191
A two-stage biomass dynamic model for Bay of Biscay
anchovy: a Bayesian approach
Leire Ibaibarriaga, Carmen Fernández, Andrés Uriarte, and Beatriz A. Roel
Ibaibarriaga, L., Fernández, C., Uriarte, A., and Roel, B. A. 2008. A two-stage biomass dynamic model for Bay of Biscay anchovy: a Bayesian
approach. – ICES Journal of Marine Science, 65: 191 – 205.
A two-stage biomass-based state-space model with stochastic recruitment processes and deterministic dynamics was developed for
the Bay of Biscay anchovy population. It is fitted in a Bayesian context with posterior computations carried out using Markov
chain Monte Carlo techniques. The model is tested first on a simulated dataset and the effects of different modelling assumptions
and of missing values evaluated. Then, it is applied to a real historical series of commercial catch and survey data from 1987 to
2006. Results are compared with those obtained by the standard assessment model for this stock, integrated catch-at-age analysis
(ICA). From the posterior distribution of biomass in the latest year (2006), the distribution of unexploited biomass in 2007 can be
derived assuming the distribution of recruitment in 2007 to be a mixture of the posterior distributions of past series recruitment.
Hence, the effect of different catch options on future biomass levels can be quantified in probabilistic terms. Finally, directions for
possible further improvements are indicated.
Keywords: anchovy, Bayesian, Markov chain Monte Carlo, mixture distribution, state-space model, stock assessment.
Received 2 March 2007; accepted 23 December 2007.
L. Ibaibarriaga: AZTI-Tecnalia, Txatxarramendi Ugartea z/g, 48395 Sukarrieta, Spain, and Mathematics and Statistics, Lancaster University,
Lancaster, LA1 4YF, UK. C. Fernández: IEO, Centro Oceanográfico de Vigo, Cabo Estai—Canido, Apartado 1552, 36200 Vigo, Spain. A. Uriarte:
AZTI-Tecnalia, Herrera Kaia Portualdea z/g, 20110 Pasaia, Spain. B. A. Roel: Cefas, Pakefield Road, Lowestoft, Suffolk NR33 0HT, UK.
Correspondence to L. Ibaibarriaga: tel: þ34 94 602 94 00; fax: þ34 687 00 06; e-mail: [email protected]
Introduction
Anchovy (Engraulis encrasicolus) in the Bay of Biscay is an important commercial species and is exploited by Spanish and French
fleets (Uriarte et al., 1996). As it is a short-lived species (generally
living up to 3 and at most 5 years), the population level depends
strongly on the incoming year-class strength, which is highly variable and largely dependent on environmental factors (Motos
et al., 1996; Borja et al., 1998; Allain et al., 2003; Fréon et al.,
2005). At present, following a series of consecutive recruitment failures, the population is below the biomass reference points, which
are the limit and precautionary population levels established by
the International Council for the Exploration of the Sea (ICES) to
provide advice (ICES, 2003), following several international agreements (Doulman, 1995; FAO, 1995). In 2005, the population was
estimated to be at its lowest historical level (ICES, 2006), prompting
the closure of the fishery during the second half of the year. In 2006,
although a reduced total allowable catch (TAC) of 5000 t was set
initially, the fishery was closed again in the second half of the year
when the surveys estimated that the stock biomass was still low.
Two research surveys, conducted for the implementation of the
acoustic and the daily egg production method (DEPM), respectively, take place every year during the spawning period (May/
June) and provide indices of stock biomass and the age structure
of the population (ICES, 2006, and references therein). Until
recently, the procedure used at ICES combined these population
indices with commercial catch data to provide a synoptic estimate
of the state of the stock through application of integrated
# 2008
catch-at-age analysis (ICA; Patterson and Melvin, 1996). This
analysis models in terms of numbers-at-age, and assumes a
period in which the fishing mortality is separable into age- and
year-effects. However, because of the short lifespan of the
species, a model based on fully age-structured population
dynamics may not be the most suitable. Here, we investigate
whether simpler models can provide good alternatives to ICA
for the assessment of the stock.
The simplest models of fish population dynamics are biomassbased (Hilborn and Walters, 1992). In their most basic form,
models of this type describe the overall change in population
level from one year to the next without making any specific
assumption on recruitment, growth, or natural mortality.
Another alternative to fully age-structured models are two-stage
models (Collie and Sissenwine, 1983; Mesnil, 2003), which separate the recruits from the rest of the population in terms of
numbers. This allows tracking of the population dynamics via
recruitment, and is simpler and with lower demands on data
than fully age-structured models. For Bay of Biscay anchovy, the
acoustic and DEPM surveys provide biomass indices, and at the
same time, newly recruited fish (age 1) form the larger fraction
of the population and the main source of variability. This leads
naturally to consideration of two-stage biomass-based models,
an approach already used by Roel and Butterworth (2000) for
the South African chokka squid (Loligo vulgaris reynaudii).
State-space models provide a unified framework for describing
animal population dynamics (Buckland et al., 2004; Thomas et al.,
International Council for the Exploration of the Sea. Published by Oxford Journals. All rights reserved.
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192
L. Ibaibarriaga et al.
2005). They consist of two time-series evolving in parallel: the state
process, representing the evolution of population abundances in
time, and the observation process, modelling the relationship
between the observations and population abundance. Both these
processes are often non-linear and non-normal.
A Bayesian approach (Punt and Hilborn, 1997; Gelman et al.,
2004) permits incorporation of expert knowledge through the
prior distribution. It provides a coherent framework for uncertainty propagation, leading from prior knowledge to posterior
knowledge (taking into account what has been learnt from the
observation data) and subsequently into a predictive distribution.
It allows one to make probabilistic predictions naturally under
different management scenarios, and to compute risk in a
decision-making context. Two general computational approaches
are available for Bayesian inference: sequential Monte Carlo
(Doucet et al., 2001) and Markov chain Monte Carlo (MCMC;
Gilks et al., 1996). Both have been used successfully in fisheries
(McAllister and Ianelli, 1997; Meyer and Millar, 1999a, b; Millar
and Meyer, 2000a, b; Trenkel et al., 2000; Newman et al., 2006).
Here, we develop a two-stage biomass-based model for the Bay
of Biscay anchovy. The model can be cast in a state-space framework, though with limited stochasticity in the state equations.
We perform Bayesian inference using MCMC methods. First we
set up the state and observation equations, identify the unknown
variables, define the prior distributions, describe the MCMC
implementation, and demonstrate how to make predictions.
Then we test the performance of the model and methodology on
simulated data, evaluating the effect of different modelling
assumptions as well as of missing values, and finally apply the
method to a time-series of real commercial catch and survey
data for the years 1987–2006. We illustrate how to derive catch
options for 2007, starting from the posterior distribution of
2006 biomass and assuming a mixture distribution for the incoming recruitment based on historical estimates.
