ICES CM 2008/J:07

Not to be cited without prior reference to the author

Hierarchical modeling of temperature and habitat effects on carrying capacity and maximum reproductive rate of North Atlantic cod in the Baltic Sea, Gulf of

St. Lawrence and throughout the North Atlantic

Irene Mantzouni and Brian R. MacKenzie

National Institute of Aquatic Resources, DTU Aqua, Section of Population and Ecosystem dynamics, Kavalergården 6, 2920 Charlottenlund, Denmark. e-mail: ima@aqua.dtu.dk

ABSTRACT Stock status evaluation and recovery policies in fisheries management rely largely on reference points derived from single-stock spawner-recruit (SR) models, whose key biological parameters are maximum reproductive rate at low stock size (alpha) and habitat carrying capacity (CC). Recent studies, employing joint or meta-analytic methods, have provided evidence that these ecological parameters, or the factors controlling them, are sensitive to environmental effects. The issue is of critical importance given global ocean warming projections; better understanding of environmental impacts on key population parameters, and hence SR models will be needed. The objective of our study is to extend the commonly used Ricker and Beverton-Holt SR models to account for (i) dynamics and variability in all north Atlantic cod stocks , thus borrowing strength from each and (ii) possible ecosystem (temperature and habitat size) effects on the model parameters (alpha and CC). In order to model the variability in SR parameters across stocks and improve estimation accuracy, the models were developed employing hierarchical methods. These methods allow stock specific estimates to be derived borrowing strength from the full dataset and also the incorporation of stock-level models on the parameters. Two different and complementary hierarchical techniques were employed: mixed and Bayesian models. Results show a significant dome shaped relationship of temperature on both cod CC, and alpha, in the north. Atlantic, and that the impacts vary geographically. These patterns may have implications for ecosystem approaches to management of cod populations in the changing temperature situations expected in the 21st century. Keywords: Gadus morhua, N. Atlantic, carrying capacity, maximum reproductive rate, temperature, hierarchical modeling, spawner-recruit relationships

INTRODUCTION

Stock status evaluation and recovery policies in fisheries management rely largely on

biological reference points estimated from spawner-recruit (SR) models (Hilborn &

Walters 1992, Mace 1994, Myers et al. 1994, Myers & Mertz 1998a, Myers et al.

1999, Quinn & Deriso 1999); the maximum reproductive rate at low stock size

(alpha) and the habitat carrying capacity (CC). Recent studies, involving N. Atlantic

cod (Gadus morhua), have provided evidence that these biologically and ecologically

meaningful parameters, or the factors controlling them, are sensitive to environmental

effects (e.g., Brander 1995, Myers et al. 1997, Planque & Frédou 1999, Myers et al.

2001, Myers et al. 2002, Stige et al. 2006), a fact with considerable implications for

cod dynamics and fisheries.

Alpha, as a reference point, is of central importance to stock dynamics for estimating

overfishing and other anthropogenic mortality factors limits (Mace 1994, Myers &

Mertz 1998a), population viability analysis (Lande et al. 1997) and, most notably, the

intrinsic rate of natural increase rm (Myers et al. 1997). In the absence of depensation,

it can be interpreted as the maximum annual rate by which spawners produce

replacement recruits and thus, it is related to the density-independent mortality from

the egg to the recruitment stage. Meta-analysis has shown that alpha varies over a

rather narrow range across and among species (Myers et al. 1999). On the other hand,

there is extensive evidence that key factors controlling successful cod recruitment,

such as spawning time, reproductive capacity, growth rate of larvae and adults,

weight-at-age, distribution, prey abundance and feeding, are controlled by

environmental variables, and mainly temperature (see Sundby 2000 for a review) and

NAO (North Atlantic Oscillation), an index also related to SST (see Stige et al. 2006

for relevant studies summary). Thermal effects are particularly strong, and opposite,

at the lower (positive) and higher (negative) limits of cod temperature range and

recruitment-temperature correlations remain significant at these extremes after re-

testing with additional data (Myers 1998, Planque & Frédou 1999, Myers 2002).

Ecosystem CC is crucial to fisheries management for estimating maximum

sustainable yield (Myers et al. 2001), and determines the effectiveness of stock

rebuilding strategies (MacKenzie et al. 2003). As a dynamic quantity, it depends on

several ecological factors and is therefore expected to change in time in response to

biotic or abiotic fluctuations (Myers et al. 2001, Del Monte-Luna et al. 2004).

MacKenzie et al. (2003), employing a synthetic approach based on the analysis of

spawner-recruit data within and across stocks and species, found that ecosystems vary

in terms of recruitment productivity, standardized for differences in spawner

abundance. The variability is particularly remarkable across the N Atlantic cod range,

suggesting that stocks differ in recruitment CC and survival. Also, there is evidence

that the spatial variability in cod asymptotic recruitment (an index of cod CC) can be

partly explained by differences in bottom mean annual temperature among areas

(Myers et al. 2001, Myers 2001). The effect, when incorporated as a covariate in the

mixed Beverton-Holt model, appears to be negative especially for the Northeast

Atlantic stocks, although its significance is not readily interpretable, since the analysis

was performed after examining the results (Myers et al. 2001).

Understanding environmental effects on cod dynamics is of particular importance in

the light of the present, depleted, state of most N. Atlantic cod stocks (Brander 2007a,

Myers et al. 1996, ICES 2005b), and the accumulating indications that adverse

climatic conditions, on the top of overfishing, have leaded to (and in many cases

sustain) this situation (Lilly et al. 2008, Brander 2007b, Shelton et al. 2006,

Drinkwater 2005, Rose 2004, Stenseth et al. 2004, Brander 1995). Furthermore, the

issue becomes critical under the global ocean warming scenarios (IPCC 2001, ICES

2006) which are “moving the goalposts of fisheries management” (Brander 2006) and

call for better understanding of the environmental impacts on key fisheries

parameters, and hence on SR models (Sakuramoto 2005).

The advantages and the potentials of joint /comparative studies in fisheries science

have long been advocated (e.g., Pauly 1980, Brander 1995) and have provided

fundamental insights on spawner-recruit dynamics (Ricker 1954, Beverton & Holt

1959, Cushing 1971). Theoretical and technological advances in the recent years have

allowed the more widespread use of synthetic approaches, such as meta-analysis,

mixed (variance-components) and Bayesian models, especially in stock assessments

(e.g., Punt & Hilborn 1997) and in studies on SR dynamics (see Myers & Mertz

1998b, Myers 2001, Myers 2002 for reviews). These approaches have revealed that

stocks within species, or related species with similar life-histories, share common

population dynamics patterns and respond to environmental effects in comparable

ways (Brander 2000, Myers et al. 2002, MacKenzie et al. 2003). Consequently, it is

possible to “borrow strength” (Snijders & Bosker 1999, Myers et al. 2001) or “stand

on the shoulders of giants” (Hilborn & Liermann 1998) by combining data across

stocks. Such approaches can yield superior parameter estimates, thereby reducing

uncertainty for management reference points, allow inference at a higher level and

improve estimation for stocks with limited data (Myers et al. 2001).

Our present study aims at extending the commonly used Ricker (1954) and Beverton

& Holt (1957) SR models to account for (i) all N. Atlantic cod stocks data, thus

borrowing strength for each through meta-analytic approaches and (ii) the possible

ecosystem (temperature and habitat size) effects on the model parameters, alpha and

CC.

DATA

We have compiled a database including population and temperature time-series for

the 21 major cod stocks in N. Atlantic (Table 1). Population data include time-series

of spawner stock biomass (S) and recruitment (R). These numbers are estimated from

sequential population analysis (SPA) standardized, in most cases, with fisheries

independent (such as research trawl survey) data and were extracted from published

stock assessment reports (Table 1).

Following the standardization method used by Myers and colleagues in various meta-

analytic studies on cod stocks (e.g. Myers et al. 1999, Myers et al. 2001), we

standardized recruitment data by multiplying them with SPRF=0 (spawners produced

per recruit in absence of fishing mortality). These parameters are calculated based on

natural mortality, weight at age and age at maturity (Mace 1994, Myers & Mertz

1998). For most of the east and west N. Atlantic cod stocks, the parameter is

estimated by Goodwin et al. (2006) and Shelton et al. (2006), respectively.

Regarding the temperature time series, we used estimates of temperature at the surface

layer (0-100m) during the spawning season, i.e., spring. The time-series for the NE

Atlantic stocks were provided by the ICES (International Council for the Exploration

of the Sea) data centre (http://www.ices.dk/datacentre/). For the NW Atlantic,

temperature time series were extracted from the DFO (Fisheries and Oceans Canada)

oceanographic databases (Gregory 2004a, b, c). Some special considerations apply to

3 of the areas (Myers et al. 2001); for eastern Baltic cod stock (cod-2532) we used

temperature estimates in ICES subdivisions 25-29, since 30-32 are unfavorable for

cod due to low salinity (Nissling & Westin 1991). For Barents Sea cod (cod-arct),

given that stock distribution can be limited by cold waters (Ottersen et al. 1998), we

estimated temperature in the area below 78oN. Low water temperature can also limit

the distribution of Icelandic cod (cod-iceg) to the southern part (Brander 2005).

