Powell et al.: Multiple stable reference points in oyster populations: Crassostrea virginica in Delaware Bay 
117 
70x10 9 - 
6.0x10 9 - 
0.0x10°' 
0.0x10° 5.0x10 s 1 .0x1 0 9 1.5x10 9 2.0x10 9 2.5x10 9 3.0x10 9 3.5x10 9 4.0x10 9 4.5x10 9 5.0x10 9 
Oyster (>20 mm) abundance 
Figure 8 
The low-abundance portion of the broodstock-recruitment relationship for the 
natural oyster beds of eastern oyster ( Crassostrea virginica ) in Delaware Bay, 
1953-2006. The dotted line is the best-fitted Ricker curve (Eq. 4), also shown 
in Figure 7 for the entire data set. The solid line is a linear fit (Eq. 5) with 
zero intercept. Note that at low abundance, the linear fit travels through the 
recruitment values slightly below that traversed by the Ricker curve. 
Equation 6 has the unique property 
of eliciting both depensatory and 
compensatory trends at low abun- 
dance. Sissenwine (1984), Hilborn 
and Walters (1992), and Peterson 
et al. (2001) have provided exam- 
ples of the well-known depensatory 
process in which increased preda- 
tory mortality rate is associated 
with increased prey population 
density because of increased prey 
preference at high prey density. 
Allen (1979) provided a somewhat 
unusual case for depensation in 
oysters determined by substrate 
availability rather than by disease. 
Hilborn and Walters (1992) pro- 
vided an analogous example from 
human exploitation of declining 
fish stocks. The present case is 
unusual, however, because box- 
count mortality first increases 
with declining abundance, but this 
depensatory phase is then followed 
by compensation in the mortality 
rate as abundance continues to 
decline. 
Calculation of first passage time 
Mean first-passage times were 
calculated from Redner (2001), 
according to the methods of Roth- 
schild et al. (2005) and Rothschild 
and Mullen (1985). Input data were obtained by divid- 
ing a two-dimensional data set into quadrants by the 
medians of the x and y variables (Fig. 10). An example 
frequency table for the broodstock and recruitment 
relationship (Table 2) shows the frequency of occur- 
rence of the data from the 54-yr time series in each of 
the four quadrants, employing the quadrant number- 
ing convention depicted in Figure 10. For instance, 
years characterized by low abundance and low recruit- 
ment, thus falling into quadrant 1, occurred 32% of 
the time. Table 2 also displays one-year transition 
probabilities compiled by examining the quadrant 
location of the x-y datum in successive years. For 
example, a low-recruitment -t-low-abundance year fall- 
ing into quadrant 1, was followed one year later by a 
high-recruitment+high-abundance year, an occurrence 
falling into quadrant 4, 18.8% of the time, whereas 
50% of the time, the following year was also a low- 
recruitment+low-abundance year. Thus, given that 
quadrant 1 is the starting point, the interval of time 
in which the population finds itself back in quadrant 
1 should be a lesser number of years than the time 
required for the population to shift from quadrant 
1 to quadrant 4. Mean first passage times (Table 3) 
express the number of years likely to elapse before the 
population with the x-y relationship characteristic of 
any one quadrant is again described by the relation- 
ship characteristic of that same quadrant, or obtains 
the relationship characteristic of one of the three other 
quadrants. 
Results and discussion 
Biological relationships that determine 
population dynamics 
Broodstock and recruitment A relationship between 
broodstock and recruitment is commonly found for shell- 
fish (Hancock, 1973; Peterson and Summerson, 1992; 
McGarvey et al., 1993; Lipcius and Stockhausen, 2002; 
Kraeuter et al., 2005), although not in every case has one 
been observed (Hancock, 1973; Crocos, 1991; Honkoop et 
al., 1998; Livingston et al., 2000). Such a relationship 
is commonly assumed for population dynamics models, 
and the adequacy of these models supports the likely 
importance of such a relationship in oysters (Mann and 
Evans, 1998, 2004; Dekshenieks et al., 2000; Powell 
et al., 2003). However, empirical evidence in oysters is 
contradictory and not well documented (e.g., Hofstetter, 
1983; Mann et al., 1994; Southworth and Mann, 1998; 
Livingston et al., 2000), and the travails of larval life 
and at settlement are certainly likely to add consider- 
able uncertainty to the success of any search for such 
