a bayesian model for anchovy (engraulis encrasicolus): the

A Bayesian model for anchovy (Engraulis encrasicolus):the combined forcing of man and environment

JAVIER RUIZ,1,* RAFAEL GONZALEZ-QUIROS,2,3 LAURA PRIETO1 AND GABRIELNAVARRO1

1Departamento de Ecologıa y Gestion Costera, Instituto deCiencias Marinas de Andalucıa, Consejo Superior deInvestigaciones Cientıficas, 11510 Puerto Real, Cadiz, Spain2IFAPA Centro el Toruno, Junta de Andalucıa, C⁄Nac. IV, Km654, 11500 El Puerto de Santa Marıa, Cadiz, Spain3Empresa Publica Desarrollo Agrario y Pesquero, Junta de

Andalucıa, c ⁄ Bergantın 39, 41012 Seville, Spain

ABSTRACT

Fishery collapses frequently result from combinedpressures of the environment and man, which aredifficult to discern because of the complexitiesinvolved and our limited knowledge. Models to re-solve this complexity often become too sophisticated,with too many assumptions and, consequently, withlittle capacity to predict beyond calibration data. Inthis paper we implement a different procedure wherethe model is kept simple and uncertainty accounts forthe equation imperfectness to reproduce ecologicalcomplexity. Human and environmental forcing on ananchovy (Engraulis encrasicolus) stock are simulatedwith only six parameters plus their error terms, andthe uncertainty is computed with Bayesian methods.The simple structure is able to reproduce the majordynamical features of this species in the Gulf ofCadiz, including data on life stages and age structurethat had no contact with the model. This is a dis-tinct performance for a frugal approach working on amid-trophic species and a positive instance whereparsimony can simulate the interaction of man,fish and the environment, provided uncertainty isaccounted for in the process.

Key words: anchovy, anthropogenic forcing, Bayesianmodel, environmental forcing, Gulf of Cadiz, life cycle

INTRODUCTION

Fishery collapses frequently result from the combinedpressure of the environment and man. Although fishpopulations fluctuate in the absence of human activity(Baumgartner et al., 1992), fishing pressure increasesvulnerability to collapse (Cushing, 1996; Bakun andWeeks, 2006), in particular when combined with anadverse physical environment (Ruiz et al., 2007).Short-lived pelagic species such as clupeoids are verysensitive to their physical environment (Nakata et al.,2000; Lloret et al., 2001; Guisande et al., 2004; Erzini,2005; Basilone et al., 2006) as it affects the survival oftheir early life stages and may cause recruitment fail-ures (Cingolani et al., 1996; Dimmlich et al., 2004).Mechanistic theories merge physical (transport, tur-bulence, etc.) and biological (predation, growth, etc.)components to connect environmental fluctuationswith clupeoid recruitment (Parrish et al., 1983;Checkley et al., 1988; Cury and Roy, 1989; Bakun,1996; Bakun and Broad, 2003). Hydrodynamic pro-cesses retain (advect) early stages within (from)favourable conditions. Adequate temperature andfood availability enhance growth rates and reducecharacteristic acute mortality owing to predatorypressure and starvation. Hence, marine areas optimalfor the successful recruitment of clupeoids frequentlyhave slow currents, high primary production and warmwaters (Cole and McGlade, 1998).

The shelf zone between Capes Santa Marıa andTrafalgar in the Gulf of Cadiz embraces thesefavourable features and sustains a significant fishingactivity, anchovy (Engraulis encrasicolus) being one ofthe main resources. Bathymetry and coastline create acyclonic circulation segregated from the energeticcurrents nearby the Strait of Gibraltar (Garcıa-Lafu-ente et al., 2006). Salt marshes and river inputrespectively heat and fertilize the shelf during summer,generating a large pool of warm and chlorophyll-richwater (Navarro and Ruiz, 2006; Garcıa-Lafuente andRuiz, 2007). At that time of year the concentration offish eggs and larvae is very high (Baldo et al., 2006),particularly for anchovy (Ruiz et al., 2006). Besidesfavourable conditions for planktonic stages, the shelf isconnected to the lower reaches of the Guadalquivir

*Correspondence. e-mail: [email protected]

Received 15 April 2008

Revised version accepted 10 November 2008

FISHERIES OCEANOGRAPHY Fish. Oceanogr. 18:1, 62–76, 2009

62 doi:10.1111/j.1365-2419.2008.00497.x � 2009 The Authors.

River, a nursery area for post-larvae of anchovy (Baldoand Drake, 2002; Drake et al., 2007).

Favourable conditions for anchovy recruitment aredistorted under specific meteorological regimes at thesouthern Iberian Peninsula. Shelf currents are highlysensitive to easterly winds, which blow as intensebursts in the area. Persistent easterlies cause the off-shore spilling of waters from the shelf through CapesSanta Marıa and San Vicente (Relvas and Barton,2002). Westward advection of fish larvae under thisregime has been documented (Catalan et al., 2006). Inaddition, latent heat fluxes during easterlies cool shelfwaters and hamper anchovy spawning (Ruiz et al.,2006). Rain at the south of the Iberian Peninsulafertilizes the shelf through freshwater discharges fromthe Guadalquivir River (Navarro and Ruiz, 2006).A dam, 110 km upstream from Guadalquivir mouth,tightly regulates discharges, which are dramaticallyreduced during years of severe drought. Besides low-ering the primary production of the shelf, the agri-culture management of the dam during dry yearsmodifies the seasonal pattern of discharges and nega-tively impacts the anchovy nursery within the estuary(Drake et al., 2002).