Therefore we used temperature estimates applying to the region south of 62 oN in

ICES subdivision Va.

We have also used habitat size estimates in order to standardize carrying capacity

models for differences in region size among stocks. As in previous meta-analytic

studies on cod (MacKenzie et al. 2003, Myers et al. 2001), we assumed that habitat is

limiting at juvenile stage (Myers & Cadigan 1993). Hence we have used the area of

ocean bottom between 40-300m as representative of stock habitat, thereby excluding

the upper pelagic layer.

METHODS

Hierarchical models

Hierarchical or multi-level modeling is a rigorous probabilistic framework offering

two mutually implicative advantages: (a) the explicit incorporation and thus, isolation

of uncertainty due to observation and systematic model error (Hilborn & Walters

1992) and (b) the combination of data across various independent sources (Gelman et

al. 1995, Berliner 1996, Hilborn & Liermann 1998, Wikle 2003, Gelman & Hill

2007). The implementation is based on the model decomposition into three stages, or

levels, according to the probability theory (Berliner 1996, Wikle 2003, Clark 2007,

McCarthy 2007). On the first level, the data model describes the probability of the

data given the explanatory variables and the parameters describing the corresponding

effects (i.e., the functional form of the SR model). Secondly, the process model

describes the variation of the data model parameters. Its importance is twofold, since

at this level we define (i) the functional form of the model by incorporating ecosystem

factors that are affecting the parameters and (ii) the distribution of the SR model

parameters across the cod stocks. The latter can be extended to account for the

mechanisms generating the among stocks differences, thus this stage is also referred

to as the stock-level model. The third level is the parameter model and it concerns the

hyper-parameters, which are used to define the probability distributions of the

parameters in the previous stages. These last two levels are based on the assumption

that certain SR model parameters are connected across stocks and hence, lie in the

core of the hierarchical meta-analytic inference. The common probability distribution

and the process generating these parameters, or describing the differences among

them, both described by the hyper-parameters of the third stage, form the interface for

the combination of the individual datasets and thus, for exchange of estimation

strength across stocks (Gelman et al. 1995).

Due to their probabilistic theoretical background, the majority of hierarchical

applications have been mainly implemented under the Bayesian paradigm (Gelman et

al. 1995, Clark 2007). As explained in the next sections, hierarchical Bayesian

inference averages over the above levels of uncertainty and variability by means of

the likelihood (data model), the priors (process or stock-level models) and the

hyperpriors (parameter model), to produce the posterior distribution of the SR model

parameters. Another popular framework in this context is mixed (or variance

components) modeling (Searle et al. 1992, Snijders & Bosker 1999, Demidenko 2004,

West et al. 2006, Clark 2007, Gelman & Hill 2007). Mixed models can be regarded as

a combination of Bayesian and frequentist approaches (Demidenko 2004), with results

parallel to the empirical Bayesian inference (Robinson 1991, Snijders & Bosker

1999).

Both approaches have certain advantages regarding multi-level modeling

implementation (Clark 2007, Gelman & Hill 2007). Mixed models are usually

quickly and easily fit but estimation may fail under certain circumstances. Bayesian

hierarchical models, on the other hand, are more flexible allowing estimation also for

more complex model structures, as well as inference on the variance components

uncertainty. In our study, we employ the former approach, implemented in the R

statistical platform by the nlme library (Pinheiro & Bates 2000), to develop the

hierarchical Ricker SR model. Regarding the Beverton-Holt model, the non-linear

mixed models approach does not have a closed-form solution, leading to more

computationally intensive estimation algorithms and to less reliable inference results

(Pinheiro & Bates 2000). Therefore, the hierarchical model was developed in the

Bayesian framework and was simulated in BUGS (Lunn et al. 2000). The Bayesian

approach was also employed to explore more complex formulations of the Ricker

model. The multi-level implementation of the SR models, under the mixed and the

fully Bayesian framework, is described in the following sections.

SR models

Ricker model

Ricker SR model is one of the standard models used in fisheries science (Hilborn &

Walters 1992):

t tBSRICt tR A S e e ε−= (1)

where t denotes the year, R is the recruitment, standardized as previously described,

and S is the spawner biomass. Parameter RICA (Ricker model Alpha) represents the

slope of the curve near the origin and is thus related to the stock productivity and the

density independent survival rate. In the present case, where we are using

standardized recruitment, it can be interpreted as the average rate at which

replacement spawners are produced per spawner over its lifetime at low spawner

abundance and in the absence of fishing mortality. This rate is standardized across

stocks for differences in weight, maturity and natural mortality at age that are

incorporated in the standardization parameter, SPRF=0. Thus, as it will be shown in

detail below, Alpha can depend on a number of time varying factors, like temperature

in the present study, accounting for effects on pre-recruit stages survival rates and also

for potential fluctuations in the SPRF=0 components.

Parameter B (beta) is related to the carrying capacity, since 1/B equals to the spawner

biomass when recruitment reaches the maximum and –B represents the density

(stock)-dependent mortality dominating after this point. The parameter, thus, depends

on the habitat size, which differs across cod areas and can cause, and explain, at least

of the across stocks variability in beta. Moreover, the available habitat can also be

influenced, and thus vary in time within a given stock, by ecosystem variables (Kell et

al. 2005). The possibility for within stocks temperature effects on the parameter, as

well as the among stocks relationship between beta and habitat size, are investigated

as it will be shown in the next sections.

Apart from beta, the CC for a given stock i can be quantified using two different

definitions. CCmax is the maximum number of recruits which can be produced by the

maximum number of spawners sustained by the ecosystem (1/beta):

CCmaxi = Alphai*betai/e

CCeq is quantifying S (and R) at equilibrium (i.e., when R=S) and thus it is a useful

parameter for management, representing the minimum spawner biomass required to

produce replacement recruitment:

CCRIC,eqi = log(Alphai)*betai

The Ricker model was linearized by natural log transformation and assuming

lognormal errors:

log( / ) log RICt t t tR S A BS ε= − +

To simplify notation the Ricker model for stock i is written as:

RIC RICit i i it ity xα β ε= + + (2)

where log( / )it it ity R S= , , log ,RIC RIC RICit it i i i ix S A Bα β= = = − and i denotes the stock.

For simplicity, we will denote RICiα as alpha, understanding that alpha=log(Alpha).

The errors in this and in the following models are assumed to be stock specific.

Beverton-Holt model

The Beverton-Holt (BH) model is also broadly used for the study of SR dynamics:

1 /t

BHt

tt

A SR eS K

ε=+

(3)

Ricker and Beverton-Holt SR models display similar behaviour at the limit of low

SSB (i.e., compensation) and therefore parameter Alpha (denoted as BHA for

Beverton-Holt model) has common interpretation, and estimation, in both models

(Myers et al. 1999). Parameter K has the same dimensions as SSB and can be

interpreted as the “threshold biomass” resulting in half of the maximum recruitment,

which equals to AK. The model assumptions regarding dynamics at higher spawner

abundance differ from Ricker model. Therefore, when SSB exceeds 2K recruitment

becomes independent of the stock biomass and is instead regulated by density

(cohort)-dependence effects due to competition among the early life stages for

limiting resources, like food and settlement habitat (Hilborn & Walters 1992).

The BH model cannot be linearized and, as discussed above, hierarchical development

was implemented employing the Bayesian framework. In this context, a useful

reformulation is the following:

log( ) log( )BH BH BHit i i i it ity xα β β ε= + − + + (4)

where log( / )it it ity R S= , , log ,BH BH BH BHit it i i i ix S A Kα β= = = and i denotes the stock.

In this form, the model is linear to BHiα , which has the same interpretation as RIC

iα of

the Ricker model in [2] and will also be denoted as alpha, and BHiβ is comparable

to RICiβ , since it corresponds to the SSB resulting in half of the maximum recruitment,

given by exp( )*BH BHi iα β . We can also estimate CC at equilibrium, CCeq, for the

Beverton-Holt model as:

CCBHeqi = exp( )( 1)BH BH

i iα β −

Recruitment CC indices, under the two SR models, are estimated as a function of

alpha and beta, and thus depend on both density independent and dependent processes

represented by each parameter, respectively.

Multi-level SR models

This section outlines the conceptual and methodological framework under both the

mixed and the Bayesian modeling approaches. We use the Ricker model to present

this part, simplifying notation by dropping the SR model identifiers, i.e., using iα and

iβ in place of RICiα and RIC

iβ , respectively. However the same approaches, apply to

the Beverton-Holt model and its parameters BHiα and BH

iβ . Estimation for this model

is described in the “Bayesian inference” section.

A convenient way to conceptualize the multi-level SR framework is by starting with a

simple regression model fit to all stocks (Gelman & Hill 2007). This type of model is

referred to as a complete-pooling model and is based on the “extreme” assumption

that each parameter is fixed to a certain value, common across stocks. For Ricker

model, the complete-pooling model can be simply written as:

it it ity xα β ε= + +

with 2N(0, )iid

it y ~ε σ . The previous model can be written in another generalized way as

in [2]. In this form it corresponds to the no-pooling model, a classic regression model

which can be estimated for each stock separately, using indicators and assuming that

parameters are completely independent across stocks. In the Bayesian framework, the

data-level models represent the likelihood, describing the distribution of the data

given the model. In other words, they convey information about the range of

parameter values that are most consistent (likely) with the data of each stock (Gelman

& Hill 2007).

