Powell et al. : Multiple stable reference points in oyster populations 
139 
point, at V=1.93 x 10 9 . This is a 
multiple-stable-point system with 
two carrying capacities, one at K H 
and one at K L . Note that the lower 
surplus production maximum is 
closer to K L than expected by the 
Schaefer relationship: N L msy > ^ (Fig. 
7). This representation of oyster 
population dynamics also gener- 
ates a type-IV reference point at 
A/=3.03xl0 9 . Type IV, like type I, 
is characterized by S.-O and 
0, but in this case -ttt >0 (Table 2). 
Figure 8 presents a stylized version 
of the surplus production trajectory 
of Figure 7. Note that the type-I 
reference points are points of con- 
vergence. Abundance rising above 
this value will produce negative 
surplus production and a return to 
the abundance level and vice ver- 
sa for a decline in abundance. On 
the other hand, type-IV reference 
points are divergences or points of 
population instability. They mark 
thresholds for population collapse. 
The divergence that is the type-IV 
reference point is maintained by 
the competing rates of box-count 
mortality and recruitment that 
switch in dominance at this point 
(Fig. 8). A population reaching a 
type-IV reference point as abun- 
dance declines will see a rapid fur- 
ther decline. Once below this point, 
the likelihood becomes very low that the population 
can cross the gulf and re-acquire its high-abundance 
trajectory. 
Reference-point-based management 
Carrying capacity Perusal of the time series suggests 
that population abundances above about 12xl0 9 are 
unstable. The analyses provided using Equation 14 
return this same expectation, that carrying capacity is 
about 9.3 x 10 9 . This explains the stability of population 
abundance during the 1970s as the population was at or 
near carrying capacity (Fig. 9). Abundance rose above 
this point a number of times between 1970 and 1985, 
but higher abundances were not sustainable. Interest- 
ingly, this carrying capacity is a carrying capacity for 
a population enzootic for MSX disease. The natural 
mortality rate during the 1970s is not much different 
from the few measures that exist for the time frame 
pre-1957 and the pre-MSX years are not outliers on the 
broodstock-recruitment diagram. So, MSX was not a 
significant agent of mortality during this period. Hence, 
predisease carrying capacity for which no empirical 
quantitative record exists is likely to have been similar 
to abundances during the 1970s, with the observed dif- 
2 4 6 8 10 
Abundance (billions) 
Figure 4 
The relationship of surplus production (Eq. 14), the rates of recruitment, 
unrecorded mortality, box-count mortality, and a conditional estimate of catch 
expressed as the fraction of the stock, for parameters defined by, for recruit- 
ment, T ( from Equation 10, m 0 from Equation 5 using the 54-year average 
4> 0 , and m bc from Equation 12. This simulation assumes compensation in the 
broodstock-recruitment curve, average unrecorded (mostly juvenile) mortal- 
ity, and a box-count mortality rate that emphasizes epizootic mortality at low 
abundance. Catch estimates are conditional on the assumption of long-term 
persistence of a chosen abundance level and distribution of the entire stock 
in habitats permitting growth to market size. 
Table 2 
The surplus production values associated with the types I, 
II, III, and where applicable, IV reference points depicted 
in the referenced figures and the defining characteris- 
tics of each reference point type. Surplus production is 
expressed in billions of oysters. NA=not applicable. 
Figure 
number 
Type I 
Carrying 
capacity 
(. K ) 
Type 
II 
N H 
msy 
Type 
III 
c 
^ min 
Type 
II 
N L 
msy 
Type IV 
Point 
of no 
return 
Surplus production 
4 
0.0 
0.665 
0.167 
0.319 
NA 
5 
0.0 
0.511 
0.103 
0.275 
NA 
6 
0.0 
0.519 
0.297 
0.318 
NA 
7 
0.0 
0.511 
-0.094 
0.112 
0.0 
c 
dS 
d 2 S 
dN 
dN 2 
Defining characteristics 
Type I 
= 0 
<0 
~0 
Type II 
>0 
= 0 
<0 
Type III 
>0 or <0 
= 0 
>0 
Type IV 
= 0 
>0 
~0 