Low recruitment under adverse meteorology, in-tense easterlies and low precipitation, is thought todecrease anchovy landings at the Gulf of Cadiz (Ruizet al., 2006). Although similar features are alsoobserved in other anchovy fisheries (Motos et al.,1996), they are seldom included in managementmodels (Freon et al., 2005). Lack of an underlyinghypothesis and neglect of inferences from data areamong the flaws leading to this failure to connectfisheries assessment and ecosystem functioning(Barange, 2001). In addition, uncertainty owing toimperfect model representation of recruitment and itsconnection with the environment is barely accountedfor in these approaches. Uncertainty is a key compo-nent to compute in the implementation of any scienceaimed at supporting decision-making (Lindley, 1971).When applied to fishery research, Bayesian approachescompute this uncertainty as a natural output afterapplication of Bayes’ theorem (Punt and Hilborn,1997). Implementation of state-space schemes of stockdynamics under the Bayesian approach accounts forboth observational and hypothesis uncertainty (Meyerand Millar, 1999; Rivot et al., 2004). As a result, theyelude the dichotomy between mechanistic (e.g., Fas-ham-like ecosystem models) and purely data-driven(e.g., generalized additive models) inference tech-niques in addition to assimilating all available infor-mation to reduce uncertainty. This is particularlyuseful when connecting information sources with

different origin and format as are available for smallpelagic fisheries. For a small pelagic like anchovy, theBayesian approach straightforwardly applies these dif-ferent sources to provide consistent simulation of stockdynamics as well as projections of future scenarios(Ibaibarriaga et al., 2008).

Available knowledge of stock dynamics mayinclude hypotheses concerning recruitment and itsinteraction with the environment but also fishery dataof varying quality and resolution as well as estimates ofthe stock by acoustic or other fishery-independentmethods. The former involves latent variables (sensuCongdon, 2006; i.e., state variables that drive thedynamics of the model but cannot be observed; e.g.,spawning, recruitment or stock size) and accuratemeasurements (e.g., wind, rain, currents, etc.) inter-connected by processes consistent with an underlyinghypothesis where the environment forces populationdynamics. Therefore, error sources come from animperfect hypothesis or its inadequate mathematicalrepresentation. The latter involves errors both in theobservations and in the model connecting these withthe latent variables, e.g., catch per unit effort is lessaccurately measured than water temperature and itsconnection with stock size frequently assumes a con-stant catchability. The Bayesian approach easily andconsistently accommodates this disparity of knowledge(e.g., hypothesis versus data, records of differentaccuracy or frequency, observations connected tolatent variables) when modelling the dynamics ofsmall fish.

This paper implements this approach for a state-space model of anchovy life stages. The model is used toinfer 17 years of stock size in the Gulf of Cadiz. As thisanchovy stock has proven to be sensitive to favourableand adverse environmental conditions (Ruiz et al.,2006), its population dynamics is modelled under theinfluence of the physical environment and connectedto available observations of sea surface temperature,river discharge, wind, catches, catch per unit effort, andacoustic records, as available. The model diagnosesvalues that are consistent with independent observa-tions of anchovy early life stages in the Gulf of Cadiz. Itis also able to explain the main crises historicallyrecorded for this fishery in the region.

MODEL

Process model

The model is applied for the 204 months betweenJanuary 1988 and December 2004. Anchovy abun-dance at various life stages is estimated at this monthly

A Bayesian model for anchovy 63

� 2009 The Authors, Fish. Oceanogr., 18:1, 62–76.

resolution to the maximum age of 24 months.Anchovies of the 2+ year group are very rarelyobserved in the fishery (Anonymous, 2006) and arenot included in the model. Therefore, the modelconsiders 0, 1, 2, …, and 23 monthly age groups. Thefirst age group (0) is considered eggs (H) and thesecond, larvae (L).

Spawning peaks in May–June, and in March–Aprilof the next year most of the population is alreadymature (Millan, 1999). On that account, eggs areintroduced in the model as a proportion of the biomassof individuals beyond age group 9,

PR, provided the

water is warmer than 16�C and the temperature (T)has increased at least 1� in the last month. Thisapproach follows other attempts to model the popula-tion dynamics of Engraulis encrasicolus where egg pro-duction is proportional to reproductive biomass with afactor that accounts for the proportion of females in thepopulation, the biomass invested by them in the pro-duction of eggs, and the proportion of that biomassresulting in healthy eggs (Oguz et al., 2008). Equa-tion 1 summarizes all these factors into a singleparameter (a, see Tables 1 and 2 for model nomen-clature and life-stages respectively) whose a posterioriprobability distribution is obtained after implementa-tion of the Bayesian approach. Limiting egg productionto time periods when a minimum temperature is ob-tained and to the warming phase of the seasonal cyclefollows observations made for this species (Garcıa andPalomera, 1996; Motos et al., 1996) and particularly for

its spawning in the Gulf of Cadiz (Ruiz et al., 2006).The state equation for eggs is:

Ht ¼ aX

Rt if Tt>16 and ðTt � Tt�1Þ>1;

or 0 otherwise:

Ht � NðHt; SHÞ ð1Þ

N stands for the normal distribution of egg abun-dance at month t (Ht) with mean and standard devi-ation (accounting for process error) equal to Ht andSH, respectively.