Hierarchical modeling is a compromise between these two extremes, imposing a “soft

constraint” on the stock-specific parameters by assuming that they are derived from a

common probability distribution (Gelman & Hill 2007). At this step, we extend the

data-level model by introducing the next level of complexity, i.e., the process-level

models, specifying the distribution of the data model parameters across stocks. These

across stocks distributions of the process (i.e., the Ricker SR model) parameters

represent the stock-level models and are thus estimated from the full dataset, so that

strength is borrowed. Usually, Gaussian distributions are assumed:

2~ N( , )i α αα μ σ (5.1) and

2~ N( , )i β ββ μ σ (5.2)

where αμ and 2ασ (or βμ and 2

βσ ) is the mean and the variance of the parameter alpha

(or beta) distribution, respectively. The distribution means, αμ and βμ , are called

fixed-effects in the mixed models terminology and represent the average value of the

corresponding parameter across all stocks, while 2ασ and 2

βσ are the variance

components (Searle et al. 1992, Pinheiro & Bates 2000). The previous stock-level

models [5.1] and [5.2] can also be written as:

i iaαα μ= + (6.1) and

i ibββ μ= + (6.2)

where 2~ N(0, )ia ασ and 2~ N(0, )ib βσ are the stock-level models errors, called

random effects, and represent the deviation of stock i parameter, iα or iβ , from the

corresponding across-stocks mean. Jointly, they are represented by a multivariate

distribution:

2

2~ N ,i a a b

i a b b

α

β

μα σ ρσ σμβ ρσ σ σ

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

where ρ is the correlation between the random effects. In the Bayesian context, the

stock-level models represent the priors, which usually convey the existing (occurring

before the data under study are seen) knowledge about the parameters and are thus

used to update or weight the likelihood. In the present context, however, whereby

priors are common to all stocks, they are used to incorporate the information on the

distribution of the parameters across stocks, which is relevant to obtain the individual

estimates.

Consequently, the Ricker data-level model incorporating the stock-level models is:

( )it i i it ity a b xα βμ μ ε= + + + + (7)

It is assumed that the residuals 2N(0, )it yi ~ε σ associated with each stock are

independent and that the residuals and the random effects are independent of each

other. The parameter 2yiσ is also a variance component (Searle et al. 1992).

The stock-level models are partially pooling the parameters towards the mean of the

distribution, and hence the estimates are called shrinkage estimates (Gelman & Hill

2007). For instance, the estimate of iα can be expressed as a weighted average of the

no pooling, stock specific model i iy xβ− , corresponding to RICiα in [2], and its mean

across stocks (the fixed effect) αμ :

2 2

2 2 2 2

1

ˆ ( )1 1

i

yi ai i i

i i

yi a yi a

n

y xn n α

σ σα β μ

σ σ σ σ

≈ − ++ +

(8)

where 2yiσ is the stock specific model variance and in is the number of observations for

stock i. The pooling is stronger when 2aσ is small and for stocks with fewer

observations and higher variability ( 2yiσ ).

Incorporating ecosystem effects

Having introduced the hierarchical model as composed of uncertainty/variation levels;

within (data-level) and across (process and parameter-level) stocks, it is

straightforward to extend the previous model in order to incorporate ecosystem (log

transformed habitat size, H and temperature, T) effects on the parameters. As it will be

shown, the stock-level models [6.1] and [6.2], describing the across stocks variation in

the parameters, are appropriately modified to account for the effects of these

predictors. Thus, stock- specific parameters are standardized for differences in known

characteristics of the ecosystems occupied by the cod stocks.

(i) Temperature

Initially, it is assumed that both parameters alpha and beta are temperature dependent

and hence, stock specific SR parameters are now also time- varying. The process-

level models can be updated to incorporate these effects in the functional form of the

model. The relationships are assumed to be quadratic in order to allow for non-linear

impacts, whereby the size and magnitude of the effect depends on the temperature

range. Thus:

2it oi T1 it T2 it= c +c T +c Tα (9.1) and

2it oi T1 it T2 it= d +d T +d Tβ (9.2)

It should be noted that the temperature time-series were centered to the overall (across

stocks) mean in order to remove the correlation between the first and second order

term estimates. Thus, the intercepts, oic and oid , in the above relationships represent

the stock specific values of alpha and beta at mean temperature. These among stocks

differences remain after accounting for temperature effects and thus, it can be

attributed to additional ecosystem factors affecting the SR model parameters in a

relatively stable manner. The intercepts can be modeled by assuming across stocks

distributions (stock-level models):

2~ N( , )o ooi c cc μ σ (10.1) and

2~ N( , )o ooi d dd μ σ (10.2)

These models describe the (constant in time) divergence between the stock-specific

alpha or beta estimate and the across stocks grand mean ocμ or

odμ , respectively. The

coefficients describing the temperature effect on alpha and beta, ( T1c , T2c ) and

( T1d , T2d ), respectively, are assumed to be common across stocks. This assumption

can be relaxed by imposing stock-level models on the coefficients. For instance, T1c

in [9.1] can be substituted by the stock specific terms 1 1

2~ N( , )T TT1i c cc μ σ .

Consequently, the Ricker SR data- level model, updated to incorporate temperature

effects on alpha and beta, is:

it it it it ity xα β ε= + + (11)

where itα and itβ are given by [9.1] and [9.2] , respectively.

(ii) Habitat size

As discussed previously, the Ricker slope is a proxy to the carrying capacity of the

ecosystem for the given stock i. CC is expected to be positively related to the size of

the stock specific habitat, defined as the juvenile feeding ground, where space can be

a limiting factor causing density dependent effects (Myers et al. 2001, Myers 2002).

Initially, a quadratic relationship is assumed in order to allow for the possibility of

non-liner effects. Differences in habitat size can, therefore, explain part of the across-

stocks variability in beta, which is represented by the distribution of intercepts, oid , in

the stock-level model [10.2]. Thus, H (natural log of habitat area, centered to the

mean in order to eliminate correlation between the coefficients) is included as

predictor in [10.2] the beta stock-level model becomes:

2~ N( , )o

2oi o H1 i H2 i kd k + f H + f H σ (12)

Combining [9.1], [9.2], [10.1], [11] and [12] the final model formulation is:

( )o

2 2 2it c oi T1 it T2 it o oi H1 i H2 i T1 it T2 it it ity +c T +c T k + f H + f H +d T +d T xμ γ κ ε= + + + + (13)

with random effects 2~ N(0, )ooi cγ σ and 2~ N(0, )

ooi kκ σ .

Bayesian Inference

Setting up the multi-level Beverton-Holt SR model [4] in a parallel fashion, we obtain

the following hierarchical structure:

log( ) log( )BH BH BHit it it it it ity xα β β ε= + − + + (14)

with 2N(0, )iid

it yi ~ε σ and parameters alpha ( BHitα ) and beta ( BH

itβ ) depending on

stock specific T time-series and thus, being time-varying:

it T1 T2

BH BH BH BH 2oi it it= c +c T +c Tα (15.1)

it oi T1 T2

BH BH BH BH 2it it= d +d T +d Tβ (15.2)

Equation [14], incorporating the above relationships, is the data- level model,

represented by the likelihood in the Bayesian framework:

2~ N( log( ) log( ), )yi

BH BH BHit it it it ity xα β β σ+ − +

The likelihood function expresses the probability of observing the data given the

functional model and its parameters. Thus, in terms of a joint conditional probability

function for all (n) stocks, it can be written as:

1

[ | , , , , , ]T1 T2 oi T1 T2

nBH BH BH BH BH BH

i oii

p y c c c d d d=∏ (16)

At the next step, stock-level models are incorporated, acting as priors for the

coefficients in relationships [15.1] and [15.2]. Models for the temperature-related

terms allow for the possibility that alpha and beta have different degrees of sensitivity

to temperature effects in the individual stocks. These across stocks distributions are of

the form:

1 11 1

2~ N( , ( ) ) [ | , ]T1i c T T1i c TT T

BH BH BH BH BH BHc cc p cμ σ μ σ= (17)

where T1i

BHc are assumed to be independent.

The stock-level models for the intercepts account for among stocks differences in

Beverton-Holt alpha and beta parameters, arising from additional effects not included

in the data-level model. For parameter beta, representing CC, variation is partly due to

differences in the habitat size occupied by the individual stocks. Therefore, the

intercepts in [15.2] can be modeled as a function of H and the corresponding priors

become stock-specific:

2~ N( , ( ) ) [ | , , , ]oi o H1 H2 o oi o H1 H2 o

BH BH BH BH 2 BH BH BH BH BH BHi i d dd k + f H + f H p d k f fσ σ= (18)

where T1i

BHd are assumed to be independent.

The parameters describing the prior distributions in [17] and [18] are referred to as

hyperparameters in the Bayesian framework, and the uncertainty of the latter is

accounted for by the hyperpriors. In all cases, we use uninformative hyperpriors,

imposing prior distributions N(0, 1000) on the hyperparameters.

