xf,t(0))
and incorporating the covariate species in Eq. 1b to obtainDerivation of Eqs. 1 through 4.
We now derive Eqs. 2a–2c from
1a–1c. First, we generalize Eqs. 1a–1c by replacing Zf,t
by (Zf,t + xf,t(0))
and incorporating the covariate species in Eq. 1b to obtain
|
Xs,t = (Zf,t+
|
(A.1a) |
|
Ys,t = Xs,t-1exp(- |
(A.1b) |
|
Zf,t = |
(A.1c) |
In the log-scale, the preceding equations become
|
xs,t = log(exp(zf,t)+
|
(A.2a) |
|
ys,t = (1- |
(A.2b) |
|
zf,t = log( |
(A.2c) |
where xs,t = log(Xs,t
), ys,t = log(Ys,t), and
zf,t = log(zf,t); all summations sums over
all sites s in a fjord f. Note that mt = m exp(-F1,t-1)
and 
t
= exp(-F2,t-1),
where the F’s denote fishing mortality. Consider the case that
= 0,
F1,t-1
0,
F2,t-10,
and assume the simplified model is stationarity, with 
Y,s
and mZ,f
being the stationary means of Ys,t and Zf,t,
respectively. Also, write y,s
= log(Y,s
) and z,f
= log(Z,f
). For the simplified model, taking expectation on both sides of Eq.
A.1c yields
Z,f
= 
s mY,s
+ Z,f.
Hence, (1-)Z,f
= s
mY,s,
which will be useful below. The basic idea for deriving Eqs. 2a–2c is to
approximate the right side of Eqs. A.2a–A.2c by a first order Taylor expansion
around =
0, F1,t-1 = 0,
F2,t-1 = 0,
ys,t-1 = y,s,
z f,t-1 = z,f
and z f,t = z,f
. Equations 2a–2c are obtained by noting that for Eq. A.2a, 
xs,t
/zf,t
= 1 and xs,t
/l
= x(0)f,t /z,f,
where all derivatives are evaluated at
= 0, F1,t-1 =
0, F2,t-1
= 0, ys,t-1 = y,s,
z f,t-1 = z,f
and z f,t = z,f.
For Eq. A.2c,
zf,t /F1,t-1
= -(
mY,s)/(s
mY,s
+ Z,f
) = -(1-
), zf,t
/F2,t-1
= -Z,f
/(s
mY,s+Z,f
)
= -,
zf,t
/
ys,t-1 = mY,s
/(s
mY,s
+ Z,f
), and
zf,t /
z f,t-1 = Z,f
/(s
mY,s
+ Z,f
) = .
Define cs = mY,s
/[(s
mY,s
+ Z,f
)(1- )].
Then it can be readily checked that s
cs = 1. Hence, Eqs. A.2a–A.2c approximately
equal
|
xs,t = zf,t
+ |
(2a) |
|
ys,t = (1- |
(2b) |
|
zf,t = df
+ (1- |
(2c) |
where df =z,f
-(1-)csy,s
-z,f
is a constant, and 
f
= /Z,f.
Consequently, we have
|
xf,t = zf,t
+ |
(3a) |
|
yf,t = (1- |
(3b) |
|
zf,t = df
+ (1-q |
(3c) |
where the last equation [Eq. 3c]
is obtained by re-labeling some of the variables in Eqs. 2 so as to make all
the model-variables fjord-specific (i.e., xf,t = csxs,t
and similarly we can define yf,t).
Next, we outline the derivation
of Eq. 4. Adding to the left side of Eq. 3c the product of 
times the lag 1 of the left side of Eq. 3c and doing likewise to the right side
of Eq. 3c eliminates the y’s from the equation, because yf,t
+ yf,t-1
= (1-
)xf,t-1
+
wf,t-1,
owing to Eq. 3b. Specifically,
|
zf,t + = (1+ + = (1 + –[(1- |
(A.3) |
Equation 3a implies that zf,t
= xf,t –( fx(0)f,t
+ 
f +
t +
f,t
). Upon substituting this expression into Eq. A.3, we obtain Eq. A.4 after some
algebra and noting that t
is modeled as a linear combination of water temperature and the NAO:
|
xf,t = ( + + constf
+ + {-[(1- + weathert |
(A.4)
|
where weathert = 
0Tt
+ 1Tt-1
+ 2Tt-2
+ k0naot + k1naot-1
+ k2naot-2 (where further 0
= ,
1 = (-)
and
2
= -,
and k0 = k, k1 = (-)k
and k2 = -gk)
and constf = (1 + )df
+ (1-)(1+
)f.
Table A1a. A complete list of variables and their description.
|
Variables |
|
|
Symbol |
Description |
|
|
stochastic effect (white noise) on cod reproduction at site s in year t |
|
|
(weighted) stochastic effect (white noise) on cod reproduction in fjord f in year t |
|
It |
dummy variable of the 1988 bloom; equals 1 for t = 1988 and 0 otherwise |
|
naot |
Northern Atlantic Oscillation (NAO) index in year t |
|
Tt |
spring water temperature in year t |
|
ws,t |
log abundance of covariate species at site s in year t |
|
wf,t |
(weighted) log abundance of covariate species in fjord f in year t |
|
xf,t(0) |
amount (in million) of larvae released in fjord f in year t |
|
Xs,t |
0-group cod abundance at site s (within fjord f) in year t |
|
xs,t |
log 0-group cod abundance at site s (within fjord f) in year t |
|
xf,t |
(weighted) log 0-group cod abundance in fjord f and in year t |
|
Ys,t |
1-group cod abundance at site s (within fjord f) in year t |
|
ys,t |
log 1-group cod abundance at site s (within fjord f) in year t |
|
yf,t |
(weighted) log 1-group cod abundance in fjord f and in year t |
|
Zf,t |
abundance of adult and mature cod in fjord f in year t |
|
zf,t |
log abundance of adult and mature cod in fjord f in year t |
TableA1b. A complete list of variables and their description.
