70 
Fishery Bulletin 1 12(1) 
Appendix 2 
A prior probability distribution was developed for 
the intrinsic rate of population increase for Atlantic 
Croaker ( Micropogonias undulatus ) off the U.S. Atlan- 
tic coast. 
The basic demographics of Atlantic Croaker (Ap- 
pendix 3) were combined in a Leslie matrix projection 
framework (McAllister et ah, 2001) to construct a prior 
probability density function for the intrinsic rate of 
population increase, r (Table 1). The mean reproduc- 
tive rate-at-age (R a ) was scaled by 10 -5 to ensure that 
for all year (50)xreplicate (2000) combinations, the 
resulting r values ranged from 0.01 to 1.5, as seems 
plausible for marine fish populations (Vasconcellos and 
Haimovici, 2006; Jensen et al., 2012): 
Ra = 10- 5 (p a SR a F a ). (Al) 
where p a , SR a , and F a are age-specific proportion of 
mature individuals, sex-ratio (assumed to be 0.5), and 
mean fecundity, respectively. Implicit in the scaling 
factor of 10~ 5 for R a was the assumption that about 
11.513 represented the cumulative mortality from egg 
fertilization to the recruiting age-group (here age-0) in 
the Leslie population model. Fecundity-at-age was es- 
timated as 
F a = yS{L oo [l-exp(-A'(«-« 0 ))]} / , (A2) 
where and y are parameters of the fecundity (number 
of eggs)-total length (mm) relationship; L„,K, and ao 
are parameters of the von Bertalanffy growth function. 
Lognormal distributions were assumed for both nat- 
ural mortality and reproductive rate-at-age (i.e., they 
were log-transformed and treated as expected means). 
Monte Carlo samplings were performed with R soft- 
ware (vers. 2.15.3; R Development Core Team, 2013) 
with an age-independent coefficient of variation (CV) 
equal to 0.3 for both parameters (however, a CV=0.3 
for the reproductive rate was insensitive to the natural 
mortality CV G [0.1, 1.0] in terms of r summary sta- 
tistics and distributions). The CVs used were a single 
realization of all possible CV combinations for natural 
mortality and reproductive rate. They were preferred 
because, unlike the r estimates in many other trials, 
the corresponding r estimates fell within and spanned 
the 0.01-1.5 interval (note: reproductive rate CVs<0.3 
yielded truncated r distributions; reproductive rate 
CVs>0.3 led to r distributions with long tails far be- 
yond 1.5). The stochasticity introduced in natural mor- 
tality rates was subsequently propagated into the sur- 
vival rate and expected survivorship-at-age. 
For each replicate, the scalar number for the initial 
(year-1) population-at-age of female Atlantic Croaker 
was 1000. McAllister et al.’s (2001) Equations 9-14 or 
Hammond and Ellis’ (2005) Equations 1-5 were ap- 
plied as appropriate. During the sampling, there were 
negative values of r. These values usually result from 
generating stochastic reproductive rates and survivor- 
ship values independently of one another, including 
coupled low values of these parameters, the combina- 
tion of which can lead to a population that cannot sus- 
tain itself (McAllister et al., 2001; Hammond and Ellis, 
2005). The final prior probability density function for 
r (Table 1) was developed after discarding those nega- 
tive r values and ensuring that the age structure of 
the projected population was stable (population stabil- 
ity occurred since year-3). 
Appendix 3 
Demographic inputs for the construction of the prior prob- 
ability distribution of the intrinsic rate of population 
increase for Atlantic Croaker ( Micropogonias undulatus) 
off the U.S. Atlantic coast. The von Bertalanffy growth 
parameters are the asymptotic length (Loo), the growth 
rate (K), and the theoretical age when length is zero (ao). 
The parameters of the fecundity (number of eggs (-length 
relationship are the coefficient (/?) and the exponent (y). 
Attribute 
Value or range 
Source 
Age (a, years) 
0-15+ 
ASMFC 1 
von Bertalanffy growth parameters: 
ASMFC 1 
Loo ( mm ) 
431 
K (mmxyear -1 ) 
0.214 
ao (years) 
-2.35 
Fecundity (F)-length(L) relationship: F = (3L y 
Morse (1980) 
P 
0.002594179 
y 
3.361 
Maturity-at-age (ages: 0-15+ years) 
0, 0.9, 1 
Barbieri et al. (1994) 
Natural mortality-at-age (year -1 ; age: 0-15+) 
0.461, 0.374, 0.324, 0.293, 0.272, 0,257, 0.246, 0.238, 
0.232, 0.227, 0.223, 0.220, 0.218, 0.216, 0.215, 0.214 
ASMFC 1 