The data-level model and the stock-level models, together with the hyperpriors,

represent the uncertainty and variability sources addressed by the hierarchical model.

By expressing them as conditional probability models, they can be combined to

produce the joint posterior distribution of the parameters in the previous levels

(Berliner 1996, Wikle 2003):

p[model (process and parameters)| data] ∝ p [data| process, data parameters]

p[process| process parameters] p[parameters]

The left hand side in the above expression is the joint posterior of all parameters. On

the right side, the first conditional is the joint likelihood function (i.e., the data-level

model) in [16]. The second conditional is represented by the stock- level models, such

as [17] and [18], and the third term corresponds to the hyperpriors.

In practice, posterior estimation in the Bayesian framework is implemented using

iterative algorithms of the Metropolis-Hastings family (Gilks et al. 1996). The basic

idea of the method is based on variance partition and conditioning as presented above

(Gelman & Hill 2007). The algorithm combines partial pooling and classic regression

estimation to update iteratively the parameters, individually or in groups. The

conditional iterative procedure produces sequences (chains) of simulations which

capture all the levels of uncertainty in the estimation of each parameter. Also, it

should be noted that, in Bayesian inference, confidence intervals (known as credibility

intervals) are estimated using the posterior distribution of a given term. Thus, they

have a different, more intuitive interpretation and describe the probability that the true

value of a parameter is within a certain range.

RESULTS

A. Ricker multi-level SR model

We use both multi-level modeling methods, mixed models and Bayesian inference, to

construct, test and explore the Ricker SR model. The former approach is more

convenient in terms of comparing different model formulations, and thus it is mainly

used to specify the final model structure, as it will be described next in more detail.

Subsequently, we employ the more flexible Bayesian framework to estimate

inferential uncertainty about all the model levels parameters and present the key

results.

Ricker mixed models

The first step is to determine the appropriate model formulation by examining both

the random and the fixed components structure. We started with the simplest model

form (equation [7]) as the basis to build up the final model by identifying: (i) the

structure of the random effects variance-covariance matrix DBo B, i.e., whether the

random variables should be assumed independent (diagonal matrix) or correlated, (ii)

the structure of the RBi B matrix, i.e., whether the residual variance should be assumed

common or stock specific, and (iii) the significance and the patterns of the

temperature and the habitat size effects. Likelihood ratio tests (lrt) were employed to

compare models fitted using REML or ML depending on whether the test applies to

random or fixed components, respectively. The lrt results agree in general with the

AIC (Akaike's Information Criterion) comparisons. The model testing procedure is

summarised in Table 2.

It should be noted that fitting the MR3 model, with stock specific residual variance,

was problematic, and the model produced non-positive definite approximate variance-

covariance parameters. Thus, the lrt test results showing that MR3 model is superior

(Table 2), should be interpreted with caution. Nevertheless, we use both approaches

for the fixed effects testing and then employ the stock specific error structure in the

Bayesian implementation of the model.

Auto-correlation

Auto-correlation at lag 1 in log(RBt B/SBt B) was found significant for certain stocks (Table

1). In order to identify whether auto-correlation was responsible for the improved

goodness of fit of the MR2.H2.Ta2 model including the T effect on the alpha

parameter, we employ two different approaches. Initially, we use first-differencing for

these stocks by introducing log(RBt-1 B/SBt-1 B) as a covariate in the model and allowing the

corresponding coefficients to be stock specific. We then fit the ML models with and

without the T effect (MR2.H2.Ta2.AC and MR2.H2.AC, respectively) and compare

them with lrt. The test shows that the T related terms remain significant (Table 2),

even though the first-differencing is known to decrease the statistical power by

increasing the Type-II error rate (Pyper & Peterman 1998).

Our second approach involved fitting MR2.H2 and MR2.H2.Ta2 only to the 10 stocks

exhibiting a low degree of auto-correlation. However, reducing the number of stocks

resulted in non-significant second order terms related to the T and H effect, T2c and

T2d , respectively. Thus, instead, we compared the models MR2.H1r and

MR2.H1.Ta1r including only the linear terms. The lrt revealed that, also in this case,

including the T effect would improve the model fit with p=0.07 (Table 2). It should be

noted that under both approaches similar results were obtained using the stock

specific error variance.

Ricker Bayesian models

We developed the final mixed model MR2.H2.Ta2 in the Bayesian framework using

both common and stock specific error structure (models BR2.H2.Ta2 and

BR3.H2.Ta2, respectively). As it was expected, the models yielded results similar to

the empirical Bayesian estimates obtained from the corresponding mixed models. The

Deviance Information Criterion (DIC), a generalisation of the AIC, was used to

compare the different model formulations (Spiegelhalter et al. 2002). The DIC is

estimated as:

DIC= mean deviance + 2pBd

where mean deviance is estimated as -2 times the log likelihood averaged over the

number of simulations and thus quantifying the lack of model fit, and pBdB is the

effective number of parameters. It was found that the two models produced similar

results regarding the coefficients but BR3.H2.Ta2 with the heterogeneous, stock-

specific error variances had considerably lower DIC (1571 versus 1678). CThe

estimated iσ ’s are comparable in most cases but the recruitment survival of certain

stocks (cod 3m, cod-3ps, cod-4vsw and cod-via) is exhibiting considerably higher

variability (Fig. 1) C. The model BR3.H2.Ta2 was extended to incorporate stock level

models on the T related terms T1c and T2c . The resulting model (BR3.H2.Ta2RS) had

a lower DIC (1560) and was thus chosen as the final Ricker multi-level model. For

comparison, we fit also the model omitting the T effect on alpha. The DIC of this

model is 1567 and the mean deviance is substantially higher (1489 versus 1461 of the

BR3.H2.Ta2RS). The more flexible Bayesian framework allows also the H effects to

be introduced directly on -1/beta (i.e., on S at maximum R) and thus the final

Bayesian model, used to produce the results presented next, becomes:

([ ] ) /[ ]o

2 2it c oi T1i it T2i it it o oi H1 i H2 i ity +c T +c T x k + f H + f Hμ γ κ ε= + − + + (19)

with 2~ N(0, )ooi cγ σ , 2~ N(0, )

ooi kκ σ and 2N(0, )iid

it yi ~ε σ . The brackets in the above

formula contain the stock level models on alpha and beta, respectively. In addition,

the stock level models also include the across stocks distributions of the T related

terms,1 1

2~ N( , )T TT1i c cc μ σ and 2~ N( , )

T2 T2T2i c cc μ σ , describing the dependence of RICiα ’s

on T.

The dependence of Ricker beta on habitat size

The model was first explored in terms of the pooling introduced in the beta parameter

by the stock level model, given by the second bracket in [19]. In Fig. 2a the individual

-1/ RICiβ ’s, estimated as a function of H, and also the corresponding coefficients

obtained from the no-pooling Ricker models fitted separately to each stock, are

plotted against log habitat size. The plot reveals that the parameter is relatively

constant when log habitat size is below the across stocks mean and then increases

exponentially. The pooling of the individual estimates towards the stock-level model

predictions is stronger in the cases where less information is available, i.e., for stocks

with lower sample size (Fig. 2c, equation [8]). The pooling (or shrinkage) results also

in plausible estimates of the parameters (positive values) for all stocks, even when the

individual SR model gives meaningless results (cod-coas, cod-3m and cod-3no).

We can also estimate the amount of across stocks variation in beta explained by

differences in the habitat size, using the RP

2 based statistic (Gelman & Hill 2007):

2 E(V )R 1 ˆE(V( 1/ ))i

oiRIC

κβ

= −−

where ˆ RIC 2i o H1 i H2 ik f H + f Hβ = + and V is the finite-sample variance operator across

stocks. The numerator represents the average variance in oiκ ’s, i.e., in the average

variance of RICiβ ’s left unexplained by H and the denominator the average variance

among the stock specific RICiβ ’s (see model in [19]). It should be noted that the

expectations are averaging over the uncertainty of the model using the posterior

simulations, thus leading to a lower estimate, comparable to the traditional adjusted

RP

2P (Gelman & Hill 2007). The intermediate value of RP

2 obtained (0.48) reveals that

habitat size can explain almost half of the observed variation in beta among cod

stocks but this CC related parameter is determined also by additional ecosystem

characteristics. In particular, the beta’s for stocks located in areas with low or

intermediate average T tend to be higher than those predicted by the model (Fig. 2a).

This can be also shown by plotting the -1/ RICiβ ’s estimated on a per unit area basis

(Fig. 4a). The plot reveals that the stocks with the higher estimates are located in

waters with intermediate mean temperature.

Temperature effects on Ricker alpha

The average, species level, functional relationship between alpha and T, obtained

using the means of the alpha related terms, ocμ ,

1Tcμ and 2Tcμ is presented in Figure

5a. The temperature effect is positive below ~5 P

oPC and becomes negative above this

“critical” point. It is noteworthy that this point is close to the mean of the cod thermal

distribution. Also, the curve is nearly flat, showing no T effect, between 4.5-6P

oPC,

roughly corresponding to the first and third quartiles of cod pawning season T. Thus,

in agreement with previous evidence, extreme temperatures exert stronger effects on

cod recruitment. The itα time-series of Ricker SR model estimated as a function of T

(equation [9.1]) are shown in Figures 4b-c and the mean, minimum and maximum

estimates are presented in Table 3.