Table 1. Symbols for the parameters and variables implemented in the model.

Symbol Description Units Prior

a Parameter for egg inputs Month)1 Normalq Parameter for the effect of discharges Month)1 Normals Parameter for the effect of sea temperature �C)1 Normalk Parameter for the effect of easterlies Day)1 Normall Residual mortality after fishing and environmental

(temperature, wind and discharges) lossesMonth)1 Normal

q Catchability Fishing trips NormalSH Standard deviation of egg input process model n GammaSL Standard deviation of larval survival process model n GammaSJ Standard deviation of juvenile survival process model n GammaSYR Standard deviation of young and reproductive survival process model n GammaSCPUE Standard deviation of CPUE observational model Tons ⁄ fishing trip GammaSA Standard deviation of acoustics observational model n GammaCPUE Catch per unit effort Tons ⁄ fishing tripA Acoustic estimate of stock size n

C Catches of anchovy in the Gulf of Cadiz nT Sea surface temperature �CD Monthly discharges from Alcala del Rıo dam hm3

W Days per month with easterlies >30 km h)1 Days

Table 2. Life stages considered in the model.

Age (months) Stage

0–1 (0) H (egg)1–2 (1) L1 (larvae 1st month)2–3 (2) L2 (larvae 2nd month)3–4 (3) J1 (juvenile 1st month)4–5 (4) J2 (juvenile 2nd month)5–6 (5) J3 (juvenile 3rd month)6–7 (6) Y1 (young 1st month)7–8 (7) Y2 (young 2nd month)8–9 (8) Y3 (young 3rd month)9–10 (9) R1 (reproductive 1st month)10–11 (10) R2 (reproductive 2nd month)... ...23–24 (23) R15 (reproductive 15th month)

64 J. Ruiz et al.


Larval survival is controlled by the intensity ofeasterlies, due to its potential to advect the larvaeaway from favourable conditions in the shelf zone(Catalan et al., 2006; Ruiz et al., 2006). This survivalis also considered to improve with increased tempe-ratures because less time is needed to grow and,as a consequence, predation can be avoided more(Regner, 1985). The state equation for the first stage oflarvae is:

L 1t ¼ 1� e�sTt

� �e�kWt Ht�1

L1t � N L1

t ; SL

� �ð2Þ

where Tt and Wt are respectively the average sea surfacetemperature in the shelf and the time (number of days)that strong easterlies (>30 km h)1) have blown duringthe month number t. Parameters s and k modulatethermal and wind effects, respectively. As for a, their aposteriori probability distribution is obtained afterimplementation of the Bayesian approach. L1

t and SL

are the mean and standard deviation (accounting forprocess error) of larval abundance.

In the second larval stage�L2

t

�are individuals of

the month 2 age group with sizes above 3 cm. They arepost-flexion larvae, although still undergoing meta-morphosis (Arias and Drake, 1990) and have a certainswimming capacity but not fully enough to controltheir horizontal position in the shelf, where currentsare of the order of tens of centimetres per second(Sanchez et al., 2006). Following this limited motility,anchovy of this size can be found inside the Guadal-quivir River estuary, although in lower numbers thanthe massive appearance of larger sizes (Drake et al.,2007). Their survival is, therefore, conditioned by amixture of the physical environment on the shelf(wind and temperature) and in the estuary (freshwaterregime).

Survival of anchovy early life stages in the estuary isconnected to the freshwater regulation in the dam ofAlcala del Rıo (Drake et al., 2002). Lack of discharge(Dt) implies lack of fertilization and low production ofthe estuary with less food for anchovy early stages.Excessive discharges, conversely, lower the salinity tounsuitable values for anchovy, and individuals leavethe estuary or die (Fernandez-Delgado et al., 2007).Figure 1 shows the abundance of anchovy early stagesin the estuary versus discharges from Alcala del Rıo.Variability is high because the data mixes seasonswhen early stages are abundant because of the repro-duction cycle (e.g., mid-summer) with periods whenthey are not (e.g., early spring). However, maximumvalues are enveloped within the standardized normal

function F(2(log(Dt) ) 2)), which identifies thehighest abundances achievable at the estuary for agiven discharge. This feature is used to model theinfluence of discharge on the survival of anchovystages. As explained above, for the second larval stagethis influence is mixed with that of the physicalenvironment on the shelf and modelled as:

L2t ¼ q 1� e�sTt

� �e�kLt U 2 log Dt � 2ð Þð ÞL1

t�1

L2t � N L2

t ; SL

� �ð3Þ

where q makes survival proportional to F.The next stages are individuals of age group 3, large

enough to move into the estuary. Anchovy of this andlarger sizes appear in large numbers within the estuaryuntil they are about 7 cm and leave for open waters(Drake et al., 2007). The growth equation of von-Bertalanffy applied to the anchovy of the Gulf of Cadiz(Bellido et al., 2000) assigns this to size five individuals.Accordingly, juvenile stages are modelled as:

J1t ¼ qU 2 log Dt � 2ð Þð Þ L2

t�1

J it ¼ qU 2 log Dt � 2ð Þð Þ Ji�1

t�1 for i ¼ 2 and 3

Jit � N Ji

t; SJ

� �for i ¼ 1; 2 and 3 ð4Þ

When juveniles leave the estuary, they swimenough to avoid adverse environments and survival ofstages beyond 5 is therefore assumed to be indepen-dent of physical conditions. They also leave theestuarine protection from predation and fishery

