Rincon et al.: A Bayesian model with dual-time resolution for estimating abundance of Engraulis encras/colus 
39 
where L, x and O’, = the asymptotic expected value 
and standard deviation, respectively, for the 
length distribution and g is somatic growth 
rate. 
The mean value of L k is an application of the von Ber- 
talanffy growth equation (Quinn and Deriso, 1998) 
starting from /t +' 2 f/k . Each individual has the possibility 
to stay in the same length class or move to a larger 
length class. This transfer is modeled through a tran¬ 
sition matrix G = (£k,i)6x6> where jg k \ denotes the prob¬ 
ability that an individual in class J k grows enough in 
a month to move to class J\. 
The size distribution at month t after growth is giv¬ 
en by l t (G) = Gl t . 
Calculation of o, andg k ] is described in Appendix 2 
of Mantyniemi et al. (2015). 
Mortality and survival A fish in length class k after 
growth has been completed has three possibilities: to 
die naturally, be caught, or to survive ( p l k , Y, k , and p lk , 
respectively), with respective probabilities given by 
Baranov (Baranov, 1976; Quinn and Deriso, 1998) as 
follows: 
(6) 
(7) 
P* = e <F '“' C, 
(8) 
with F and M denoting fishing-induced and natural 
mortality, respectively (where M and F are assumed 
as constant in time for all individuals available to the 
fishery, i.e., bigger than 10 cm TL). Then the propor¬ 
tion of caught fish in length class k from all dead fish 
is computed as 
The number of surviving anchovies at time t deter¬ 
mined by the probability p x = A,k given in Equation 
8, together with the environmentally forced recruits at 
time t+ 1 computed in the first part, constitute the pop¬ 
ulation available to fisheries in the stock for time t +1 
(Eq. 3). The procedure is repeated for each time step. 
Length distribution Accordingly, the length distribu¬ 
tion in the next time step will be defined as the weight¬ 
ed sum of the known length distribution of recruits (be¬ 
tween 10 and 12 cm TL) and the length distribution of 
surviving individuals (l' t S) ), as follows: 
l t+1 =(l-^)l ( t S) + ^*(l,0,0,0,0,0) T , 
N tn ^, + i 
where 1J S1 = p t l ( t G) . 
Observation model 
Catch in numbers Anchovy catches at time t, c t , 
are assumed to follow a beta-binomial distribution, as 
follows: 
c^d-Betabiniqpi^-q^ri^d^ r = 1,...,222, (9) 
where = Xk=iAk’ d t =N ) representing all dead 
individuals (by natural and fishing-induced 
mortality) at time t and jj [ as defined in the 
supplementary text (online only). 
This choice corresponds to an overdispersed alterna¬ 
tive to a binomial distribution (Gelman et al., 2013). 
A binomial distribution is too restrictive, considering 
the schooling and clustering behavior of anchovy (Man¬ 
tyniemi and Romakkaniemi, 2002), and a beta-binomi¬ 
al distribution reflects the fact that all individuals do 
not have the same probability of being captured. Note 
that the expected value of the distribution (q t d t ) cor¬ 
responds to Baranov’s catch equation. 
CPUE The catch per unit of effort at time t, CPUE t 
is applied only for the spawning times, i.e when the 
mean SST of the month is higher than 16°C and it has 
increased at least 1°C from the mean temperature of 
the last month. As extensively discussed in Ruiz et al. 
(2009), this value is reported yearly but most of the 
CPUE signal is produced during the spawning period. 
CPUE t is normally distributed with a mean propor¬ 
tional to the stock size at time t (N t ,) and a very high 
variance that reflects a vague knowledge about this 
variable, as follows: 
CPUE x \Q,N t ~N(QN x ,\ 0 5 ), (10) 
where Q, = the catchability coefficient. 
Acoustic surveys Estimations of the population size 
by means of acoustic surveys are available for two 
specific months: June 1993 and June 2004, as in Ruiz 
et al. (2009). These acoustic data for stock size, A t , t 
- 71,215, provide abundance estimates. They are as¬ 
sumed to follow a normal distribution with unknown 
variance o \, that is transformed to a lognormal distri¬ 
bution as follows: 
\ | A t ,cx A ~ LyV(log(7V^)-0.5cr^,log(l + CL 2 )) t = 71,215, 
where CV = the coefficient of variation of N t . 
Those acoustic surveys are the only “contact” of the 
model with nonfishery estimates of stock size. We de¬ 
cided to let variance be a nonfixed parameter to be 
determined a posteriori by the Gibbs sampler. This 
decision imposes an additional numerical burden on 
the exercise, but not an excessive one owing to the low 
number of estimates, and is justified given the impor¬ 
tant role of this fishery-independent source of data. 
Directed acyclic graph (DAG) 
Figure 1 shows the directed acyclic graph (DAG) for 
the model. The DAG represents random quantities as 
elliptical nodes connected by arrows that indicate con¬ 
ditional dependencies. Data are introduced as rectan¬ 
gles and the arrows could be dotted or solid lines that 