Methods
For modelling the dynamics of Bay of Biscay anchovy, we consider two periods within each year. The first begins on 1 January,
when it is assumed that age incrementation occurs and age 1
recruits enter the exploitable population, and runs to the date
when the monitoring research surveys (acoustics and DEPM)
take place. The timing of the surveys varies slightly from year to
year, but for the purposes of this model, it is assumed that both
surveys take place on 15 May. The second period covers the
rest of the year. The fraction of the year corresponding to
the first period is denoted by f (hence, 15 May corresponds to
f ¼ 0.375). The fractions from the beginning of year y to the
time point within each period when commercial catch is
assumed to take place are denoted by h1( y) and h2( y), respectively,
with 0 , h1(y) , f , h2(y) , 1. Figure 1 provides a schematic
display of the timing of events throughout the year.
Let B(s( y), a) and C(s(y), a) denote population biomass and
catch (in tonnes), respectively, of age class a (where class a+ will
denote individuals aged a and older) at time instant s of year y.
Recruitment in year y corresponds to age 1 biomass at the beginning of the year, and it is assumed to be equal in logarithmic scale
to average recruitment (mR) plus some normally distributed
process error with precision (inverse of variance) cR, i.e.
logðRy Þ ¼ logðBð0ðyÞ ; 1ÞÞ NðmR ;
1
Þ:
cR
ð3Þ
Taking into account the rate of biomass decrease, g, and the age 1
catch taken during the first period, age 1 biomass at survey time f is
Bð fðyÞ ; 1Þ ¼ Ry expfgf g Cðh1ðyÞ ; 1Þ expfgð f h1ðyÞ Þg: ð4Þ
Total biomass at survey time is the sum of age 1 biomass and the
biomass surviving from the survey period of the previous year,
taking into account the rate of biomass decrease and the catch
taken:
Bð fðyÞ ; 1þÞ ¼ Bð fðyÞ ; 1Þ þ Bð fðyÞ ; 2þÞ
State equations
The basic processes controlling the dynamics of closed (nonmigrating) and exploited fish populations are growth, recruitment,
natural mortality, and fishing mortality or catch (Hilborn and
Walters, 1992). Our model for the Bay of Biscay anchovy is
based on that developed by Roel and Butterworth (2000) for the
South African chokka squid. The population dynamics are
described in terms of biomass with two distinct age groups,
recruits or fish aged 1 year, and fish that are 2 or more years
old. It is assumed that recruitment and catch occur instantaneously as pulses, whereas growth and natural mortality
operate continuously in time, as described by the following deterministic differential equation:
dBðtÞ ¼ gBðtÞdt;
¼ Bð fðyÞ ; 1Þ þ Bð fðy1Þ ; 1þÞ expfgg
Cðh2ðy1Þ ; 1þÞ expfgð1 h2ðy1Þ þ f Þg
ð5Þ
Cðh1ðyÞ ; 2þÞ expfgð f h1ðyÞ Þg:
ð1Þ
where B(t) denotes the biomass at time t, and g is an instantaneous
rate of biomass decrease accounting for intrinsic rates of growth
(G) and natural mortality (M ), where g ¼ M 2 G, which are
assumed year- and age-invariant. Integrating Equation (1) over
the time interval (t0, t1) without recruitment and catch gives
Bðt1 Þ ¼ Bðt0 Þ expfgðt1 t0 Þg:
ð2Þ
Figure 1. Timing of events taking place throughout the year for the
Bay of Biscay anchovy population.
193
Two-stage biomass dynamic model for Bay of Biscay anchovy
Applying Equations (4) and (5) repeatedly, the total biomass at
survey time in any year y can be expressed as a function of the
initial biomass, defined as the total biomass at the beginning of
the second period of year 0, i.e.
B0 ¼ Bð fð0Þ ; 1þÞ;
ð6Þ
and all previous recruitment and catch values:
Bð fðyÞ ; 1þÞ ¼ B0 expfgyg þ
y
X
Rj expfgð f þy jÞg
j¼1
y
X
Cðh2ð j1Þ ; 1þÞ expfgð f þ1h2ð j1Þ þyjÞg
j¼1
y
X
The Beta distribution allows a variety of shapes for its density
function, making it a flexible tool for modelling proportions.
The Beta distribution in Equation (9) has as mean the age 1
biomass proportion in the population at survey time, P( f( y)),
and as variance (1 + ejsurv)21 P( f(y))(1 2 P( f(y))). This variance
function agrees with the values of the standard errors estimated
for the historical series of age 1 biomass proportion estimates
from the DEPM surveys (standard errors are not available for
the age 1 biomass proportion estimates arising from acoustic
surveys).
For each year and survey, it is assumed that observation
Equations (8) and (9) are independent of each other.
Independence of these equations between years y ¼ 1, . . . , Y and
surveys surv = depm, ac is also assumed.
Parameters and prior distributions
Cðh1ð jÞ ; 1þÞ expfgð f h1ð jÞ þ y jÞg:
j¼1
ð7Þ
We interpret Equation (3) as a stochastic state equation incorporating process error for recruitment, whereas Equations (4) and (7)
are deterministic state equations, expressing age 1 and total
biomass at survey time each year as a function of the unknown
total initial biomass, B0, annual recruitments, Ry, and the rate of
biomass decrease, g.
The unknown parameters are the rate of biomass decrease, g, the
initial total biomass, B0, the average log-recruitment level, mR,
the precision of the normal process error for log-recruitment,
cR, the survey catchability parameters, qdepm and qac, and the parameters defining (or intervening in) the precisions of the observation equations, cdepm, cac, jdepm, and jac. The series of catches
are assumed known. As a Bayesian analysis will be conducted, a
prior distribution on the unknown parameters has to be elicited.
We assume that all are independent a priori, so that the joint
prior distribution is the product of the individual prior distributions, which are chosen to be
Observation equations
The biomass index from a survey is assumed to be proportional to
the true population biomass at the time of the survey, with constant of proportionality q, referred to as the catchability coefficient. The survey is said to provide an absolute biomass index if
q ¼ 1, and a relative index otherwise. In statistical terms, the
most common error distribution assumption for survey indices
is Lognormal (Hilborn and Walters, 1992), which is adequate
for their non-negative real values. For each of the two surveys
(surv=depm, ac) considered, we take as observation equation:
1
logðBsurv ð fðyÞ ; 1þÞÞ Normal logðqsurv ÞþlogðBð fðyÞ ; 1þÞÞ;
;
csurv
ð8Þ
where the quantity on the left hand side is the logarithm of the
survey surv index of total biomass at the time of the survey, f.
The parameters qdepm and qac denote catchability of DEPM and
acoustic surveys, respectively, and are assumed constant through
time. The parameters cdepm and cac are the precisions (inverses
of
variances) of
the Normal distributions (so that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
expð1=csurv Þ 1 for surv = depm, ac are the coefficients of
variation of the indices in the original scale, before taking logs).
In addition to total biomass indices, the DEPM and acoustic
surveys provide estimates of the age structure of the population.