In order to estimate the amount of variability in the log(R/S) data explained by using

the T dependent alpha’s for each stock, we can use the statistic:

2 E(V( ))R 1E(V )

it it

it

yyα−

= −

The individual estimates are presented in Table 4 and the overall RP

2 equals 0.5. The

RP

2 quantifying the total data variability explained by the final Ricker model is 0.61

and the stock specific statistics are presented in Table 5.

No significant differences were identified among the slopes T1ic and/or T2ic

describing the non-linear dependence of alpha on T (Fig. 5a-b). Thus also the

“critical” temperature, the point after which the negative effects on alpha prevail,

estimated as - T1ic /2* T2ic , does not differ significantly across stocks and can be

considered equal to the species estimate of ~5P

oPC (Fig. 4a). However, stocks are

expected to differ in the alpha rate of change, given by:

2itT1i p T2i

d c T cdTα

= +

where pT is their current temperature. Accordingly, we can estimate the expected

proportional change in itα , itdα , resulting from a 3P

oPC increase in the current average

temperature of each stock (Fig. 6a, Table 3). It is revealed that the rate is positive for

stocks with current mean spring temperature below 4P

oPC and becomes increasingly

negative above around 5P

oPC.

The stock specific intercepts oic in equation [9.1] represent the among-stocks

differences in alpha left unexplained by the T variability. It is revealed that there are

significant differences between the stocks; cod-coas, cod-3no and cod-3pn4rs are

exhibiting the lowest while cod-347d, cod-arct, cod-2j3kl and cod-iceg the highest

estimates compared to the across stocks mean,ocμ (Fig. 3b). It is also demonstrated

that the deviations are stronger, and negative in most cases, for stocks at intermediate

or lower mean spring temperatures. Thus, additional ecosystem (biotic and/or abiotic)

factors are affecting the cod maximum reproductive rate, as represented by the Ricker

model alpha.

Effects of temperature and habitat size on carrying capacity

The T effects have also implications for the CC related parameters CCBmaxB and CCBeq B,

which depend on both alpha and beta Ricker parameters, as previously described.

Thus, for a given stock, CC is time-varying following the fluctuations of alpha with

T. The mean, minimum and maximum stock specific values of CCBmaxB and CCBeq per

unit area are presented in Figures 3c, d and in Tables 6-7. The proportional change in

the average CCBmaxB induced by 3P

oPC increase in the mean stock specific temperatures is

given by exp( itdα ) and is shown in Figure 6b and Table 7. The change in CCBeq B is

equal to itdα (Fig. 6a, Table 3). It follows that the across stocks pattern is similar to

the one revealed for /itd dTα , with the impact becoming progressively more negative

in areas with higher present mean temperatures, while for certain stocks inhabiting

colder waters, it is expected that the change will be positive, other factors remaining

equal.

On a per unit area basis, it is evident that there are extensive differences in CC across

stocks (about 30 fold). However, the functional form of the relationship between CC

and habitat size is determined by the beta-H stock-level model. Thus, CC is expected

to depend non-linearly on log habitat size. Indeed, the mean estimates are shown to

follow closely the curve predicted using and the beta sub-model (Fig. 7a).

Nevertheless, cod-347d and cod-iceg tend to deviate from this pattern. The divergence

is driven mainly by the across stocks differences in the alpha related intercepts oic ,

since these stocks are among those displaying the highest estimates in the parameter

(Fig. 3b).

Evidence for temperature effects in the no-pooling Ricker SR models

For comparison, we also fit individual stock (no-pooling) models, assuming either

linear or quadratic dependence of alpha on T. In the first case, the effect was found

significant (p<0.1) for cod-347d, cod-4vsw and cod-coas and close to significant

(p<0.2) for cod-arct, cod farp and cod-2j3kl. In the second case, significance was

obtained for cod-arct, cod-via, cod-4vsw and cod-gom (p<0.1), and for cod-coas, cod-

viia, cod-3no, cod-4x and cod-3m (p<0.2). The majority of these stocks (excluding

cod-arct and cod-2j3kl) are located in the upper T range. It is also notable, that the

effect was shown non-significant for most stocks with a considerable proportion of

observations in the middle range 4.5-6P

oPC. Especially for the linear no-pooling models,

the p-value of the T related term was negatively correlated with the sample size (p=.1,

Fig. 8), indicating that the power of the test is low for stocks with low number of

observations. Thus, the pooling of the T related slopes in the multi-level Bayesian

model is stronger in these cases (Fig. 9a-c).

B. Beverton-Holt multi-level SR model

The Beverton-Holt (BH), analogous to the final Ricker SR model given by [19], is the

following:

([ ] ) log( ) log( )o

2 BH BHit c oi T1i it T2i it i i it ity +c T +c T xμ γ β β ε= + + − + + (20)

where 2~ N( , )o

BH 2i o H1 i H2 i kk f H + f Hβ σ+ , 2~ N(0, )

ooi cγ σ ,1 1

2~ N( , )T TT1i c cc μ σ and

2~ N( , )T2 T2T2i c cc μ σ .

We first compared the Ricker and BH models in terms of the data variability

explained by each, using the RP

2P statistic described previously. The BH fit seems to

have a slightly better performance for most stocks, but the difference is pronounced

only for cod-2j3kl (Fig. 10a, Table 5). Also, the DIC of BH model was lower (1534)

compared to the Ricker DIC (1560). However, the variance explained by alpha is in

general higher for the Ricker model, and the difference is mostly evident for cod-farp

and cod-3ps (Fig. 10b, Table 4).

The two SR models provided similar estimates for the alpha related

terms,ooi c oic μ γ= + , T1ic and T2ic (Fig. 11a-c). Consequently, also the critical T and

the itdα , depending on the previous terms, are not significantly different. In addition,

the form of the stock-level model between BHiβ and log habitat size is analogous to

the corresponding Ricker sub-model (Fig. 2b). However, the model displays better fit,

quantified by the RP

2P statistic, explaining about 70% of the across stocks variability in

the parameter. The BH model provided, in general, higher point estimates for beta,

representing SSB resulting into maximum replacement spawners (CCBmaxB) in the

absence of fishing mortality. The divergence, however, is considerable only in few

cases, namely for cod-iceg, cod-2j3kl and cod-2532 (Fig. 11d). Following the pattern

in beta, there are substantial differences in the mean estimates of CCBeq B provided by

the two SR models (Table 6), and to a lesser extent in CCBmax (Table 7). Also, as in the

Ricker model, the relationship between CC and log habitat is non-linear (Fig. 7b), and

thus, per-unit area comparisons of the parameters are not straightforward.

Finally, we present the fitted Ricker and BH SR models corresponding to the mean,

minimum and maximum alpha estimates (Fig. 12). The resulting curves are similar

for most stocks, whereas for cod-iceg, cod-2j3kl and cod-2532, for which the two SR

models provided considerably different beta estimates, the BH model predicts higher

recruitment survival at upper stock sizes. It is notable that for most stocks, the upper

and lower curves, corresponding to the highest and lowest alpha estimates, seem to fit

well with extreme observations (e.g. cod-7ek, cod-coas, cod-farp, cod-iceg, cod-kat,

cod-2j3kl, cod-3pn4rs, cod-3ps, cod-gb, cod-viia). In other cases (e.g. cod-2224, cod-

347d, cod-3m, cod-3no, cod-4tvn), however, the models are less sensitive to these.

This can be an effect of partial pooling which, by definition, gives more weight to

observations closer to the mean. Alternatively, for those stocks, T and S fluctuations

cannot explain effectively the variation in recruitment and other factors are driving the

patterns. C

We also use cumulative z scores, in order to illustrate the recruitment survival patterns

in relation to temperature and spawner biomass, and mainly to assess the models

capacity to capture effectively these trends (Fig. 13). The standardized z scores of a

given time-series are calculated as anomalies (deviations) from the mean divided by

the standard deviation and are especially useful for illustrating the degree of

coherence between patterns of different variables (e.g. Molinero et al. 2005). In this

context, it is evident that the fitted values, in most cases, follow consistently the

general trends of the observed log(R/S). The performance of the two SR models is

equivalent and improved especially for stocks or for periods when recruitment

survival is most strongly (positively or negatively) affected by temperature. C

DISCUSSION

The main objective of our study is to investigate the effects of both the spawning

season temperature and the nursery grounds size on cod alpha and beta SR

parameters, representing maximum reproductive rate, recruitment survival, density

dependent mortality and carrying capacity, across the species N Atlantic distribution.

Thus, we employed two complementary hierarchical methods, i.e., mixed modeling

and Bayesian inference, in order to combine data on all cod stocks in a meta-analysis

of both their Ricker and Beverton-Holt SR relationships. Our key conclusion, under

both SR models, is that, at the species level, there is a dome-shaped relationship

between alpha, and thus also CC, and temperature. Consequently, T effect is positive

in colder waters up to ~5oC, which is close to the mean of the cod spawning season

thermal range, and negative for stocks inhabiting warmer areas. Regarding the effect

of habitat, it is demonstrated that beta, which is also a CC component and describes

density dependent regulation, is dependent on area size in a non-linear way. These

findings imply that ocean warming will cause, and is already imposing, considerable

impacts on both alpha and CC, and thus on cod SR dynamics. More importantly, the

models allow inference on both the sign and the extent of these effects both at the

species and the individual stocks level.

