Munyandorero: Climate effects on Micropogonicis undu/atus 
53 
MOitlfit,©) 
Observation process PDF 
(i = 1, ..., 2; t = 1972, 2008). (lc) 
Specifically, C = R U W; R = (i? t K the set of total 
fishery removals U=1972, 2008); and W = {W t (, the 
set of MWET time series ( t= 972, 2008). For Ml, B t 
was exposed only to fishing (C = R). For M2, B t was af- 
fected by both fishing and water winter temperature (C 
= R U W). For simplicity, the fisheries removals were 
assumed to be known perfectly. The SEAMAP index 
during 1972-89 was treated as an unobserved random 
variable because it was unavailable across that period. 
The deterministic, time-discrete part of biomass ex- 
pectation in Ml and M2 is expressed as 
E(B l+1 ) = B t+1 = fl t +G t -Ii2ft, (2) 
f 
where f - a subscript for fishery and, during year t\ 
Gt = production that quantifies the overall change 
in biomass due to somatic growth, re- 
cruitment, and natural mortality; and 
Rf t = fishery-specific removals. 
Gt is a function of B t , the intrinsic rate of popula- 
tion increase (r), and the carrying capacity (B «,). The 
Graham-Schaefer (or logistic) form was chosen to quan- 
tify Gt because of its simplicity (it has 2 parameters, r 
and Boo) and because it is a central case among possi- 
ble shapes of production models (Prager, 1994). There- 
fore, for Ml, 
G t = rS t[l-J L j- (3a) 
In biomass dynamic modeling with environmental 
effects, environmental factors can act on the stock pro- 
ductivity (i.e., on r, B^, or both), the fisheries’ or sur- 
veys’ catchabilities, or both (Freon, 1988; Jacobson et 
al., 2005; Jensen, 2002, 2005). MWET was normalized 
and introduced into the parameter r because MWET 
affects Atlantic Croaker productivity through growth 
or recruitment during the prerecruit stage (Hare and 
Able, 2007). The approach followed the framework 
of log-linearly adding environmental covariates into 
fisheries models (e.g., Hilborn and Walters, 1992) and 
assuming implicit controlling effects of MWET on re- 
cruitment (lies and Beverton, 1998; Levi et al., 2003). 
Therefore, for M2, the year-specific intrinsic rate of in- 
crease (rt) is 
r t = r 0 e 
aWt 
(3b) 
where a is a coefficient controlling (linearly) the influ- 
ence of MWET on Atlantic Croaker productivity and 
ro is a scaling parameter. In common with similar ap- 
plications (e.g., Maunder and Watters, 2003), a was 
limited to values greater than zero because MWET is 
positively correlated with juvenile production (Norcross 
and Austin 2 ; Hare and Able, 2007). 
To improve the efficiency of the Markov Chain Mon- 
te Carlo (MCMC) estimation algorithm implemented 
in BUGS, the state-space formulations of Ml and M2 
were expressed in terms of depletions, b t (b t = B t /B oa ), 
herein considered to be “true” and assumed to have log- 
normal distributions (Meyer and Millar, 1999; Millar 
and Meyer, 2000): 
^1972 ~ LN log (&1972 ) Up j 
for Ml and M2, (4a) 
(1 + r)b t -rb^-b oa 'ZRf t 
f 
b t +i~LN\\og 
for Ml (t = 1973, ..., 2008), 4 (4b) 
:2 
6 t+ i ~ LN < log 
(1 + r t )b t - r t b t -b^Rtt 
f 
for M2 (t = 1973, ..., 2008), 
(4c) 
where b t = the expected depletion in year t, treated as 
deterministic; 
6oo = 1 /B„; and 
Tp = the precision (inverse of the variance, 
CTp ) of the process error. 
For the observation error model (Eq. lc), each bio- 
mass index (z'tj ) was assumed to be proportional to the 
year- and period (y)-specific biomass and to be log- 
normally distributed about its expected, model esti- 
mate ( iq ): 
it j ~LAl[log(itj),T i j] (5a) 5 
hj A\{l\b t B,„. 
(5b) 
For the NEFSC index, j=l (1972-93) when the index 
varied at low levels with no obvious trend or 2 (1994- 
2008) when the index showed an overall increasing 
trend (Fig. IB). In fact, this index indicates that At- 
lantic Croaker accessibility and vulnerability changed 
between these periods. For the SEAMAP index, j = 1 
(1990-2008) because the index varied without trend 
(Fig. IB). In Equation 5a, 
xjj = the observation error precision (xjj = l/a| , 
is the observation error variance) by period. 
In Equation 5b, 
4 Equation 4 a-c corresponds with BUGS parameteriza- 
tions and code. The usual stochastic formulation of Equ- 
tion 4b, for example, is h t+ i — 
(1 + r)b t -rb? Rft 
f 
where e t ~A(o,Opj and (1 + r)b t - rb% = 6 t+ i- The 
expected (deterministic) biomass ( B t+ i ) ahd the stochas- 
tic (true) biomass (B t +U 411 y ear f +l are B t+1 =6 t+1 B co and 
A+ 1 = The same formulation applies for 
Equation 4, a and c. 
5 The usual stochastic formulation of Equation 5a is U = i tJ e a>1 j t , 
with ® ijt ~JV(0,<T{j). 