Let P(s(y)) ¼ B(s( y), 1)/B(s(y), 1+) denote the age 1 biomass proportion in the population at time instant s of year y. The observation equation for the age 1 biomass proportion estimated
from each survey, Psurv ( f( y)) for surv = depm, ac, is taken as:
Psurv ð fðyÞ Þ Beta ejsurv Pð fðyÞ Þ; ejsurv ð1 Pð fðyÞ ÞÞ :
ð9Þ
logðqdepm Þ Normal mqdepm ;
logðqac Þ Normal mqac ;
1
cqac
1
!
cqdepm
!
Gamma acdepm ; bcdepm
ac
cdepm
with mean ¼ depm
bcdepm
ac
cac Gamma acac ; bcac
with mean ¼ ac
bcac
!
1
jdepm Normal mjdepm ;
cjdepm
!
1
jac Normal mjac ;
cjac
1
logðB0 Þ Normal mB0 ;
cB0
!
1
mR Normal mmR ;
cmR
ac
cR Gamma acR ; bcR
with mean ¼ R
bcR
!
1
logðgÞ Normal mg ;
:
cg
!
ð10Þ
The hyperparameters of these prior distributions and corresponding prior medians and 95% probability intervals are listed
in Table 1. The prior distributions were centred at values that
194
L. Ibaibarriaga et al.
were considered realistic and chosen to have substantial but not
unreasonably large dispersion. In particular, qdepm and qac have
prior median 1, corresponding to absolute abundance indices.
The Gamma prior distributions of cdepm and cac have median
10, which corresponds to a coefficient of variation of around
32% for the Lognormal observation equations, whereas jdepm
and jac are centred at 5. The prior median value assumed for g
is 0.7, computed as g ¼ M 2 G, based on a natural mortality
rate of M ¼ 1.2, and an average biomass growth rate G ¼ 0.5, estimated from weight-at-age data (ICES, 2006). The prior median of
cR is 1.8, leading to a coefficient of variation for the recruitment
process of around 85%. To analyse the sensitivity of posterior
inference to prior assumptions, two sets of hyperparameter
values are considered for the logarithm of initial biomass,
log (B0), and the average of the log-recruitment series, mR. For
the first set of priors, mB0 is set equal to the midpoint of the
range of observed (for the real dataset) DEPM and acoustic total
biomass indices (in log scale) and mmR is set equal to the midpoint
of the range of the observed (for the real dataset) age 1 biomass
indices (in log scale) projected backwards to the beginning of
the year using the prior median value of g ¼ 0.7. For the second
set of priors, we consider larger means. The variances of log (B0)
and mR are fixed at 1 and 2, respectively, leading to a wide range
of plausible values, as can be seen from the prior credible intervals
listed in Table 1.
Joint posterior distribution
From Bayes’ theorem, the joint posterior probability density function (hereafter pdf) of the unknowns (parameters and states) in a
state-space model is proportional to the product of the pdfs of
observations, states and priors:
pð param, statesjobservÞ
/ pðobservj param, statesÞpðstatesj paramÞpð paramÞ:
ð11Þ
Substituting in Expression (11) the pdf of the observation
equations in (8) and (9), the pdf of the state equations in (3)
and the product of the prior pdfs in (10), the joint posterior pdf is
Y Y
fN ðlogðBsurv ð fðyÞ ; 1þÞÞj logðqsurv Þ
surv y[Isurv;1þ
1
þ log ðBð fðyÞ ; 1þÞÞ;
csurv
Y Y
jsurv
fB ðPsurv ð fðyÞ Þje Pð fðyÞ Þ; ejsurv ð1 Pð fðyÞ ÞÞÞ
surv y[Isurv;prop
1
fN logðRy ÞjmR ;
cR
y¼1
Y
Y
fN ðlogðgÞÞfN ðlogðB0 ÞÞfN ðmR ÞfG ðcR Þ
Y
fN ðlogðqsurv ÞÞfG ðcsurv ÞfN ðjsurv Þ
surv
Y
Y
Y
I½Bð fðyÞ ; aÞ . 0;
ð12Þ
y¼1 a¼1;2þ
where Isurv,1+ and Isurv,prop are vectors indexing the years y for
which there is a total biomass index and an age 1 biomass proportion estimate from the appropriate survey (to exclude
missing values). The first three lines in Equation (12) are the
observation equations, the fourth line contains the state process
for recruitment, and the fifth and sixth lines correspond to the
prior distribution of the parameters. The last line in Equation
(12) is a product of indicator functions, which are equal to 1 if
the restriction within the brackets is fulfilled, and to 0 otherwise.
From Equations (4) and (7), this implies that the rate of
biomass decrease g must be small enough, and initial total
biomass B0 and annual recruitments Ry large enough, to support
the recorded catch levels across years. The restrictions written in
Table 1. Hyper-parameters specifying the two prior distributions and corresponding medians and 95% central credible intervals for survey
catchabilities qsurv, the parameters defining and intervening in the precision of the observation equations csurv, and jsurv for surv¼depm,ac,
the initial biomass B0, the average log-recruitment level mR, the precision of the normal process error for log-recruitment cR, and the rate
of biomass decrease g.
Parameter
Prior 1
Prior 2
Hyper-parameters
Median (95% CI)
Hyper-parameters
Median (95% CI)
m
¼
0
1
(0.1,
16.0)
m
¼
0
1 (0.1, 16.0)
q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . surv
. . . . . surv
c
¼
0.5
c
¼
0.5
q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .q. surv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . surv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
csurv
ac . . . . .¼
0.8
10 (0.2, 65.1)
ac . . . . .¼
0.8
10 (0.2, 65.1)
. . . . . surv
...............................................
. . . . . surv
...............................................
¼
0.05
b
¼
0.05
b
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . surv
. . surv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
jsurv
mj . . . . .¼
5
5 (0.6, 9.4)
mj . . . . .¼
5
5 (0.6, 9.4)
. . . . . surv
...............................................
. . . . . surv
...............................................
cj . . . . .¼
0.2
cj . . . . .¼
0.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . surv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . surv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m
¼
10.5
36
316
(5
116,
257
806)
m
¼
10.9
54 176 (7 631, 384 602)
B0
B
B
0
0
.........................................................
.........................................................
c
¼
1.0
c
¼
1.0
B
B
0
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
mR
mm ¼ 9.8
9.8 (7.0, 12.6)
mm ¼ 10.7
10.7 (7.9, 13.5)
. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
¼
0.5
c
¼
0.5
m
m
R
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
cR
ac ¼ 4
1.8 (0.5, 4. 4)
ac ¼ 4
1.8 (0.5, 4. 4)
. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
¼
2
b
¼
2
b
c
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
mg ¼ log (0.7)
0.7 (0.1, 5.0)
mg ¼ log (0.7)
0.7 (0.1, 5.0)
g
.........................................................