|
Parameters |
|
| Symbol |
Description |
|
a0, a1, a2 |
common direct (lag1, lag2) bloom effect on the (weighted) log 0-group abundance of any given fjord |
|
Af |
direct bloom effect on the (weighted) log 0 -group abundance in fjord f |
|
|
site effect in cod reproduction |
|
|
year effect in cod reproduction |
|
ar1, ar2 |
Lag 1 and lag 2 autoregressive coefficients in Eq. 5 |
|
|
within-cohort intraspecific effect |
|
Bf |
lag-1 bloom effect on the (weighted) log 0 -group abundance in fjord f |
|
cs |
relative 1-group cod abundance at site s within fjord f |
|
Bf |
lag-2 bloom effect on the (weighted) log 0 -group abundance in fjord f |
|
dt |
intervention effect function of the 1988 bloom |
|
|
covariate species effects |
|
|
fraction of the natural spawning population size that gives rise to 1 million larvae in fjord f |
|
|
equals |
|
|
water temperature effects from current year, last year and two years ago. |
|
F1,t |
fishing mortality rate of the 1-group cod |
|
F2,t |
fishing mortality rate of the mature cod |
|
|
Between-cohort intraspecific effect |
|
|
equals |
|
|
NAO effects from current year, last year and two years ago. |
|
|
average number of mature cod needed to spawn 1 million larvae |
|
ma1, ma2 |
lag 1 and lag 2 moving-average coefficients in Eq. 5 |
| Y,s
|
stationary mean 1-group abundance at site s under the simplified (1a-1b) with no trends nor covariates |
|
y,s
|
equals log( |
|
Z,s
|
stationary mean mature cod abundance in fjord f under the simplified (1a-1b) with no trends nor covariates |
| z,s
|
equals log(µZ,f) |
|
mt |
survival rate from the 1-group cod to 2-group |
|
|
annual rate of change in fishing mortality of mature cod |
|
|
common direct bloom effect on the 1-group cod |
|
|
common indirect bloom effect on the 1-group cod |
|
|
baseline survival rate of the mature cod (in year 1970) |
|
|
survival rate of adult and mature cod |
|
|
common bloom effect on the 0-group cod |
Table A2a. Parameter estimates obtained
through the model fitting assuming, for reference, no effect of the algae bloom
(see Chan et al. 2002b); f
= /Z,f.
Bold P values represent significance at the 5% level. Larvae releases
are only performed in some fjords.
|
Parameter
|
Estimate
|
SE
|
Ratio
|
P
value
|
Fjord
name
|
| ma1 | -0.13 | 0.105 | -1.23 | 0.226 | |
| ar2 | 0.45 | 0.090 | 5.02 | 0.000 | |
| ma2 | -0.36 | 0.098 | -3.68 | 0.001 | |
 1 |
-0.0073 | 0.0019 | -3.90 | 0.000 | |
| k4 | -0.12 | 0.048 | -2.53 | 0.015 | |
| 0 |
-0.073 | 0.054 | -1.37 | 0.177 | |
| 1 |
0.098 | 0.056 | 1.75 | 0.086 | |
| 2 |
0.13 | 0.058 | 2.21 | 0.032 | |
 0 |
0.043 | 0.031 | 1.41 | 0.164 | |
| 1 |
-0.068 | 0.033 | -2.02 | 0.049 | |
| 2 |
-0.055 | 0.037 | -1.49 | 0.143 | |
| 1 |
0.13 | 0.054 | 2.33 | 0.024 | Torvefjord |
| 2 |
-0.041 | 0.030 | -1.39 | 0.171 | Topdalsfjord |
| 3 |
0.036 | 0.017 | 2.13 | 0.038 | Høvåg |
| 4 |
0.045 | 0.014 | 3.30 | 0.002 | Bufjord |
| 5 |
-0.0033 | 0.0097 | -0.34 | 0.735 | Flødevigen |
| 7 |
-0.0047 | 0.0086 | -0.54 | 0.590 | Sandnesfjord |
| 8 |
0.021 | 0.018 | 1.21 | 0.234 | Søndeledfjord |
| 11 |
0.016 | 0.27 | 0.57 | 0.569 | Kilsfjord |
| 16 |
-0.011 | 0.010 | -1.11 | 0.272 | Nøtterø |
| 17 |
0.018 | 0.0061 | 2.99 | 0.004 | Holmestrand area |
| 20 |
-0.0086 | 0.0046 | -1.88 | 0.067 | Hvaler |
Table A2b. Parameter estimates of the ecological parameters obtained from the equating the coefficients in Eqs. 4 and 5. Bold P values represent significance at the 5% level.
| Term | Estimate | SE | Ratio | P value |
|
0.67 | 0.071 | 9.49 | Term |
|
0.54 | 0.11 | 4.77 | Term |
|
0.73 | 0.094 | 7.80 | Term |
| Pollack old | -0.37 | 0.16 | -2.35 | Term |
