ( Crassostrea virginica ) 
population in each quadrant, after an infinite number of 
steps were calculated from the Delaware Bay oyster brood- 
stock-recruitment distribution (Fig. 7). The observed 
distribution of occurrences is given in Table 2. Arrows 
indicate trajectories between quadrants. Quadrants are 
defined in Figure 10. 
Quadrant 
1 
2 
3 
4 
Mean first passage 
time (yr) 
1 — > 
3.25 
6.06 
6.45 
5.00 
2 — > 
3.60 
5.78 
7.82 
4.14 
3 -> 
4.20 
4.56 
5.78 
5.24 
4 — > 
5.40 
6.26 
6.65 
2.89 
Distribution after an 
infinite number of steps 
0.308 
0.173 
0.173 
0.346 
6 Anecdotal information indicates that numbers were low in 
the 1960s as well. 
decline in abundance on leased grounds after 1985 does 
not generate a perceptible change in the broodstock-re- 
cruitment relationship. 
Oyster larvae tend to set preferentially on live oysters 
and boxes rather than on cultch (shell clumps, shells, 