.........................................................
cg ¼ 1
cg ¼ 1
qsurv
195
Two-stage biomass dynamic model for Bay of Biscay anchovy
Equation (12) indirectly imply that we treat the catch as dissagregated by age class in the first period of the year, but without
disaggregation in the second period. If treating the catch as dissagregated by age in the second period also was considered more
suitable, it would be enough to replace the current restrictions
by the stronger ones Pa¼ 1,2+ I[B(1(y), a).0], which correspond
to changing the time point for the restrictions from survey time
f to the end of the year.
From model Equations (3), (4), and (7) –(9), it is clear that
likelihood-based estimators of the catchabilities qdepm and qac
will be positively correlated among themselves, and negatively correlated with those for initial biomass B0, average log-recruitment
mR, and biomass decrease rate g. At the same time, estimators of
the last three parameters will be positively correlated among themselves. Note that if catches were all set to zero, there would be a
complete lack of identifiability in the likelihood, and it would be
impossible to estimate the parameters uniquely, because multiplying B0 and exp(mR) by any positive constant while dividing qdepm
and qac by that same constant would leave the observation
equations invariant. This problem is mitigated by the fact that
there are positive catches, and by the prior distribution in a
Bayesian context (because it centres the values of the unknown
quantities around some levels, although there can be substantial
prior uncertainty). Nevertheless, strong correlations will remain
a posteriori.
The general problem of strong correlation between catchability
and recruitment estimators has already been noted by the ICES
Working Group on Assessment of Mackerel, Horse mackerel,
Sardine, and Anchovy (WGMHSA; see ICES, 2006), whose usual
response has been to consider that the DEPM provides absolute
indices (qdepm ¼ 1; note that this is interpreted as perfect prior
knowledge in a Bayesian context). The reasoning underlying this
procedure is that for the implementation of the DEPM, all biological parameters involved in the estimation process are measured
and are assumed to be estimated without bias from the surveys.
Similarly, the issue of correlation between estimators of the rate
of biomass decrease, g, and average log-recruitment, mR, could
be addressed by assuming that one of them is known without
error. A common assumption in most of the assessment models
is that natural mortality is known. Therefore, at present it would
seem more plausible to fix g than mR.
MCMC implementation
We sample from the joint posterior distribution in Equation (12)
using MCMC techniques. A MCMC algorithm to sample from a
multidimensional distribution (such as the posterior pdf here)
works as follows. It is started from any arbitrary vector. Then,
each component (or subset of components) of the multidimensional vector is sampled in turn, conditioning on the current
values of the other components. Repeating the latter step a large
number of times, the draws can for most purposes be used as if
they were from the joint multidimensional distribution. Thus,
instead of sampling from the original high-dimensional distribution, one samples from lower-dimensional conditional distributions derived from it, a much simpler problem. MCMC is
particularly well suited to the Bayesian setting, where often
one encounters complex high-dimensional posterior distributions
which need to be explored via simulation. The book by
Gilks et al. (1996) provides an accessible introduction to MCMC.
A main issue with MCMC is to decide at which point the chain
has converged (i.e. when the samples are truly representative of
the multidimensional distribution). The draws have to explore
the whole domain, so the chain has to be long enough and the
draws kept must be independent of the starting vector. Start-up
effects and autocorrelation within the chain are usually mitigated
by discarding an initial number of draws (the so-called burn-in
period), and by keeping only one draw every several iterations
(referred to as thinning), respectively. The length of the chain,
the burn-in period, and the thinning interval should all be long
enough to ensure convergence. Cowles and Carlin (1996)
present a comparative review of commonly used convergence
diagnostics. The free software CODA (Convergence Diagnostics
and Output Analysis; Best et al., 1997) provides a suite of functions
to examine MCMC draws. Visual inspection of the traces is also
important to understand the behaviour of the chain (Gilks et al.,
1996). MCMC algorithms should always be run several times
from different starting vectors, checking that posterior results
from the different runs are practically indistinguishable.
BUGS (Bayesian inference Using Gibbs Sampling) is a software that implements Bayesian analysis using MCMC
(Spiegelhalter et al., 1996; Gentleman, 1997; Lunn et al., 2000).
It can be freely downloaded from http://www.mrc-bsu.cam.ac.
uk/bugs/. The sampling method for each conditional distribution is chosen automatically by the program, depending on
conjugacy, concavity, or other properties (Lunn et al., 2000).
BUGS is flexible and easy to use, avoiding the need to write
specific computer code in a low-level language and facilitating
the routine implementation of MCMC methods. Examples of
the use of BUGS in fisheries can be found in Meyer and Millar
(1999b), Mäntyniemi and Romakkaniemi (2002), and
Michielsens et al. (2006).
The BUGS code for our model consists of observation
Equations (8) and (9), Equations (3), (4), and (7) describing the
population dynamics, and the prior distributions in Equation
(10). The restrictions on g, B0, and Ry imposed by the positive
catch values are handled using the following programming trick:
auxiliary Bernoulli variables are introduced, where the probability
of the assumed realized values is 1 if the restrictions are fulfilled,
and 0 otherwise (this is coded using the so-called step function).
From Equation (12), the conditional posterior distributions are
Normal for log(qdepm) and log(qac) and Gamma for cdepm, cac,
and cR, so these parameters are sampled directly. The conditional
posterior distributions of the rest of parameters are non-standard,
and Metropolis –Hastings steps are used.
Prediction
Predicting the level of the population in the year after the last
observation year, Y + 1, can be crucial to provide appropriate management advice. Biomass at the beginning of year Y + 1 is composed of the new recruits entering the population and the
survivors from year Y.
The survivors from year Y are obtained by projecting the posterior distribution of total biomass at survey time, B( f(Y ), 1+),
forward to the beginning of year Y + 1 by the equation
Bð0ðYþ1Þ ; 2þÞ ¼ Bð fðYÞ ; 1þÞ expfgð1 f Þg
Cðh2ðYÞ ; 1þÞ expfgð1 h2ðYÞ Þg;
ð13Þ
where C(h2(Y ), 1+) is the commercial catch in the second period of
year Y. In Equation (13), g and B( f(Y ), 1+) must be drawn from the
posterior distribution, where the draws of B( f(Y ), 1+) are obtained
196
L. Ibaibarriaga et al.
applying Equation (7), using the posterior draws of g, B0, and R1,
. . . , RY.
According to the process model in Equation (3), the predictive
distribution of log-recruitment in year Y + 1 is log (RY+ 1) N(mR,
1/cR), with (mR, cR) drawn from the posterior distribution.
However, in practice, it may be more appealing to assume for
the predictive distribution of recruitment in year Y + 1 a mixture
of the posterior distributions of the past recruitment series:
RYþ1 Y
X
wy py ðjobservÞ;
ð14Þ
y¼1
where py(. j observ) denotes the posterior distribution of Ry, and
PY
wy are the weights of the mixture distribution, such that
y=1
wy ¼ 1. These weights can be set based on information on incoming recruitment or on assumptions regarding future recruitment
scenarios. Otherwise, all Y years could be weighted equally.