Hierarchical modeling

The hierarchical, multi-level approach offers a number of advantages, which have

been demonstrated to be particularly useful for the analyses of fisheries data (e.g.

Hilborn & Liermann 1998, Myers 2002). The methods are based on stock–level

models describing the variation among the stock specific parameters across the

species range. For beta, these models are extended to include habitat size as an

individual stock predictor, which can partly explain the observed variation. In the

Bayesian framework, the stock-level models, or priors, can also be used to introduce

existing knowledge in the model (Gelman & Hill 2007). In the present study,

however, we use an empirical Bayesian approach, wherein priors are uninformative

or, in the case of beta, depend on H (log of habitat size).

An alternative method to model across stocks variation in the parameters would have

been to fit separate (no-pooling) models to each stock and then model the stock-

specific parameters. Multi-level methods, however, combine these two stages in a

single model, wherein inference is based on both the within and the among stocks

variability, incorporating uncertainty in all parameters. Thus, the stock-level models

are used to convey information about the probability distribution of the parameter

estimated by all stocks data. Since the inference of single stock parameters is based on

these priors, strength is “borrowed” and the “loan” (pooling) is higher for “poorer”

(limited or highly variable observations) stocks. Consequently, the empirical Bayesian

inference is superior to the no-pooling models, especially in case the latter provide

implausible estimates or lack the required power to demonstrate the significance of

certain terms.

Effects of temperature on alpha

Bringing together data, and particularly temperature, across the entire cod N Atlantic

distribution, has allowed inference on the functional form of the relationship between

alpha and T at the species level. Temperature has opposite effects at the upper

(negative) and the lower (positive) thermal extremes, roughly corresponding to waters

with T above or below 5oC, respectively. Due to the quadratic form, the strength of

the effect is stronger for temperatures closer to the extremes, and weakest at the

middle, “neutral” 4.5-6oC interval. Similar geographic patterns have been observed

for the response of cod recruitment to temperature across certain stocks (Planque &

Frédou 1999). Consequently, the effect of a potential increase in current mean

temperature will be more pronounced for stocks inhabiting areas closer to the

distribution limits. Accordingly, alpha for cod-3pn4rs, with the lowest mean T, is

expected to increase by more than 30%, while in Celtic Sea, where mean T is the

highest, the decrease will be about 20%.

No significant differences were found among the stock specific T related slopes, even

though the pooling of these terms towards the species mean is considerable only for

stocks with limited data. Nevertheless, we allowed the relevant terms to be stock

specific in the models, by introducing a stock–level model on the corresponding

parameters, a choice which was also supported by the model selection criteria (lrt,

AIC or DIC). In addition, statistical significance should not be used as a measure in

multi-level models to determine whether a parameter should be considered fixed or

random (Gelman & Hill 2007). Rather, we allow the model to estimate the best

possible parameters for each stock, while accounting for uncertainty. In effect, it was

shown that T impacts, also for single stocks, are adequately described by the mean,

species-specific relationship. Consequently, the estimated species-level sub-model can

be useful for inference in single stock studies, particularly for the prediction of

potential ocean warming effects in areas where T variation is limited or within the

“neutral” range.

It is noteworthy, that the T effect remains significant, even after first order

differentiation, although this method is known to increase the type-II error probability,

when employed for low frequency environmental signals (Pyper & Peterman 1998).

Conversely, inference is obscured in single stock (no-pooling) Ricker models, where

the significance of the T related terms depends on the amount of available data and

also on the thermal range of a given population, with stocks near the limits having

lower p-values. This can be an explanation for the failure of recruitment-temperature

correlations after more data is obtained, especially for stocks near the center of the

range, since significance is less likely for populations with a considerable amount of

observations in the “neutral” interval, where the alpha-T curve is almost flat.

In the present context, wherein we use standardized recruitment time-series

(multiplied by SPRF=0) and allow for T effects on the parameter, alpha can be

interpreted as the maximum rate at which spawners produce replacement spawners, in

the absence of fishing mortality, given the temperature conditions during the

spawning season in a particular year. Therefore “maximum” should not be interpreted

in absolute terms, since it has a temporal, temperature dependent component.

Accordingly, we can produce a set of SR curves, corresponding to the mean,

minimum and maximum alpha estimates, which show the most pronounced

differences. Stock productivity depends on spawners reproductive potential and also

on natural mortality from the egg to the adult stage (recruitment converted to SSB at

F=0). Mortality during the pre-recruit stage is the primary source of recruitment

variability (Leggett & DeBlois 1994) and can highly obscure the spawner-recruit

relationship (Megrey et al. 2005), while good recruitment year classes remain good

after settlement (Hjort 1914, Myers 2001, Myers 2002). In this sense, temperature

dependent alpha can explain 50-97% or 40-96% of the log(R/S) variation within

stocks, using the Ricker or the BH multi-level models, respectively.

It was also demonstrated that apart from T, additional factors are influencing alpha,

causing the across stocks variability in the intercepts of the alpha-T relationship. The

differences are more pronounced among stocks located in colder waters, while in

warmer areas it seems that T is the limiting factor. Indeed there is extensive evidence

that environmental variables like NAO (Stige et al. 2006), are also affecting cod

recruitment across N. Atlantic. Especially for Baltic Sea stocks, salinity and oxygen

are the strongest factors determining recruitment success (MacKenzie et al. 2007). In

addition, fishing can impact on the spawning potential of a stock by altering age and

size at maturity and/or growth rate (Heino et al. 2002, Marteinsdottir et al. 2005,

Ottersen et al. 2006). Biotic interactions with prey, predator or competitive species

have been shown to affect the productivity and survival rates of both early and adult

cod stages (Lilly et al. 2008). The reasons for the among stocks variability in alpha,

whether of local or of species level importance, bear further investigation. It would

also be pertinent to employ similar methods in order to investigate cod response to

other forcing factors and to identify whether these impacts would be additive or

superior to T effects.

Finally, it should be noted that our results regarding parameter alpha are in agreement

with previous studies employing similar meta-analytic models to study SR dynamics

(Myers et al. 1999, Myers et al. 2001), and advocating that alpha is relatively constant

across cod stocks. The average variation among the stock-specific parameters is ~7

fold, and the mean (corresponding to mean T) estimates are comparable to those

obtained by Myers et al. (2001) employing a non-linear mixed BH model to a quite

overlapping set of cod stocks and using slightly different SPRF=0 estimates. Besides,

our findings are in accordance with similar studies providing evidence that the across

cod stocks differences in alpha are related to mean bottom temperature in each region

(Myers et al. 2001, Myers et al. 2002).

The dependence of beta on habitat size

We have assumed that the availability of juvenile habitat, defined as the area between

40-300m (Myers et al. 2001), is shaping the density dependent mechanisms

controlling cod recruit survival. Thus, we have included H as a predictor in the

relevant stock-level models on beta, describing across stocks variability in the

parameter. This way we are controlling (standardizing) for differences in habitat size

when estimating the density-dependent parameters. Also, the approach is flexible,

allowing the H parameters to be readily updated or substituted by time-varying

estimates, if appropriate. The dependence on the log transformed habitat size, under

both SR models, was shown to be non-linear and following an exponential pattern.

However, the BH level model has a bitter fit, explaining 70% of the across stocks

variability, compared to 50% of the corresponding Ricker sub-model. In both cases, it

seems that the employed definition of H is effectively representing the density-

dependent processes, accounting for at least half of the variation in beta. However,

stocks inhabiting waters with lower or intermediate temperatures tend to have higher

estimates than those predicted by the models.

There was also some evidence that beta is affected by temperature. Although the

effect of temperature on this SR parameter is not usually considered (but see Kell et

al. 2005), stochastic factors have been shown to affect density dependent mortality in

cod (Fromentin et al. 2001). Density-dependent habitat selection by juvenile cod has

been documented in North Sea, where, as postulated by the ideal free distribution

theory (Fretwell & Lucas 1970), in years of low biomass the individuals tend to

concentrate in areas with optimal temperatures (Blanchard et al. 2005). Also,

observations during past warming periods have demonstrated that cod reacts quickly

to increased temperature, shifting or expanding spawning and feeding locations

northwards (Drinkwater 2005, Rose 2005). In any case, it was shown that introducing

the effect of T on alpha rather than beta or both parameters, would better describe the

observed data and provide superior predictions.

Effects of temperature and habitat on carrying capacity

CC, either defined as the equilibrium SSB (CCeq) or as the SSB producing the

maximum replacement spawner biomass (CCmax), is determined by both SR

parameters, alpha and beta, and thus, by the factors controlling them; T and H,

respectively. Due to the dependence on temperature, within stocks CC is not a fixed

quantity, but a dynamic variable varying temporally according to the alpha

fluctuations driven by T. Further, excluding CCeq obtained by the Ricker model, CC is

an exponential function of T. Therefore, temperature effects, approximated by the

proportional change following an increase in current T, are more severe for CC. For

example, CCmax for cod-2j3kl is expected to increase by more than 80%, compared to

~20% increase in alpha. In Celtic Sea, which is expected to have the strongest

negative effect, the reduction in CC will be above 40%, while the decrease in alpha

was estimated at 23%.

