38 
Fishery Bulletin 116(1) 
calculation will be explained in the second part of the 
model. The variance is large enough to avoid any con¬ 
straint on parameters while the Gibbs sampling algo¬ 
rithm explores the space of possible values for the pa¬ 
rameters. Note that corresponds with the assumption 
of birth in the jth week of month t— 2. 
The regulation of freshwater in the dam of Alcala 
del Rio has a significant impact on the survival of ju¬ 
veniles at their second (3-4 months old), third (4-5 
months old) and fourth (5-6 months old) stages (Ruiz 
et al., 2009). A lack of discharge, D t , implies low fer¬ 
tilization of the water, and excessive discharges cause 
individuals to leave the estuary and cause a loss of the 
protection it provides. The concentration of anchovy as 
a function of discharges can be described by the stan¬ 
dardized normal density function as (p(ln(D t ) - 4.6052) 
and by taking the maximum value when 7) t =100 hm 3 
(Ruiz et ah, 2009); consequently, the model for second-, 
third-, and fourth-stage juveniles (3-, 4-, and 5-month- 
old fish, respectively, at time t+ 1, t+ 2, and t+ 3, respec¬ 
tively S t+1 (3), 7? t+2 (4) and 7? t+3 (5) is formulated as 
5 t+i - 2 (i)~iV(O), 2000 : )l{0 < 5 t+1 _ 2 (i) < 1)} 
with 
5~7(I) = R t+i _ 3 (/-l)p(p(ln(D,)-4.6052) *=3,4,5, (2) 
where parameter p accounts for the effect of monthly 
discharges D t . The variance chosen (2000 2 ) is large 
enough not to constrain the model while the model 
searches for appropriate parameter values and keeps 
the population within sensible limits. 
The number of individuals in age group 5 (B t+3 ( 5)) 
could be considered as a function of the number of eggs. 
According to Maunder and Deriso (2003), and Man- 
tyniemi et al. (2013), recruitment process error is ac¬ 
counted with a lognormal process: 
ln( R , +3 ) ~ N(\n(B ll3 (5))~ 0.5c^, <t£), 
where <7 K = ln(C^ +1), and C r is a parameter represent¬ 
ing the coefficient of variation of fi t+3 (5). 
Therefore, R t acts as the link between the environ¬ 
mental forcing of anchovy life stages (Ruiz et al., 2009) 
and the dynamics of the stock as modeled in Mantyni- 
emi et al. (2013), where the change in total population 
size equation is defined as 
-^t+i = s t^t + B t+ i, (3) 
where s t = the proportion of the population that sur¬ 
vives to the next time step (see supplemen¬ 
tary text (online only) for the formal definition 
of the survival process error); 
N t = the stock population size (individuals with 
more than 5 months of age) in relation to 
the population size at the beginning of the 
first time step, N*, i.e., N t = 1 and absolute 
population size is calculated as N t N* ; and 
f? t+ i = the number of recruits at time £+1 in rela¬ 
tion to TV* 
Size-structured stock dynamics Population of the stock, 
whose length range is herein defined as 10-22 cm in 
total length (TL) according to the ICES reports (see for 
example ICES 2 ’ 3 ’ 4 ’ 5 ’ 6 ), was assumed to be gathered in 6 
length intervals of 2 cm TL each, J k = [/ k , 7 k+1 ), &=!,..., 
6, where 7 k and 7 k+1 are the size class limits. 
Spawning As an approximation for the three days 
of spawning frequency, and in order to keep compu¬ 
tational time reasonable, the female population is as¬ 
sumed to spawn once each week if suitable conditions 
exist. They are supposed to produce a number of eggs 
computed with the following dot product: 
Eggs t = (l t • N t e), 
where l t and e are column vectors with 6 components. 
The first column vector denotes the size class frequen¬ 
cies after recruitment, allocating the relative popula¬ 
tion TV t in the 6 length intervals defined above. The 
second column vector represents the number of eggs 
per gram spawned by a mature female in each length 
class. Each component of column vector e is calculated 
as the product of the fixed parameters fee, mat k , sexr, 
and w k , defined in Table 1. Note that this number of 
eggs only becomes effective if spawning conditions set¬ 
tled in Equation 1 are satisfied. 
Growth The growth of individuals is assumed to 
take place instantly at the beginning of each month. 
The length of individuals at time t (individuals that 
at time 7-1 were in length class k (L k )), is assumed 
normally distributed with an expected value and stan¬ 
dard deviation calculated as follows: 
JU U =LJl-e s )+ /k, '^ +/k e~ s 14) 
and 
<7 /k = a, V \-e 2g , (5) 
2 ICES (International Council for the Exploration of the 
Sea). 2005. Report of the working group on the assess¬ 
ment of mackerel, horse mackerel, sardine and anchovy 
(WGMHSA), 6-15 September 2005, Vigo, Spain. ICES CM 
2006/ACFM:08, 615 p. [Available from website.] 
3 ICES (International Council for the Exploration of the 
Sea). 2010. Report of the working group on anchovy and 
sardine (WGANSA), 24-28 June 2010, Vigo, Spain. ICES 
CM 2010/ACOM:16, 289 p. [Available from website.] 
4 ICES (International Council for the Exploration of the 
Sea). 2012. Report of the working group on southern horse 
mackerel, anchovy and sardine (WGHANSA), 23-28 June 
2012, Azores (Horta), Portugal. ICES CM 2012/ACOM:16, 
544 p. [Available from website.] 
5 ICES (International Council for the Exploration of the 
Sea). 2014. Report of the working group on southern horse 
mackerel, anchovy and sardine (WGHANSA), 20-25 June 
2014. Copenhagen, Denmark. ICES CM 2014/ACOM:16, 
599 p. [Available from website.] 
6 ICES (International Council for the Exploration of the 
Sea). 2016. Report of the working group on southern 
horse mackerel, anchovy and sardine (WGHANSA), 24-29 
June 2014, Lorient, France. ICES CM 2016/ACOM:17, 588 
p. [Available from website.] 
