Rincon et al.: A Bayesian model with dual-time resolution for estimating abundance of Engraulis encrasicolus 
37 
Figure 1 
Directed acyclic graph (DAG) of the Bayesian state-space 
model used to incorporate an environmentally forced recruit¬ 
ment of European anchovy (Engraulis encrasicolus) in the 
Gulf of Cadiz, Spain. The DAG represents random quantities 
as elliptical nodes that are connected by arrows indicating 
conditional dependencies. Rectangular nodes represent known 
data. The nodes are separated into 2 groups (represented by 
the large rectangular frames) that determine the life cycle: 
in the bottom frame are the nodes needed to define processes 
that take place before recruitment and in the top frame are 
those used to define processes that take place after recruit¬ 
ment. A description of each node can be found in Table 1. 
can advect eggs and larvae away from favorable 
conditions (Catalan et al., 2006). Hence, the num¬ 
ber of individuals that survive is negatively influ¬ 
enced by the occurrence of easterlies during the 
following two months after spawning. After these 
two months, individuals are better able to control 
their position in the water and are less vulner¬ 
able to currents. 
The model resolves the triggering of spawning 
by SST and egg and larval survival that depend 
on easterlies through weekly time steps over two 
months (see Eq.l). Juvenile and adult stages are 
modeled by using monthly time steps that ad¬ 
equately resolve the dynamics of the population 
(see Ruiz et al. [2009] for a detailed description of 
anchovy life stages). Therefore the model resolu¬ 
tion changes from weekly to monthly time steps 
as the development of anchovy individuals moves 
from egg and larval toward juvenile and recruit 
stages. This coupling between two different time 
resolutions is achieved by converting the eggs 
produced in the first, second, third, and fourth 
week to juveniles at the 9th, 10th, 11th and 12th 
week, respectively. Consequently, B t ( 2), defined 
as the number of individuals that are two months 
old in month t (first juveniles), is modeled by fol¬ 
lowing a truncated normal distribution and using 
data from weekly SST at the times of spawning 
(9, 10, 11 or 12 weeks before) and the weekly ef¬ 
fect of easterlies in the preceding 2 months, as 
follows: 
5,(2) ~N{Bj 2],200000 2 ) I { 5, (2) > 0}, 
where I = an indicator variable; and 5,(2)is de¬ 
fined as follows: 
5j2) = Eggs t _ 3 { M, 1 + M; + M, 3 + M, 4 ), (i) 
where 
M\ = 
. J e - w V- - +<i+HA+■ ) 
if T'_ 2 > 16, T'_ 2 - 
T 4 > 
i t-3 ^ 
•0.25, 
0 
otherwise 
A/, 2 = ' 
f ■- A ( W ,'+»£,+. +W^2 +. ,.+wl 
:> if 7j 2 2 > 16, T t 2 2 
-T x 
1 1-2 
>0.25, 
1 0 
otherwise 
’ - X ( W} +wl l +w*,+. .+H' 4 , +w, 
5j) if Tl 2 >\6Jl 2 
-T l 
1 1-2 
>0.25, 
0 
otherwise 
and 
is above 16°C (Ruiz et ah, 2006, 2009). To resolve the 
weekly heat-triggered egg production we assume that 
it occurs when there is an increment of at least a quar¬ 
ter degree in a week, and therefore consistent with the 
proposal of a degree per month in Ruiz et al. (2009). 
Empirical data from other areas suggest that females 
may spawn on average every three days under suit¬ 
able temperature conditions (Somarakis et ah, 2004). 
After spawning, strong currents created by easterlies 
M 
J -Mn , , 1 +w, 2 +iv l '+H', t 2 +.,.+wl ,+tr, 4 ,) 
I 6 
0 
if 7^ 2 > 16, T* 2 - T'_ 2 > 0.25, 
otherwise 
where T x ] and W t ’ = the temperature and the number 
of days that strong easterlies (>30 
km/h) have blown, respectively, dur¬ 
ing week j in month t. 
Parameter A accounts for the effect of strong winds and 
Eggs t _3 is the number of eggs at time t-3 in relation 
to the population size at the first time step (AT), whose 