Habitat size was shown to effectively explain a substantial part of variability in beta,

and thus, also in CC, across stocks. On a per unit area basis, however, there is

considerable variability in CC among populations, a pattern also recognized in a

previous meta-analytic study on cod (Myers et al. 2001). The differences across areas

are ~30 fold under Ricker and lower (~20 fold) under the BH model, which was

shown to have a higher R2 in the beta-H stock-level model. These pronounced

differences can be attributed to the non-linear, exponential nature of the relationship

between beta and log habitat size. In addition, CC indices, excluding Ricker CCeq,

depend on the exponent of alpha (Alpha), representing maximum reproductive

potential and temperature dependent pre-recruit survival. Alpha is magnifying the

across stocks differences, which are suppressed using the variance stabilizing log

transformed parameter, and thus, contributes to the variability among the stock

specific CC’s. Consequently, the relationship between CC and habitat size is complex

and cannot be simply assessed on a per unit area basis.

Perspectives

Multi-level models are especially useful for identifying and quantifying processes

acting on a broad scale, determining fish population dynamics across or in specific

regions of their distribution. Thus, it has been possible to describe the response of cod

recruitment and CC to temperature, while allowing for the effect of habitat size on the

density-dependent mechanisms. It would be highly interesting to develop similar

approaches in order to make inference for other fish species, especially those with

sufficient variability among stocks. Also, studying environmental impacts on cod key

forage or competitor species, either separately or combined in multi-species models,

would reinforce the predictions on the implications of ocean warming for the

structuring of local ecosystems.

The development of mathematical or empirical models describing mechanisms

through which environmental impacts operate on the stock or the species level, is

fundamental for the quantification of background processes linking climatic factors,

and their variability, to life history parameters and hence to population dynamics. It is

also essential, to incorporate the various sources of uncertainty, to discern and

illustrate patterns despite ecosystem complexity. A further advantage of such models

is that their parameters have a meaningful interpretation and, thus, population traits of

interest, and their dynamics and/or variability, can be estimated directly. However,

process modeling, usually, involves non-linear relationships, hampering empirical

estimation, especially when the data series are short and noisy and the inter-annual

variability in the forcing environmental factor low, a situation not uncommon to fish

populations. To couple the needs both for mechanistic, possibly non-linear, models

and robust parameter estimation, hierarchical or multi-level models can provide a

useful and flexible toolbox, describing stochastic processes of various forms and

allowing for inference across or within stocks (Gelman & Hill 2007).

Further, the obtained species-level patterns can be combined with models developed

for individual stocks, describing mechanisms specific to local ecosystems. For

instance, SR models coupled with models for different aspects of fish biology/ and or

ecology, can be used to improve stock assessment and management, under the

influence of environmental forcing (Kell et al. 2005). The parameterization of such

models with patterns obtained using meta-analytic methods, could provide further

insights and reduce uncertainty, borrowing the strength gained by the combination of

more extensive datasets. In addition, the flexibility of the Bayesian framework, apart

from multi-level structures, allows the simulative implementation of mechanistic

models, incorporating complex processes also for single stocks

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cod-2224cod-2532cod-347dcod-7e-kcod-arctcod-coascod-farpcod-icegcod-katcod2j3klcod3mcod3nocod3pn4rscod3pscod4tvncod4vswcod4xcodgbcodgomcodviacodviia

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Figure 1. Stock specific residual standard errors and 95% confidence intervals obtained from the Ricker Bayesian model plotted versus sample size.

10 11 12 13

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Log habitat size (km2)

Bet

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lvel

eR

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el (0

00’s

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)

c

Figure 2. The fitted stock-level model of beta as a function of H (log habitat size) in the Ricker (a) and in the B-H (b) multi-level Bayesian models. c: The ratio of beta estimated from the Bayesian Ricker model and the corresponding estimate obtained from individual, no pooling models plotted versus sample size. The ratio is closer to 1 (representing weaker pooling) for stocks with higher number of observations.

0.000 0.005 0.010 0.015

Beta per unit area (000’s tons/ km2)

cod3pn4rscod4tvncod2j3klcod3ps

cod-arctcod-2532cod-2224

codgomcod3no

cod4xcod-kat

cod-coascod-347d

codgbcod-iceg

codviiacod-farpcod4vsw

cod3mcodvia

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a

1 2 3 4

Stock specific Intercept

cod3pn4rscod4tvncod2j3klcod3ps

cod-arctcod-2532cod-2224

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codgbcod-iceg

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c

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CCmax per unit area (000’s tons/ km2)

cod3pn4rscod4tvncod2j3klcod3ps

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cod-347dcodgb

cod-iceg

codviiacod-farp

cod4vswcod3m

codvia

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Mean CCeq per unit area (000’s tons/ km2)

d

Figure 3a: The beta parameter estimates (±se) obtained from the Bayesian Ricker model (1000’s tons/km2) ordered by increasing mean temperature (bottom –up). b: The stock specific intercepts [ ]

oc oiμ γ+ (±se) estimated from the Bayesian Ricker model ordered by increasing mean temperature (bottom –up). c: The stock specific CCmax (±se), estimated as tons/km2, obtained from the Bayesian Ricker model ordered by increasing mean temperature (bottom –up). d: The stock specific CCeq (±se), estimated as tons/km2, obtained from the Bayesian Ricker model ordered by increasing mean temperature (bottom –up).

0 5 10 15

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Temperature (oC)

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ha

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01

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Temperature (oC)

Alp

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Figure 4a: The species level relationship between alpha and temperature (black line), estimated by the Ricker Bayesian model. The grey lines correspond to the 95% credibility intervals. The stock specific relationships between alpha and temperature estimated by the Ricker Bayesian model, for the stocks in the lower (b) and upper (c) temperature range. Credibility intervals of the upper and lower estimates are also plotted.

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

cod3pn4rscod4tvncod2j3klcod3ps

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stock specific T related slopes ct1i

a

-0.06 -0.04 -0.02 0.00 0.02

cod3pn4rscod4tvncod2j3klcod3ps

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codgbcod-iceg

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stock specific T related slopes ct2i

b

Figure 5: The stock specific T related slopes T1ic (a) and 2T ic (b) (±se) describing the relationship between alpha and temperature, obtained from the Bayesian Ricker model ordered by increasing mean temperature (bottom –up).

2 4 6 8 10

0.0

0.5

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Current mean T (oC)

Alph

a at

incr

ease

dT/

Cur

rent

mea

n al

pha

2 4 6 8 10

0.5

1.0

1.5

2.0

2.5

Current mean T (oC)

CC

max

at in

crea

sed

T/ C

urre

nt m

ean

CC

max

a b

Figure 6a: The stock specific ratios (±se) between mean alpha and alpha corresponding to a 3oC increase in the current mean temperature plotted against current mean temperature. The same result applies also to CCeq. b: Corresponding plot for CCmax.

10 11 12 13

050

0010

000

1500

020

000

10 11 12 13

050

0010

000

1500

020

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3500

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Log habitat size (km2)Log habitat size (km2)

Mea

n C

Cm

ax(0

00’s

tons

)a b

Figure 7: Mean CCmax and CI’s estimated by the Ricker (a) and the BH (b) models, using mean alpha and beta estimates. The curves correspond to the estimates obtained using beta as predicted by the beta-H models.

20 30 40 50

0.0

0.2

0.4

0.6

0.8

Sample size

P-va

lue

Figure 8: The p-values of the T related term describing the linear dependence of alpha on temperature obtained from the no-pooling Ricker models plotted against sample size. The correlation is negative (p=0.1).

T re

late

d sl

opes

ct1

i Rat

io (q

uadr

atic

mod

el)

20 30 40 50

-40

-20

020

40

20 30 40 50

-20

020

4060

20 30 40 50

-10

010

2030

4050

T re

late

d sl

opes

ct2

i Rat

io (q

uadr

atic

mod

el)

T re

late

d sl

opes

ct1

i Rat

io (l

inea

r mod

el)

a b c

Figure 9: The ratio between the T related slopes obtained from the no-pooling Ricker models assuming quadratic dependence of alpha on T and the corresponding slopes obtained from the Bayesian Ricker model (a: T1ic , b: 2T ic ) plotted versus sample size. The pooling is less (ratio close to 1) for stocks with more data. c: The corresponding plot between the T related slopes obtained from no-pooling Ricker models assuming linear dependence and T1ic .

a

0.80 0.85 0.90 0.95 1.00

0.80

0.85

0.90

0.95

1.00

Ove

rall R

2BH

mod

el

Overall R2 Ricker model

0.2 0.4 0.6 0.8 1.0

0.2

0.4

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0.8

1.0

Alph

a R

2BH

mod

elAlpha R2 Ricker model

b

Figure 10a: The R2 quantifying the total data variability explained by the B-H model versus corresponding Ricker R2.

b: The R2 quantifying the total data variability explained by temperature dependent alpha in the B-H model versus corresponding Ricker R2.