Making assumptions on commercial catch in year Y + 1, a predictive distribution of biomass at any time point in that year or at
the beginning of year Y + 2 can be derived. These biomass predictive distributions permit the evaluation of a variety of catch
options under different recruitment scenarios, allowing managers
and stakeholders to establish appropriate exploitation levels for the
population in accordance with specific management targets.
Results
To assess the properties of the model and the methodology, we first
considered simulated data. Only then did we analyse the real
dataset of commercial catch and DEPM and acoustic survey
indices for the years 1987–2006 (Table 2). For both simulated
and real data, the two surveys (DEPM and acoustic) were
assumed to provide estimates for 15 May in all years, so f ¼
0.375. The values of h1( y) and h2(y) were estimated for each year
as the average over the months in the corresponding period of
the annual fractions from the beginning of the year to the
middle of the month, weighted by the total monthly catch.
All posterior results presented were based on MCMC runs with
a burn-in period of 50 000 iterations, followed by 100 000
additional iterations, of which every tenth draw was kept (thinning
to reduce autocorrelation). Each of these runs took 310 min on a
PC with a 1.7 GHz Pentium processor. Runs from different starting vectors led to similar results. Because of the strong correlations
a posteriori, the chains’ mixing was expected to be worse for the
case when both surveys’ catchabilities were treated as unknown,
compared with the case where qdepm ¼ 1 was assumed. This was
confirmed by visual inspection of traces and plots of autocorrelation functions. Further examination of the chains’ behaviour was
carried out with the convergence diagnostics provided in CODA.
In particular, cumulative plots of quantiles showed stability after
the burn-in period, and the retained draws passed the Geweke’s
and the Heidelberger and Welch’s convergence diagnostics.
Despite the generally high autocorrelation within the chains, the
Raftery and Lewis diagnostic confirmed that the burn-in period
and the chain length were sufficient to estimate the median and
the posterior 95% credible intervals of the parameters with the
reported accuracy. Greater accuracy could be obtained by
running longer chains.
Simulated data
A simulated scenario consisting of population biomass, parameter
values, and survey observations was generated trying to emulate
the main features of real data. Commercial catches and their
timing through the years were taken to be equal to those in the
real dataset (Table 2). The annual rate of biomass decrease was
chosen as g ¼ 0.68 (ICES, 2006). Initial biomass (B0), i.e. total
biomass at the beginning of the second period in 1986, was
taken to be 46 000 t. This was derived starting from the real
DEPM age 2+ survey index for 1987 (no acoustic index was available in 1987; see Table 2) and projecting it backwards in time
taking into account g and the catches taken in the intervening
period. The annual recruitments, Ry, were taken to be the values
estimated by the ICA model for the period 1992–2006, because
the ICA estimates are only considered to be reliable for the past
15 years, the years it uses for model fitting. For earlier years, Ry
was derived from the average of the real DEPM and acoustic age
1 indices in that year (Table 2) after accounting for the corresponding survey catchability estimates (qdepm ¼ 1, qac ¼ 1.3,
based on the ICA estimates), and projecting the resulting value
back to the beginning of the year taking into account g and the
age 1 catch taken before the survey of that year. From these
chosen values of g, B0, and Ry and from the catch data, time-series
of age 1 and total biomass were calculated using the deterministic
state Equations (4) and (7). The chosen values of B0 and Ry were
sufficiently large and g sufficiently small to support the level of
catches through the time-series. The parameters intervening in
the precision of the observation equations were chosen as
cdepm ¼ cac ¼ 10, jdepm ¼ jac ¼ 4.72, values derived from the
estimated standard errors of the real DEPM survey indices (currently there is no estimate of the precision of the acoustic
surveys). Simulated DEPM and acoustic total biomass indices
and age 1 proportion estimates were drawn from observation
Equations (8) and (9), conditioning on the “true” (chosen as
explained above) age 1 and total biomass values and the “true”
values of the parameters qdepm, qac, cdepm, cac, jdepm, and jac.
The prior distributions that we specified (summarized in Table
1) do not take into account the restrictions on g, B0, or the annual
recruitments Ry attributable to the positive catches. Truncating the
prior distributions to the permissible area has the effect of shifting
g towards smaller values and B0 and Ry towards larger values.
Figure 2 illustrates this for g, Ry in 2005, which has low catches,
and Ry in 2001, which has large catches, for Priors 1 (upper
panels Figure 2) and 2 (lower panels). The larger the catches, the
larger the effect of the restrictions.
Inference was conducted for the simulated dataset using Priors
1 and 2. Two different settings were explored, depending on
whether the DEPM survey was assumed to provide an absolute
or a relative abundance index (qdepm ¼ 1 or unknown,
respectively).
Table 3 summarizes the posterior distributions of the parameters in the model, and Figure 3 displays the time-series of posterior medians and 95% credible intervals of recruitment and
compares it with the “true” values (black dots) for Prior 1
(dashed lines) and Prior 2 (continuous lines). When catchabilities
are treated as unknown for both surveys, they are underestimated.
Owing to posterior correlations, this is reflected in overestimation
of g, B0, and the recruitments Ry (left panel of Figure 3). Prior 2,
which has larger means for recruitment and initial biomass,
leads to slightly higher recruitments a posteriori. When the
197
Two-stage biomass dynamic model for Bay of Biscay anchovy
Table 2. Historical dataseries of catch data (age 1 and total) by period with the corresponding annual timings (h1( y) and h2( y)) and DEPM
and acoustic age 1 and total biomass indices.
y
h1( y)
h2( y)
C(h1( y), a)
C(h2( y), a)
Bdepm ( f(y), a)
Bac ( f(y), a)
a51
a 5 1+
a 5 1+
a51
a 5 1+
a51
a 5 1+
1986
–
0.596
–
–
5
080
–
–
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1987
0.307
0.569
2
711
8
318
6
543
14
235
29
365
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1988
0.325
0.552
2
602
3
864
10
954
53
087
63
500
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1989
0.282
0.608
1 723
3 876
4 442
7 282
16 720
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1990
0.307
0.581
9
314
10
573
23
574
90
650
97
239
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1991
0.235
0.573
3
903
10
191
8
196
11
271
19
276
28
322
64 000
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1992
0.254
0.593
11
933
16
366
21
026
85
571
90
720
84
439
89
000
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1993
0.237
0.613
6
414
14
177
25
431
–
–
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1994
0.233
0.580
3 795
13 602
20 150
34 674
60 062
–
35 000
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1995
0.292
0.550
5
718
14
550
14
815
42
906
54
700
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1996
0.276
0.573
4 570
9 246
23 833
–
39 545
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1997
0.208
0.637
4
323
7
235
13
256
38
536
51
176
38
498
63 000
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1998
0.199
0.632
5
898
7
988
23
588
80
357
101
976
–
57 000
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
1999
0.230
0.638
2
067
10
895
15
511
–
69
074
–
–
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2000
0.257
0.575
6 298
12 010
24 882
–
44 973
–
98 484
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2001
0.298
0.595
5
481
11
468
28
671
73
198
124
132
90
928
137
200
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2002
0.183
0.614
1
962
7
738
9
754
6
352
30
697
17
723
97
051
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2003
0.300
0.654
625
2 379
8 101
16 575
23 962
15 732
29 430
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2004
0.299
0.588
2
754
4
623
11
657
14
649
19
498
37
124
46 018
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2005
0.114
0.449
102
790
372
2 063
8 002
2 405
15 603
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. .
