Powell et al.: Multiple stable reference points in oyster populations: Crassostrea virginica in Delaware Bay 
121 
Table 4 
One-year transition probabilities, as well as the fre- 
quency of occurrence, of the eastern oyster ( Crassostrea 
virginica) population in each quadrant over the 54-yr 
time series were calculated from the Delaware Bay oyster 
broodstock-mortality distribution (Fig. 9). Median abun- 
dance was 2.64xl0 9 and the median mortality fraction 
was 0.127. Arrows indicate trajectories between quad- 
rants. Quadrants are defined in Figure 10. 
Quadrant 
1 
2 
3 
4 
1 — > 
0.222 
0.444 
0.222 
0.111 
2 — > 
0.125 
0.500 
0.063 
0.313 
3 — > 
0.059 
0.059 
0.647 
0.233 
4 — > 
0.300 
0.400 
0.300 
0.000 
Frequency of 
occurrence 
0.170 
0.320 
0.320 
0.189 
Number of years 
9 
17 
17 
10 
medians allocates most, but not all, epizootic years in 
the abundance range of 1.5xl0 9 to 3 x 10 9 to a single 
quadrant. 
Nevertheless, even with this ambiguity, high-mortal- 
ity events were more likely with low abundance and 
some transitions were more likely to occur than oth- 
ers. Mean first-passage times were particularly long 
for transitions to quadrant 1 (low-mortality+low-abun- 
dance), always exceeding 6 years (Table 5). Mean first- 
passage times were also long for most transitions to 
quadrant 3, the low-mortality+high-abundance quad- 
rant, with the exception of those with quadrant 3 as 
the initial state. By contrast, the population was likely 
to return to quadrant 2 (high-mortality+low-abundance) 
from most quadrants in about 3-4 years (Table 5). This 
return interval is an expression of the relative fre- 
quency of Dermo epizootics. Interestingly, the tendency 
to return to quadrant 2 (high-mortality+low-abundance) 
was distinctly less from quadrant 3 (low-mortality and 
high abundance) than from other quadrants. High-mor- 
tality events were unlikely to occur when abundance 
was high. The distribution of first-passage times again 
supports the presence of multiple stable states for the 
Delaware Bay oyster population. 
The distribution of mortality with abundance is not 
constant, nor does it display a simple density depen- 
dency. Epizootics occurred less often at high abundance 
and near lowest abundance. Decreased mortality at low 
abundance was not unexpected for a population exposed 
to a disease that generates epizootic conditions (Gill, 
1928; Ackerman et al., 1984; Kermack and McKend- 
rick, 1991). Normally, transmission rates of disease 
decline with decreased host density because contact 
rates decrease (Black, 1966; Andreasen and Pugliese, 
1995; Godfray and Briggs, 1995; Heesterbeek and Rob- 
erts, 1995) and this leads to lower rates of mortality. 
This decline in transmission rates is true for nearly 
all diseases but does not seem to be the case for MSX 
Table 5 
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 
broodstock-mortality distribution (Fig. 9). The observed 
distribution of occurrences is given in Table 4. Arrows 
indicate trajectories between quadrants. Quadrants are 
defined in Figure 10. 
Quadrant 
1 
2 
3 
4 
Mean first passage 
time (yr) 
1 — > 
6.54 
3.55 
6.14 
4.59 
2 
6.87 
3.03 
7.09 
3.67 
3 — > 
8.10 
6.00 
3.11 
4.21 
4 — * 
6.18 
3.88 
5.68 
5.11 
Distribution after 
infinite steps 
0.153 
0.339 
0.321 
0.196 
or Dermo, which are characterized by inherently high 
transmission rates over a wide range of abundance 
(Hofmann et al., 1995; Powell et al., 1996, 1999). In the 
Delaware Bay oyster stock, the declining frequency of 
epizootics at low abundance originates in the dynam- 
ics of stock dispersion. A contraction of the stock away 
from areas of highest disease mortality normally is as- 
sociated with low abundance. Thus, epizootics are most 
likely to occur in a narrow window of abundance as the 
stock expands from its habitat of refuge on the medium- 
mortality beds, thereby leaving a greater proportion of 
the stock once again on the medium-mortality beds. 
This stock contraction, consequently, mitigates against 
a recurrence of the high-mortality event. Depensation in 
the mortality rate as abundance declines is, of course, 
an extinction scenario, were it to continue. The coun- 
tervailing compensatory process of stock contraction is 
the dominant protective action against local extinction, 
rather than a decline in host density that reduces dis- 
ease transmission rates. 
What is unusual is the low probability of epizootics at 
high abundance. Mortality rates are often assumed to 
be invariant over a wide abundance range for marine 
species (e.g., Paloheimo, 1980; Hoenig, 1983; Vetter, 
1987; Clark, 1999) and, contrariwise, increased mor- 
tality at high abundance is expected of most popula- 
tions exposed to epizootic disease (e.g., Anderson and 
Gordon, 1982; Andreasen and Pugliese, 1995; Godfray 
and Briggs, 1995; Jaenike, 1998). Neither expectation 
conforms to what has been observed. Thus, one of the 
interesting quandaries is the maintenance of popula- 
tion abundance near the higher carrying capacity of 
the population during the 1970s-1985 high-abundance 
period. Some portion of this was caused by reference 
point-based management, which controlled fishing mor- 
tality to values normally below 5% of the stock (Pow- 
ell et al., 2008). Some portion was due to higher than 