Discharges at Alcala del Río dam (Hm3)10 100 1000

Ab

un

dan

ce (

#/10

3 m

3 )

0

20

40

60

80

100

120 0.4

0.2

0.3

0.1

Pro

bab

ility

den

sity

(1/

log

(Hm

3 ))

Figure 1. Concentration of anchovy at post-larval stage as afunction of monthly discharge from the Alcala del Rıo dam.The solid line is the function F(2(log(D)-2)), where F is theprobability density function for the standardized normal.Data derived from Drake et al. (2007).



exploitation (Drake et al., 2007). The former loss isincluded as a mortality term represented as parameterl whose a posteriori probability distribution is obtainedafter implementation of the Bayesian approach. Fish-ery losses are obtained from ICES landing data for theregion IX.a south and considered to be proportional tothe relative abundance of that stage to the total ofindividuals larger than the 5 age group(

PBt). As

Equation 5 shows, the resulting algorithm is equiva-lent to considering fishing mortality constant throughages. Although effort is frequently higher for larger(older) individuals of other fisheries, there exists noinformation on this selectivity for anchovy in the Gulfof Cadiz. No clear a priori age-selection pattern seemsto be expected for a fishery where the negligibleabundance of group 2 narrows the window of capturedsizes and the market does not particularly appreciatelarger individuals over smaller ones.

Y1t ¼ 1� lð Þ 1� Ct�1

.XBt�1

� �J3t�1

Yit ¼ 1� lð Þ 1�Ct�1

.XBt�1

� �Yi�1

t�1 for i¼ 2 and 3

R1t ¼ 1� lð Þ 1� Ct�1

.XBt�1

� �Y3

t�1

Rit ¼ 1� lð Þ 1�Ct�1

.XBt�1

� �Ri�1

t�1 for i¼ 2 to 15

Yit � N Yi

t; SYR

� �for i ¼ 1; 2 and 3

Rit � N Ri

t; SYR

� �for i ¼ 1 to 15 ð5Þ

where Yit and Ri

t are respectively the mean of young(6 < age < 10) and reproductive (age > 9) stages ofanchovy outside the estuary. SYR is the standarddeviation accounting for process error.

The normal distributions in Equations 1–5 wererestricted to the positive domain while sampling theposterior probabilities with Gibbs techniques (seebelow). This follows the logical lack of negative valuesfor the abundance of individuals.

Observational model

The model connecting the observations to the latentvariables (life stages of the process model) assumesthat catch per unit effort (CPUE) equals stock size(P

Bt) divided by a catchability coefficient (q):

CPUEt ¼X

Bt

.q

CPUEt � N CPUEt; SCPUE

� �ð6Þ

where SCPUE is the standard deviation accountingfor errors in the observation of CPUE data. Themodel assumes a proportionality between CPUE andstock abundance with a coefficient, q, that does notvary over time. Changes in fleet efficiency ordynamics, species targeting and environment vari-ability among other factors may cause q to vary overthe exploitation history of a fishery (Maunder et al.,2006). In addition, school dynamics of small pelagicsmay result in density-dependent catchabilities (Freonand Misund, 1999) and produce nonlinear connec-tions between CPUE and the stock in a dynamicsconditioned by vessel type (Bertrand et al., 2004;Parente, 2004). These components may have limitedimpact on q variability for a stock such as anchovyin the Gulf of Cadiz that is spatially constrained toa small area and for a CPUE data set (see below)based on a homogeneous fraction of the purse-seinefleet. Still, q has most probably differed from con-stancy in a time span where the fleet has undergonea process of significant modernization and in ascenario where stock collapses and societal conflictsare present. However, lack of detailed informationabout the fleet as well as scarce fishery-independentdata on the stock prevented us from time-resolving qvariability. We rather allow the unresolved dynamicsto be part of the uncertainty accounted for by themodel.

ICES data (Anonymous, 2006) provide acousticestimates of the stock size (A) for the June of years1993 and 2004. These are included in the observa-tional model as:

At � NX

Bt; SA

� �for t ¼ 66 and 198 (June 1993

and 2004, respectively) ð7Þ

where SA is the standard deviation accounting forerrors in the observation of the acoustic estimates.Preliminary tests (not shown) showed little sensitivityof the model output to the inclusion of a ‘catchability’parameter for acoustic data (i.e., to formulate theobservational model for acoustic as Equation 6 ratherthan 7). Therefore, the simpler Equation 7 waspreferred to save one parameter in the modelimplementation.

Directed acyclic graph

A directed acyclic graph (DAG) representation ofthe model is shown in Fig. 2. Following Millar andMeyer (2000), hollow and solid arrows representlogical functions and stochastic dependencies,

66 J. Ruiz et al.


respectively. Data are represented as rectangleswhereas oval nodes represent stochastic variables.Structures of the state-space model that are repeatedfrom month 2 to 204 are enveloped within theheavy-line rectangle. The different stages Ht, Lt

1, Lt2,

Jt1,…correspond to 0, 1, 2, 3, … (see Table 2)

monthly age groups, respectively. Despite its appar-ent complexity, the DAG incorporates the effect ofthe environment on the dynamics of 24 life stages ofanchovy during 204 months and connects it withobservations using only six parameters (a, s, k, q, land q) plus sources of errors (SH, SL,SJ, SYR, SCPUE

and SA).