2006
0.327
–
287
598
–
15 280
21 436
16 686
30 649
Figure 2. Prior distributions of g and recruitment (grey), and the effect of imposing restrictions through catches (black). Catches were low in
2005, and high in 2001. The top panels correspond to Prior 1 and the bottom panels to Prior 2. Vertical dashed lines indicate the medians of
the distributions.
198
L. Ibaibarriaga et al.
Table 3. Results for simulated data without missing values.
Parameter
DEPM relative
DEPM absolute
Prior 1
Prior 2
Prior 1
Prior 2
(1)
0.5
(0.3,
0.9)
0.5
(0.3,
0.7)
1.0
(1.0,
1.0)
1.0 (1.0, 1.0)
q. . .depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.7
(0.4,
1.2)
0.
6
(0.3,
0.9)
1.2
(0.9,
1.5)
1.2
(0.9, 1.5)
q. . .ac. . . .(1.3)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
(10)
7.3
(3.4,
13.4)
7.1
(3.4,
13.1)
7.6
(3.5,
14.3)
7.5
(3.4, 14.1)
depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
(10)
6.6
(3.2,
11.8)
6.5
(3.1,
11.6)
7.2
(3.5,
13.0)
7.2
(3.5,
13.0)
ac
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . depm
(4.72)
5.
9
(4.1,
9.3)
6.1
(4.2,
9.6)
4.7
(3.7,
7.8)
4.7
(3.7,
7.7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . ac
(4.72)
4.3
(3.5,
6.1)
4.3
(3.5,
5.4)
4.9
(3.7,
8.7)
4.9
(3.7,
8.4)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
000)
101
424
(50
057,
214
726)
120
746
(70
215,
245
734)
50
361
(38
556,
66
644)
51
387
(39
448,
68 495)
B. . .0. . .(46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m
(
–)
11.3
(10.6,
12.0)
11.5
(10.9,
12.1)
10.7
(10.3,
11.1)
10.7
(10.3,
11.1)
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
1.7 (0.9, 2.9)
1.7 (0.9, 2.9)
1.6 (0.9, 2.7)
1.6 (0.9, 2.7)
R (–)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
g (0.68)
1.0 (0.7, 1.1)
1.0 (0.8, 1.2)
0.7 (0.6, 0.8)
0.7 (0.6, 0.8)
Posterior median and 95% credible intervals, depending on the prior and on assumptions regarding the catchability of the surveys. “True” values are given in
brackets next to the parameter names.
catchability of the DEPM survey is fixed at its “true” value of
qdepm ¼ 1, posterior inference improves considerably, with posterior distributions that are closer to the “true” values and have
less spread. In this case, inference on recruitment is not dependent
on the prior chosen, and the “true” recruitment values are within
the 95% posterior credible intervals in most of the years (right
panel of Figure 3).
Figure 4 provides some pairwise scatterplots of the posterior
distribution under Prior 1 with DEPM taken as relative. As
expected, these show positive correlation between the catchabilities qdepm and qac, negative correlation between qdepm and the
initial biomass B0, the average log-recruitment level mR, and the
biomass decreasing rate parameter g, although these last three parameters are positively correlated among themselves.
Table 4 presents posterior inference when g is fixed to its “true”
value of 0.68. With DEPM as relative, fixing g improves inference
considerably, as can be seen by comparison with the results in
Table 3. With DEPM as absolute, posterior inference was already
good with unknown g (Table 3), so fixing g only improves the
results marginally. The same conclusion emerges from Figure 5,
which compares, under Prior 1, posterior inference on recruitment
with g estimated (dashed lines) or fixed (continuous lines).
Missing data are common in research surveys at sea. For the real
data provided in Table 2, 30% of the survey indices are missing. To
study the effect of missing data, the model was applied to the
simulated dataset with the same missing values as in the real
data. Figure 6 displays “true” values (black dots), and posterior
medians and 95% credible intervals of recruitment for the
Figure 3. Results for simulated data without missing values, with g estimated: posterior median and 95% credible intervals of recruitment
under the two priors (dashed line and open square for Prior 1 and solid line and cross for Prior 2). The left panel is for the case in which DEPM
is taken as relative and the right panel when it is absolute. The black dots represent the “true” values of recruitment.
199
Two-stage biomass dynamic model for Bay of Biscay anchovy
Figure 4. Results for simulated data without missing values, g estimated, Prior 1, taking DEPM as relative: pairwise scatterplots of the posterior
distribution of some parameters.
simulated dataset without (dashed lines) and with (continuous
lines) missing values, under Prior 1 (with g estimated). Clearly,
the presence of missing values leads to larger uncertainty in the
results. The differences are smaller in the last few years, because
there are no missing values from 2001.
Real data
The real historical series of the Bay of Biscay anchovy abundance
data is presented in Table 2 (ICES, 2006). The model was run
with Priors 1 and 2 and with the DEPM index taken as
relative (qdepm unknown) or absolute (qdepm ¼ 1). Always, g
was treated as unknown and estimated. Posterior medians and
95% credible intervals are displayed in Figure 7 for the annual
recruitments (dashed and continuous lines relate to Priors 1
and 2, respectively), and in Table 5 for parameters. When the
DEPM index is taken as absolute, there is almost no difference
between the posterior distributions obtained from the two
priors, because fixing the value of a catchability parameter gets
around part of the confounding issue. When DEPM is taken as
relative, qdepm is estimated to be well below 1. This has the
effect too of lowering the estimate of qac, while increasing those
of g, B0, and the Ry.
Figure 8 shows the posterior distributions (median and 95%
credible intervals) of the spawning-stock biomass (SSB, defined
Table 4. Results for simulated data without missing values when g is fixed to its “true” value (0.68).
Parameter
DEPM relative
DEPM absolute
Prior 1
Prior 2
Prior 1
Prior 2
(1)
0.9
(0.8,
1.1)
0.9
(0.8,
1.1)
1.0
(1.0,
1.0)
1.0 (1.0, 1.0)
q. . .depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
(1.0,
1.4)
1.2
(1.0,
1.4)
1.2
(1.0,
1.5)
1.2 (1.0, 1.5)
q. . .ac. . . .(1.3)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
(10)
7.6
(3.6,
14.0)
7.6
(3.6,
14.1)
8.0
(3.9,
14.6)
8.1 (3.9, 14.7)
depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
(10)
7.2
(3.5,
13.1)
7.2
(3.6,
13.1)
7.
3
(3.6,
13.0)
7.3
(3.6, 13.1)
ac
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . depm
(4.72)
4.9
(3.8,
7.9)
4.8
(3.7,
8.3)
4.8
(3.7,
8.1)
4.7
(3.7,
7.6)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . ac
(4.72)
4.7
(3.7,
8.2)
4.7
(3.7,
7.8)
4.9
(3.7,
7.9)
4.9
(3.7,
8.1)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(46
000)
48
110
(40
438,
57
778)
48
225
(40
553,
57
949)
48
171
(40
494,
56
632)
48
473
(40
717,
820)
B. . .0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
.............
m
10.7 (10.3, 11.0)
10.7 (10.3, 11.1)
10.7 (10.3, 11.0)
10.7 (10.3, 11.0)
R ( –)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
cR (–)
1.7 (0.9, 2.8)
1.7 (0.9, 2.8)
1.6 (0. 9, 2.7)
1.6 (0.9, 2.7)
Posterior median and 95% credible intervals, depending on the prior and on assumptions regarding the catchability of the surveys. “True” values are given in
brackets next to the parameter names.
200
L. Ibaibarriaga et al.
Figure 5. Results for simulated data without missing values, Prior 1: posterior median and 95% credible intervals of recruitment under two
assumptions of g (dashed line and open square when g is estimated, solid line and cross when g is fixed at its “true” value of 0.68). The left
panel is for the case in which DEPM is taken as relative, and the right panel when it is absolute. The black dots represent the “true” values of
recruitment.
as total biomass at the peak of spawning period, which corresponds to survey time) series divided by the SSB in 1989, which
is currently the year defining the ICES biological reference
points for management (ICES, 2006, and references therein). It
is reassuring that the posterior distributions of these relative
SSBs are rather insensitive to the DEPM catchability assumption
and the priors used.
As already mentioned, the usual practice in ICES WGMHSA is
to fix qdepm ¼ 1 and to estimate just qac. The tendency also is to fix
the natural mortality rate at M ¼ 1.2. Here, we noticed with the
simulated data that it was not possible to estimate both survey
catchabilities and g, so hereafter we focus only on the case where
the DEPM index is considered as absolute (qdepm ¼ 1). On the
other hand, the rate of biomass decrease g ¼ M 2 G is estimated.
Figure 6. Results for simulated data with g estimated, Prior 1: posterior median and 95% credible intervals of recruitment (dashed line and
open square is without missing values, solid line and cross with missing values). The left panel is for the case in which DEPM is taken as relative
and the right panel when it is absolute. The black dots represent the “true” values of recruitment.
201
Two-stage biomass dynamic model for Bay of Biscay anchovy
Figure 7. Results for real data with g estimated: posterior median and 95% credible intervals of recruitment under the two priors (dashed line
and open square for Prior 1, solid line and cross for Prior 2). Left and right panels correspond to DEPM taken as relative and absolute,
respectively.
Only the results for Prior 1 are presented, because this prior is considered to be more in accordance with expert knowledge of the
stock from the historical catch series and survey data. We note,
however, that posterior results are almost identical under both
sets of priors.
Figure 9 compares the prior (continuous lines) and posterior
(dashed lines) densities of the recruitments from 1987 to 2006.
Rather different posterior distributions are obtained for different
years. Recruitment has been low since 2002, and at its lowest in
2005. The posterior medians and 95% credible intervals of SSB
are compared with the estimates resulting from ICA (ICES,
2006) in Figure 10. The results show reasonable agreement,
despite substantial differences in model formulation. There are
missing survey values in a number of years before 2001, widening
the credible intervals. Starting from 2001, however, there have
been no missing values, so taking also the low recruitment
estimates, posterior credible intervals for the final years are
narrower.
Prediction for 2007
The mixture of the posterior distributions of annual recruitments
described in Equation (14) with all the years weighted equally is
shown in Figure 11a. The density has three peaks of decreasing
height, then decreases to zero with a long tail. Low, medium,
and high recruitment regimes might be defined using the local
minima between the peaks, as depicted in Figure 11a. This partition of the mixture distribution of annual recruitments could
be used to define mixture weights for predicting under three
different scenarios: low recruitment scenario—give positive
equal weight to all years y for which the posterior median of
recruitment falls in the leftmost interval, and assign zero weight
to all other years; medium recruitment scenario—give positive
equal weight to all years y for which the posterior median of
recruitment falls in the central interval, and assign zero weight
to all other years; high recruitment scenario—give positive equal
weight to all years y for which the posterior median of recruitment
Table 5. Results for real data.
Parameter
DEPM relative
DEPM absolute
Prior 1
Prior 2
Prior 1
Prior 2
0.4
(0.2,
0.7)
0.3
(0.1,
0.7)
1.0
(1.0,
1.0)
1.0 (1.0, 1.0)
q. . .depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(0.2,
1.0)
0.4
(0.1,
0.9)
1.2
(0.8,
1.7)
1.2 (0.8, 1.7)
q. . .ac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 0.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
5.2
(2.3,
11.0)
5.1
(2.2,
11.1)
6.0
(2.6,
12.2)
5.9
(2.5, 12.1)
depm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
5.9
(2.2,
14.9)
6.1
(2.2,
15.3)
4.9
(1.7,
11.2)
4.9
(1.8, 11.3)
ac
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . depm
3.6 (2.3, 6.8)
3.6 (2.3, 6.1)
4.0 (2.4, 8.9)
4.0 (2.4, 8.0)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j. . ac
3.2
(1.9,
5.2)
3.1
(1.9,
4.9)
3.3
(2.1,
7.5)
3.3 (2.0, 6.7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
(43
889,
239
707)
112
619
(47
660,
354
122)
36
140
(26
232,
53
755)
37
009
(26 644, 57 172)
B. . .0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..91
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m
11.4
(10.
7,
12.3)
11.6
(10.8,
12.7)
10.5
(10.1,
11.0)
10.5
(10.1,
11.0)
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
1.5
(0.8,
2.5)
1.5
(0.8,
2.6)
1.3
(0.7,
2.3)
1.4
(0.7,
2.3)
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
g
0.9 (0.7, 1.1)
1.0 (0.7, 1. 2)
0.6 (0.4, 0.8)
0.6 (0.4, 0.8)
Posterior median and 95% credible intervals depending on the prior and on assumptions regarding the catchability of the surveys.
202
L. Ibaibarriaga et al.
Figure 8. Results for real data with g estimated: posterior median and 95% credible intervals of SSB relative to SSB in the reference year 1989
under the two priors (dashed line and open square for Prior 1, solid line and cross for Prior 2). Left and right panels correspond to DEPM taken
as relative and absolute, respectively.
falls in the rightmost interval, and assign zero weight to all other
years. Figures 11b –d illustrate the distribution of recruitment in
2007 for each of these scenarios.
Two quantities are central to the implementation of the precautionary approach to fisheries management by ICES: Blim, defined
as the SSB level below which the population has a high probability
Figure 9. Results for real data with g estimated, Prior 1, with DEPM taken as absolute: prior (solid line) and posterior (dashed line) densities of
annual recruitments.