Prior probability distributions

Conditional independence of state-space modelsdemands that priors for the parameters (a, s, k, q, land q), the errors (SH, SL,SJ, SYR, SCPUE and SA), andfor the first month of the state variables (H1, L1

1, L21, J1

1,…) be declared. It is commonly recommended tomake use of available information to define priors instock assessment (Punt and Hilborn, 1997; Hilbornand Liermann, 1998), although there is frequentlycriticism of altering objective inference.

We follow the rationale of Millar and Meyer (2000)and use non-informative priors for the errors and

t H t

H

1

t L

1 L t 1

H t

tR

2

t L

2 L t

1

1 L

t

1

t J

1 J t

2

1 - t L

2

t J 2 J

t

1

1 J

t

3

t J 3 J

t

2

1 J

t

1

t Y 1 Y

t

3

1 J

t

2

t Y 2 Y

t

1

1 - t Y

3

t Y 3

t Y 2

1 - t Y

1

t R 1 R

t

3

1 Y

t

2

t R 2 R

t

1

1 R

t

15

t R 15 R

t

15

1 R

t

…… …

C t

W t T t

T t-1

D t

tB

3

1 t R

2

1 t R

1

1 t R

3

1 t Y

2

1 t Y

1

1 t Y

3

1 t J

2

1 t J

1

1 t J

2

1 t L

1

1 t L

tCPUE

tCPUE

CPUE S

t A A S

t A

q

H S

L S

J S

YR S

t = 2 to 204

Figure 2. Directed acyclic graph of themodel. Hollow and solid arrows repre-sent logical functions and stochasticdependencies respectively. Data are rep-resented as rectangles and oval nodesrepresent stochastic variables. Structuresof the state-space model that is repeatedfrom month 2 to 204 are envelopedwithin the heavy-line rectangle.



diffuse normal distributions for the first month of thestate variables. As in Millar and Meyer (2000), theinverse of error priors were approximated by Gamma(0.001, 0.001) functions to avoid improper distribu-tions. Normal distributions for the initial priors ofeggs, larvae-juveniles and young-adults are, respec-tively, N(1000,10000), N(100,1000) and N(10,100),where mean and standard deviation are millions ofindividuals.

All parameters were restricted to positive values asthe sign of the process is already made explicitin Equations 1–7. Little a priori information can bederived from the values of environmental parametersin the model (s, k and q) except for the qualitativeknowledge that the environment modulates recruit-ment (Ruiz et al., 2006). This knowledge was trans-lated into priors as normal distributions with standarddeviation equal to a mean (0.5�C)1, 0.15 days)1 and0.1 month)1 for s, k and q, respectively) whosevalue significantly modifies the survival for theaverage value of the environmental variable (e.g., theproduct of the mean k and the monthly average ofeasterlies duration results in a factor of �e)1 for thewind).

Similar diffuse normal distributions were alsogiven to q, l and a. It is not straightforward to assigna prior mean to a parameter such as l, the residualmortality after environmental effects and fisherycatches. It was then decided to explore its domain(from 0 to 1) with three values (0.01, 0.5 and 0.99).A parallel procedure was implemented for a and q.The domain of a was enclosed between zero and a(unrealistically high) top where females spawn every2 days during 1 month with an exceptionally highbatch fecundity of 40 000 eggs (Motos, 1996) thatare all viable as input to the model. Similarly, thedomain of q was generously constructed after con-sidering that its value in Equation 6 can be thought(only in the algorithm world) as the number offishing trips necessary to deplete the stock. Thisproduces a minimum value of 1 (one trip captures thestock) and a maximum of about 107 (each fishing triponly captures 1 kg of anchovy). The singularities ofanchovy in the Gulf of Cadiz, heavy environmentalcontrol and reduced life span (Ruiz et al., 2007),prevented us from defining a prior for q from infor-mation of other fisheries (Hilborn and Liermann,1998). This preliminary exploration of modelbehaviour showed little sensitivity of the parameterposterior to the point of the domain explored in theprior, except for the case of q. When values of qchosen were too small, the prior prevents high valuesof the posterior, whereas avoiding these small ranges

of values results in about the same posterior meanand variance. Given our lack of prior knowledgefor q, we avoided this small end range of prior valuesfor q.

Sampling the posterior probabilities of the parameters

The Bayesian approach estimates the parameters byupdating their a priori probability with the likelihoodof the observations. Bayes’ theorem makes the a pos-teriori probability of the parameters proportional to theproduct of the priors and the likelihood:

POSTERIORI / pða; s; k; q; l; q; SH; SL; SJ; SYR;

SCPUE; SA; H1; L11; L2

1; J11; :::; R15

1 Þ � pðA66; A198;

CPUE1; :::; CPUE204 j a; s; k; q; l; q; SH; SL; SJ;

SYR; SCPUE; SA; H1; L11; L2

1; J11; :::; R15

1 Þ ð8Þ

Explicit formulation of this joint probability densityfunction is extremely laborious because of the need toconstruct the full conditional of a stage-resolvingmodel with 24 states and 204 time steps. We imple-mented the whole Bayesian model in version 1.4 ofWINBUGS (free at http://www.mrc-bsu.cam.ac.uk/bugs/). The software avoids this tedious formulation asit is designed to construct complex posteriors for theuser and to sample them by means of Gibbs techniques(Spiegelhalter et al., 1996).