203
Two-stage biomass dynamic model for Bay of Biscay anchovy
Figure 10. Results for real data. Comparison of posterior median (thick solid line) and 95% credible intervals (thin solid lines) and ICA point
estimates (dashed line) of SSB. The Bayesian results are with g estimated, Prior 1, and with DEPM taken as absolute.
of collapse, and Bpa (where the subscript pa stands for precautionary approach), defined as the level such that when SSB is estimated
to be above it, the true stock SSB has a low probability of being
below Blim. Starting from the posterior distributions of SSB
(total biomass at survey time) in 2006 and biomass decrease rate
g (under Prior 1 and taking qdepm ¼ 1) and assuming that no
catch is taken in the second period of 2006, Table 6 shows the predictive probability that SSB in 2007 will fall below Blim and Bpa
under different recruitment scenarios and catch options for the
first period of 2007. For the Bay of Biscay anchovy population,
Figure 11. Results for real data with g estimated, Prior 1, with DEPM taken as absolute. The different recruitment scenarios are
(a) undetermined, (b) low, (c) medium, and (d) high, constructed using different weights for the mixture distribution used to predict
recruitment.
204
L. Ibaibarriaga et al.
Table 6. Results for real data.
Catch
C(h1(2007),1+)
Undetermined R
Low R
Medium R
High R
P(B < Blim)
P(B < Bpa)
P(B < Blim)
P(B < Bpa)
P(B < Blim)
P(B < Bpa)
P(B < Blim)
P(B < Bpa)
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..0.10
0.31
0.29
0.85
0.00
0.03
0.00
0.02
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
5. . . .000
0.19
0.37
0.53
0.95
0.00
0.10
0.01
0.03
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
10
000
0.27
0.43
0.76
0.98
0.01
0.24
0.01
0.04
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
15
000
0.33
0.49
0.90
0.99
0.05
0.42
0.02
0.06
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
20 000
0.39
0.55
0.97
1.00
0.15
0.58
0.03
0.09
Predictive probability of the SSB in 2007 falling below the reference points Blim (21 000 t) and Bpa (33 000 t) under different recruitment scenarios and catch
options for the first period of 2007. The timing of the catches h1(2007) and the percentage split of the total catch C(h1(2007),1+) between ages 1 and 2+ are
taken as the average of the values in the historical series. B in the table denotes SSB in 2007.
Blim and Bpa are currently set by ICES at 21 000 and 33 000 t,
respectively. For a low recruitment scenario, even if no catches
are taken in the first period of 2007, the probability of SSB being
below Blim in 2007 is 30%. On the other hand, as for all shortlived species, good recruitment will drive the population back to
within safe biological limits.
Discussion
Here, we developed a two-stage biomass model for the population
dynamics of Bay of Biscay anchovy. Modelling the recruits of the
population separately captures the major changes in the population of this short-lived species, which is mainly dominated by
the incoming recruitment. On the other hand, modelling
biomass instead of numbers of fish is sensible, because the
overall population indices from the research surveys (DEPM and
acoustic) of this population, as well as commercial catch values,
are given in terms of biomass. The performance of the model
was examined with simulated data and found to be satisfactory
on the whole.
Despite the relative simplicity of the model, results for real data
were reasonably consistent with those obtained from ICA. The
over-parameterization of ICA for the assessment of Bay of Biscay
anchovy has been recognized by ICES (ICES, 2006), mainly
because of the disaggregation of the population into six age
classes, when no more than three are usually observed. The
overall agreement indicates that the bulk of the information
regarding the population dynamics is contained in the common
elements of the two models, mainly in the two age-class biomass
indices from the surveys and, to a lesser extent, in the total level
of catches. Little is gained by ICA from including catch-at-age
data into the observation equations, emphasizing the importance
of monitoring the population by direct surveying (ICES, 2006).
The model here has been set in a state-space framework
(though with limited stochasticity in the state equations) that provides a general means of including different sources of uncertainty
in the model, such as a variety of observation processes or uncertainty in the state equations. It has been fitted via Bayesian
methods, using the free software BUGS, which has helped to illustrate one of the common problems in survey-based assessments:
the near impossibility of determining the absolute level of the
population. In this case, this was shown by the high posterior correlations between the surveys’ catchability parameters, annual
recruitments, total initial biomass, and the rate of biomass
decrease. If commercial catches were equal to 0, the overall level
of these parameters could not be determined. Positive commercial
catches and the prior distributions help somewhat to identify this
level. Nonetheless, inference on recruitment levels will be
dependent on the assumptions made on the surveys’ catchabilities
and on the rate of biomass decrease, so the estimated recruitment
values should be considered as relative rather than as absolute
values. When both surveys are considered as relative (unknown
catchabilities), examination of the simulated data shows that posterior results are not reliable. Therefore, if additional external
information or expert knowledge on the level of the population
was available, it would be advisable to use it to construct the
prior distributions for initial biomass and annual recruitments.
Additionally, the restrictions imposed by the observed catches
can have non-negligible effects on the results.
As expected, missing values in the data lead to wider posterior
credible intervals for the recruitments. The Bayesian framework
allows one to quantify the increased uncertainty for missing
values, and potentially to incorporate information from additional
sources via the corresponding prior distribution.
The two-stage biomass model we have developed can be used
not only to assess current population levels, but also to provide
management advice using the predictive distribution of biomass.
Predicting recruitment remains the major hurdle. Until an efficient predictor indicator for recruitment is found, defining the
predictive distribution of recruitment as a mixture of the posterior
distributions of past recruitments provides a flexible way to
examine different recruitment scenarios while incorporating
uncertainty. Starting from the predictive distribution of
unexploited biomass (biomass without catch removals), it is
straightforward for managers to evaluate and to quantify the
effect of different catch options thereafter. The model could also
be used to evaluate harvest control rules for the population, and
to test the effectiveness of management measures such as annual
quotas, two-step quotas, area, and season or other closures.
Further work on this model that is currently being undertaken
includes generalizing the deterministic differential equation
describing biomass dynamics to a stochastic differential equation.
This leads naturally to the full inclusion of process errors into the
state equations (instead of just for recruitment), resulting in more
realistic uncertainty description and risk quantification. However,
as the model becomes more complex and the number of parameters to be estimated increases, additional sources of information, in the form of either data for the observation equations
or expert knowledge for prior specification, will be necessary for
sensible results to be obtained.
Acknowledgements
We thank the participants of the ICES Working Group on
Assessment of Mackerel, Horse Mackerel, Sardine, and Anchovy
for providing the data and encouraging this work, as well as two
Two-stage biomass dynamic model for Bay of Biscay anchovy
anonymous referees for useful comments that helped to improve
the paper. The study was partially supported by the Industry,
Commerce, and Tourism Department and by the Agriculture,
Fisheries, and Food Department of the Basque Government, and
by the Mathematics and Statistics Department at Lancaster
University, UK, where the first author is registered as a PhD
student. The paper is contribution no. 385 from AZTI-Tecnalia
(Marine Research Unit).
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doi:10.1093/icesjms/fsn002