Slow convergence is a chronic issue in state-spacemodels owing to the high correlation of variables inthe time series (Rivot et al., 2004). We followedKass et al. (1998) to ensure convergence. An initialrun (5000 iterations of burn-in period plus 10 000additional iterations) was used to select over-dispersed parameter values. Parameter values for themean as well as the 5 and 95 percentiles of thisinitial run initiated three sample chains. The con-vergence was diagnosed when Gelman and Rubin(Brooks and Gelman, 1998) statistics reached valuesbelow 1.2 for the parameters at about 100 000iterations.

Data

Three sources of data have been used throughout thispaper: environmental records as covariates, fisheriesinformation for the observational model and field dataof early life stages to validate results.

Sea surface temperature (SST) was derived fromdata obtained by the Advanced Very High ResolutionRadiometer (AVHRR) sensor. The nighttimeAVHRR PATHFINDER SST v5 monthly means with4 · 4 km2 pixel resolution were obtained from NASAPO.DAAC website (http://podaac.jpl.nasa.gov/). Our

68 J. Ruiz et al.


region of interest was extracted from the global imageand arithmetic means were calculated based on allpixels within this region. Discharges from the Alcaladel Rıo dam were provided by Confederacion Hid-rografica del Guadalquivir. They correspond to themonthly accumulated cubic hectometres that are dis-charged from the dam each month. Wind data are themonthly accumulated time (in days) that easterliesfaster than 30 km h)1 have been recorded in themeteorological station of Cadiz. Figure 3 showsthe time series of these environmental data for thetime period covered by the model.

Annual CPUE data from ICES were transformedinto monthly values by assuming the same (ICES)value for all months of that year. This approach waspreferred to more sophisticated methods that incor-porate monthly changes in CPUE. Exploratory ana-lysis of other (non-ICES) CPUE data with monthlyresolution shows signals that are more related to thedynamics of the fishing fleet than to variations of stocksize. The relative importance of anchovy and sardine(Sardina pilchardus; an alternative, lower priceresource) on the total catch of the purse-seine fleetconsistently fluctuates from sardine in winter toanchovy in spring and summer, as does the effortexpended on both species. However, the low fishingeffort expended on anchovy during winter monthsmakes occasional catches result in artifactually highCPUE. ICES acoustic data for the years 1993 and 2004were selected as reliable estimators of stock size.Although more acoustic data exist for other years, theywere not included in the model because they are underreview (Anonymous, 2006).

Average monthly egg and larval abundance at theshelf of the Gulf of Cadiz for the years 2002–2004were obtained from Ruiz et al. (2006), wheremethodological details can be found. Briefly, they

correspond to averages of 26 stations covering thewhole north-eastern section of the Gulf withmonthly sampling with double-oblique hauls usingBongo nets (40-cm mouth diameter and 200-lmmesh size). Abundance of early stages inside theGuadalquivir River estuary for the years 1997–2004were taken from Drake et al. (2007), where meth-odological details can be found. Briefly, they corre-spond to monthly sampling of the last 32 km of theestuary with passive hauls with large nets (2.5-mmouth and 1-mm mesh size) using tidal currents.

RESULTS

Figure 4 shows the a posteriori histogram of the modelerror and parameters, the latter together with theirprior probability density functions. Parameter posteri-ors are less dispersed than priors, evidence of dataadded to the priors and their effects on model para-meters. In addition, posteriors of environmentalparameters, such as the thermal (s) and discharge (q)effect, increase their mean value compared to theirpriors, whereas wind (k) shows a decrease. This indi-cates a greater environmental effect than foreseen apriori for s and q as well as less of an impact (or amodel formulation that does not grasp all the intri-cacies of the process) for k. Low values of the mortalityparameter (l) point at the environment and catches asthe main factors controlling the population. Highmortalities induced by the environment and the fish-ery reduce l to values that are negligible for thefunctioning of the process model.

Posterior probabilities for the different stages of thelife cycle show evident seasonal patterns (Fig. 5), withhigher larvae concentration usually found during thefirst half of the year when sea-water heating is manifest(Fig. 3). In addition, larvae showed conspicuous in-

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 0

100

200

300

1500

3000

Eas

terl

ies

(day

s)

Dis

char

ges

(H

m3 )

5

10

15

(a)

(b)

Tem

per

atu

re (

ºC)

14

16

18

20

22

24

Figure 3. Time series of environmentaldata-forcing processes in the model.(a) Monthly mean sea surface tempera-ture (line) at the shelf of the Gulf ofCadiz and number of days per monththat easterlies faster than 30 km h)1

have been recorded at Cadiz meteoro-logical station (bars). (b) Discharge fromthe Alcala del Rıo dam.



0 200 400 6000.000 0.003 0.006 0.009

0 4e-5 8e-5 0 40 80 120

Acoustic (a)

(b)

CPUE

0 3e-6 6e-6

Eggs

2000 4000 6000

Larvae

Juveniles Young/Adults

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 0 0.1 0.2 0.3

0 0.05 0.1 0.15 0 0.3 0.6 0.9

0 0.03 0.06 0.09

Figure 4. (a) Inverse of error posteriors.(b) Probability density functions (lines)for the priors of the model parametersand a posteriori distribution (verticalbars).

70 J. Ruiz et al.


terannual signals with low values during 1994–1995.During these years, particularly in 1995, the modelpredicts a severe decline of the juvenile populationwith a very high level of certainty. The model predictsa sudden end to this decrease of juvenile abundance in1996, when probability persistently accumulates atpopulation values above the average of the time series.The decline of juveniles in 1995 is preceded by adecline of individuals vulnerable to the fishery (

PB)

which is predicted by the model in the transitionbetween 1994 and 1995. The probability of

PB then

has low population values until 1996, when themodel predicts a recovery which, although not asevident as for juveniles in that particular year, ismanifest later on. Besides this interannual signal, theprobability of

PB shows a clear seasonal signal with

continuous declines after the recruitment occurs earlyin the year.

Seasonal and interannual fluctuations ofP

B canbe better discerned in connection with fishery

information (Fig. 6). Highs and lows ofP

B andcatches evolve consistently. Catch data for 2000 areshaped by a serious societal conflict which dramati-cally reduced the fishing effort that year. The catchper unit effort (CPUE) is less affected by this featureand its evolution is more coherent with

PB (which

it estimates) than catches for year 2000. The firstavailable CPUE data (until the early 1990s) are,nevertheless, very stable compared with catch andP

B values. These data are under consideration forreview by ICES because of inconsistencies with morerecent information obtained from the fishery(Anonymous, 2006).

The consistency of CPUE, catch andP

B evolutionin Fig. 6 partially validates the model outputs. Figure 7adds to this exercise by comparing predicted egg, lar-vae and juvenile numbers with in situ abundance datawhen available. Eggs and larvae abundance are derivedfrom Ruiz et al. (2006), and juvenile information fromDrake et al. (2007). The model reproduces the overallpattern of eggs and larvae abundance, although dis-crepancies also exist in the peak of egg abundanceduring early 2003. The overall pattern of the modelledjuvenile population is also consistent with in situ data,except for years such as 1997 when the timing iscorrect but the magnitude of the modelled peak isgreater than observed.

DISCUSSION

Uncertainty is ubiquitous in our knowledge onclupeoid dynamics. Use of fishery, acoustic or egg-production methods to estimate population size isimpaired by significant flaws (Brehmer et al., 2006;Stratoudakis et al., 2006). The dynamics itself ishighly unstable in connection with environmental andecological processes, and models do not fully resolve it(Bakun and Broad, 2003). Increased model complexityto resolve the dynamics does not decrease the uncer-tainty (Hill et al., 2007). More structure implies moreassumptions and a decreased capacity to predictbeyond calibration data (Anderson, 2005).

Mo

del

led

po

pu

lati

on

(m

illio

ns)

0

800

1600 0

1000

2000 0

2000

4000

1988 1990

1992 1994

1996 1998

2000 2002

2004

Larvae

Juveniles

(c)

(b)

(a)

Figure 5. Time series of probability for the modelled popu-lation abundance of larvae (a), juveniles (b) and stock sizeSB (c).

0

100

200

300

400

500

1988 1990 1992 1994 1996 1998 2000 2002 2004

Mo

del

led

po

pu

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(m

illio

ns)

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CP

UE

(to

ns/

fish

ing

tri

p)

Qu

arte

rly

catc

h (

mill

ion

s)

0.5

1.0

1.5Figure 6. Time series of the modelledpopulation of individuals older than6 months. Thick and thin lines arerespectively the mean and the percen-tiles (5 and 95) of the posterior distri-bution. Bars and dots are respectivelyquarterly catch and CPUE from ICESdata.



The model presented above is parsimonious. Itsstraight connection between the environment andanchovy dynamics implements previous knowledge(Ruiz et al., 2006) with simple equations and fewparameters. These elude the intricacies of biogeo-chemical cycles and trophic interactions that relatephysical forcing to anchovy fluctuations. Thoseintricacies are encapsulated in Equations 1–4. Thesimplification might generate apparent inconsistenciesbetween model output and validation data such asstock overestimation for the year 2000, egg timing for

the year 2003 or the magnitude of juvenile abundancefor the year 1997 (Fig. 7). Considering that observa-tional data are not an exact image of the real worldassists in analysing these inconsistencies. For instance,low catches in 2000 do not result from low abundancebut from societal issues in the fleet. In this case, thediscrepancy between data and model favours modelvalidation, because predictions stay close to CPUE,and the data are not affected.

Including more fishery-independent estimates ofstock size to the observational part of the model would

Lar

vae

(#m

–2)

2002 2003 2004

0

20

40

60

80

100

120

140

0

500

1000

1500

2000

1997 1998 1999 2000 2001 2002 2003 2004

Juve

nile

s (#

10–3

m–3

)

(b)

(c)

0

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800

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on

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0

2000

4000

6000

Eg

gs

(#m

–2)

0

50

100

150

200

0

1000

2000

3000

4000

(a)

Figure 7. Time series of modelled eggs(a), larvae (b) and juveniles (c) versusabundance recorded at the Gulf of Cadizshelf (eggs and larvae) and Guadalquivirestuary (juveniles). Thick and thin linesare respectively the mean and the per-centiles (5 and 95) for the posteriordistribution of variables. Bars are in situabundances.

72 J. Ruiz et al.


undoubtedly ameliorate its capacity to resolve thesediscrepancies and, therefore, the response of fish tothe environment. Lack of sensitivity to the inclusionof a ‘catchability’ coefficient in Equation 7 indicatesthat available acoustics estimates of the stock arestill too few to significantly impact the overallbehaviour of the model. Ibaibarriaga et al. (2008)showed that enough fishery-independent informationmakes Bayesian approaches sensitive to the inclusionof ‘catchability’ coefficients in the observationalmodel. Additional environmental factors could alsofurther improve the fitting of the model to the data.For instance, wind affects both the advection of eggsand larvae (Ruiz et al., 2006) as well as the trophicdynamics of the spawning area (Navarro and Ruiz,2006). The latter results from Ekman pumpingdriving coastal upwelling⁄downwelling, and the sub-sequent enrichment⁄impoverishment of nutrients inthe surface waters of the shelf. Advection and tro-phic effects of wind could be distinctly resolved ifchlorophyll concentrations derived from the remotesensing of ocean colour were included as an addi-tional covariate. This inclusion would allow simu-lation of the trophic environment of larvae, a keycomponent of their survival (Blaxter et al., 1982;Painting et al., 1998). However, sea surface colourfrom SeaWiFS data have only been available sinceSeptember 1997, and including them in the modelimplies a non-homogeneous treatment of environ-mental forcing along the time series. We avoidedthis inconsistency, which nonetheless seems todecrease the potential of the model for resolvingwind effects as the posterior of k renders the

expected influence of this covariate on stockdynamics lower than a priori.

The Bayesian framework consistently transformsinto uncertainty components of the dynamics leftunresolved by model simplification but still maintainssubstantial diagnostic power. The main stock collapsein 1995 is explained with a high level of certainty as afailure of recruitment in juveniles in that year inconnection with severe decreases of dam discharges(Figs 3 and 5). Juvenile failures were already detectedwith low uncertainty the year before, when the stockcollapse was not yet evident. The concentration ofprobability at such low values 1 year ahead of the crisissuggests that future developments of the approachcould result in some prognostic capacity. This poten-tial was further evaluated by implementing the modelwith different runs that progressively incorporate theyears of the time series (year 2–17). In each run, thedistribution of

PB⁄q for the last year of the series was

predicted without including the observational modelfor that particular year. Figure 8 shows the

PB⁄q

predicted with this sequential implementation of themodel. At the beginning of the time series, when littleinformation is available to infer parameter posteriors,the predictions significantly differ from data anduncertainty is large. However, uncertainty decreases asthe model ‘learns’ from the data and the evolution ofthe predicted

PB⁄q is coherent with the tendencies of

CPUE observed later. This coherence must, never-theless, be analysed with caution as the process modelincludes observed catch and environmental data inthe prediction. However, the coherence between thepredicted

PB⁄q and the later observed CPUE supports

1988 1990 1992 1994 1996 1998 2000 2002 2004

0.5

1.0

1.5

CP

UE

(T

on

s / F

ish

ing

tri

p)

0.5

1.0

1.5

q B

Figure 8.P

B ⁄ q predicted (lines) versusCPUE observed (bars) for the sequentialimplementation of the model along thetime series. Thick and thin lines arerespectively the mean and the percen-tiles (5 and 95).

1988 1990 1992 1994 1996 1998 2000 2002 2004

Cat

ches

at

ICE

S IX

.a

(mill

ion

s)

0

100

200

300

400

Figure 9. ICES quarterly catches ofanchovies of year group 0+ and 1+(black and grey bars, respectively).



the use of this approach to analyse the outcome ofpotential future scenarios (meteorological regime andcatches) on the evolution of the stock size.

Moreover, the model predicts the breakaway fromthe crisis with three successive years (1996–1998)when juvenile probability is concentrated at valuesabove standard levels (Fig. 5). Age composition ofcatches depicts a contribution of the 0+ year groupthat is also above the average for that period (Fig. 9),thus indicating the singular abundance of small fish atthat time. It is noteworthy that Equations 1–7 have nocontact with the age or size structure of the catch.Hence, the ability of the model to reproduce thisfeature is process- and not data-driven. Similarly,Equations 6 and 7 do not assimilate data of Fig. 7 andthe capacity of the model to reproduce the pattern ofegg, larvae or juvenile abundance is process-driven.

This model demands few process parameters incomparison with other modelling approaches. Sixparameters (a, l, s, k, q and q) plus their errorsimplement the population dynamics as well as itsconnection with the environment and observationaldata. In comparison, Fasham et al. (1990) andERSEM (Vichi et al., 2007) models demand respec-tively 45 and more than 130 parameters. For anchovyin this region, this simple structure is able to repro-duce the major dynamical features detected by theinformation available on fishery and life stages. Ruizet al. (2007) point out that this dynamics is a case ofBOT-TOP control. Under this type of control thefishing pressure restrains the population from the topand prevents adults from surviving beyond 1 year.Without sustenance of adults, the population reliestotally on recruits to persist. Owing to the vulnera-bility of early stages to ocean processes, the stock isthen totally controlled by environmental fluctuations.This is a neat case of fish dynamics forced by theenvironment and not necessarily easy to extrapolateto other anchovy populations. However, at least forthe Gulf of Cadiz, a simple Bayesian model thatrepresents our lack of knowledge as uncertaintycharacterizes anchovy dynamics well beyond thetrivial fact that fish populations fluctuate.

ACKNOWLEDGEMENTS

This work was funded by the Consejerıa de Agricultura yPesca of Andalucıa as well as by projects P06-RNM-02393 from Junta de Andalucıa and SESAME (FP6-036949). Meteorological data were provided by theAgencia Estatal de Meteorologıa. The manuscript hasbeen greatly improved during the review processthanks to the anonymous contribution of referees.

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a bayesian model for anchovy (engraulis encrasicolus): the

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