1 2 3 4 5

cod3pn4rscod4tvncod2j3klcod3ps

cod-arctcod-2532cod-2224

codgomcod3no

cod4xcod-kat

cod-coascod-347d

codgbcod-iceg

codviiacod-farpcod4vsw

cod3mcodvia

cod-7e-k

Stock specific Intercept

a

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

cod3pn4rscod4tvncod2j3klcod3ps

cod-arctcod-2532cod-2224

codgomcod3no

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codgbcod-iceg

codviiacod-farpcod4vsw

cod3mcodvia

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T1icSlope

b

-0.10 -0.05 0.00 0.05

cod3pn4rscod4tvncod2j3klcod3ps

cod-arctcod-2532cod-2224

codgomcod3no

cod4xcod-kat

cod-coascod-347d

codgbcod-iceg

codviiacod-farpcod4vsw

cod3mcodvia

cod-7e-k

2T icSlope

c

020

040

060

080

010

00B

eta

(000

’s to

ns)

d

Figure 11: Comparison between a. the alpha related intercept [ ]

oc oiμ γ+ b, c: and the T related slopes ( T1ic and 2T ic , respectively). d: beta (expressed as 2*K for the BH and as B for the Ricker model) estimates and 95% credibility intervals obtained from the Ricker (grey bars) and B-H (black bars) Bayesian SR models.

Figure 12: The final Ricker (black lines) and B-H (grey lines) SR multi-level models fitted to each stock. The bold lines correspond to mean temperature dependent alpha and the dashed or dotted lines to the curves obtained using the upper and lower alpha estimates.

Figure 13: Cumulative z scores of the observed log(R/S) (black bold line), the fitted log(R/S) estimates obtained from the multi-level Ricker (black dashed line) and B-H (black dotted line) Bayesian models, temperature (grey dashed line) and S (grey dotted line) time-series

Table 1. Cod stocks summary.

* and updates thereof.

Stock Code Time Period Area REFERENCE

cod-2224 1970 - 2005 W Baltic ICES 2004, ICES 2005a* cod-2532 1966 - 2003 E Baltic ICES 2004, ICES 2005a* cod-347d 1963 - 2005 North Sea ICES 2004, ICES 2005a* cod-7e-k 1971 - 2005 Celtic Sea ICES 2004, ICES 2005a* cod-arct 1953 - 2003 Arctic Sea ICES 2004, ICES 2005a* cod-coas 1984 - 2004 Norwegian

coastal ICES 2004, ICES 2005a*

cod-farp 1961 - 2004 Faroe Plateau ICES 2004, ICES 2005a* cod-iceg 1956 - 2003 Iceland ICES 2004, ICES 2005a* cod-kat 1971 - 2004 Kattegat ICES 2004, ICES 2005a* cod2j3kl 1962 - 1989 N

NewfoundlandBishop et al. 1993

cod3m 1972 - 2000 Flemish Cap Vázquez & Cerviño 2002 cod3no 1959 - 2004 Grand Bank Power et al. 2005

cod3pn4rs 1974 - 2003 N Gulf of St Lawrence

Fréchet et al. 2005

cod3ps 1977 - 2002 S Newfoundland

Brattey et al. 2004

cod4tvn 1953 - 2004 S Gulf of St Lawrence

Chouinard et al. 2006

cod4vsw 1970 - 2001 E Scotian Shelf

Fanning et al. 2003

cod4x 1983 - 2000 W Scotian Shelf

Clark et al. 2002

codgb 1978 - 2004 Georges Bank O'Brien et al. 2005 codgom 1982 - 2004 Gulf of Maine Mayo & Col 2005 codviia 1968 - 2005 Irish Sea ICES 2004, ICES 2005a* codvia 1978 - 2004 W Scotland ICES 2004, ICES 2005a*

Table 2: Codes and comparisons between the Ricker mixed and Bayesian models Mixed models

Code Remarks H effects T effects AIC (DIC) lrt p-value 1 MR1 Random effects

correlation 1774.3 (REML) 1 vs 2 0.5

2 MR2 1772.7 (REML) 2.vs 3 <.0001 3 MR3 Stock specific errors 1666.3 (REML) 4 MR2.H1 Linear 1755.3 4 vs 5 0.00 5 MR2.H2 fitted to stocks with

low auto-correlation Linear 1743.2 5 vs 2 0.04

6 MR2.H2.Ta1 fitted to stocks with low auto-correlation

Linear Linear on alpha

1744.5 6 vs 7 0.00

7 MR2.H2.Ta2 Quadratic 1737.9; 1797.9 (REML) 7 vs 5 0.01 8 MR2.H2.Tb1 first-differencing of

log(R/S) Quadratic 1744.8 8 vs 9 0.02

9 MR2.H2.Tb2 Quadratic Linear on alpha

1741.4 9 vs 5 0.05

10 MR2.H2.Tab2 Quadratic Quadratic on alpha

1740.9 10 vs 7 0.58

10 vs 9 0.11 11 MR2.H2.Ta2.MRS first-differencing of

log(R/S) Quadratic Quadratic

on alpha 1797.9 (REML) 11 vs 7 0.14

12 MR2.H2.AC Quadratic Quadratic on alpha / random

effects on T terms

1596.78 12 vs 13

0.00

13 MR2.H2.Ta2.AC Quadratic Quadratic on alpha and beta

1589.46

14 MR2.H1.r Quadratic Linear on beta

766.51 13 vs 15

0.07

15 MR2.H1.Ta1.r Quadratic Quadratic on beta

765.32

Bayesian models

Code Remarks H effects T effects DIC BR2.H2.Ta2 Quadratic Quadratic on

alpha 1678

BR3.H2.Ta2 Stock specific errors Quadratic Quadratic on alpha

1571

BR3.H2.Ta2RS Stock specific errors and T related slopes

Quadratic Quadratic on alpha

1560

BR3.H2 Stock specific errors Quadratic 1567

63

Table 3. Mean, maximum and minimum stock specific alpha’s estimated by the Ricker multi-level model. The change refers to the change in mean alpha induced by an increase in current mean T by 3oC.

mean min max Change cod-2224 2.83 2.67 2.88 -0.04 cod-2532 1.87 1.62 2.00 -0.09 cod-347d 4.53 4.39 4.62 -0.06 cod-7e-k 1.88 1.67 2.27 -0.23 cod-arct 3.52 3.13 3.61 0.06 cod-coas 0.64 0.50 0.82 -0.69 cod-farp 2.07 1.76 2.24 -0.13 cod-iceg 3.23 3.03 3.31 -0.08 cod-kat 1.92 1.80 1.96 -0.08 cod2j3kl 2.54 1.61 3.32 0.23 cod3m 2.71 2.31 2.80 -0.04 cod3no 1.16 0.95 1.24 0.02

cod3pn4rs 1.03 0.73 1.43 0.32 cod3ps 1.85 1.66 2.09 0.20 cod4tvn 2.04 1.97 2.07 0.02 cod4vsw 2.00 0.98 2.32 -0.19

cod4x 1.76 1.74 1.76 -0.01 codgb 2.10 1.91 2.19 -0.11

codgom 2.68 2.53 2.72 -0.04 codvia 2.25 2.15 2.42 -0.16 codviia 2.15 1.71 2.31 -0.13

64

Table 4: R2 quantifying the total data variability explained by temperature dependent alpha in the B-H model and the Ricker multi-level models. Ricker BH cod-2224 0.92 0.88 cod-2532 0.93 0.86 cod-347d 0.96 0.95 cod-7e-k 0.95 0.95 cod-arct 0.96 0.93 cod-coas 0.88 0.86 cod-farp 0.84 0.20 cod-iceg 0.97 0.81 cod-kat 0.94 0.91 cod2j3kl 0.85 0.83 cod3m 0.88 0.80 cod3no 0.95 0.95 cod3pn4rs 0.91 0.88 cod3ps 0.69 0.55 cod4tvn 0.94 0.85 cod4vsw 0.94 0.94 cod4x 0.79 0.74 codgb 0.91 0.91 codgom 0.78 0.78 codvia 0.95 0.94 codviia 0.89 0.88 ACROSS STOCKS 0.50 0.42

65

Table 5: R2 quantifying the total data variability explained the B-H model and the Ricker multi-level models.

Stock specific overall R^2 Ricker B-H

cod-2224 0.95 0.95 cod-2532 0.94 0.95 cod-347d 0.97 0.97 cod-7e-k 0.95 0.95 cod-arct 0.97 0.97 cod-coas 0.92 0.92 cod-farp 0.96 0.97 cod-iceg 0.98 0.98 cod-kat 0.95 0.94

cod2j3kl 0.88 0.93 cod3m 0.89 0.88 cod3no 0.95 0.96

cod3pn4rs 0.92 0.92 cod3ps 0.92 0.95 cod4tvn 0.96 0.97 cod4vsw 0.95 0.95

cod4x 0.88 0.88 codgb 0.93 0.93

codgom 0.93 0.94 codvia 0.95 0.95 codviia 0.94 0.95

ACROSS STOCKS 0.61 0.62

66

Table 6: Mean, maximum and minimum CCeq estimates obtained by the Ricker and the BH multi-level models.

Ricker BH mean min max mean min max

cod-2224 10.82 10.21 11.03 35.2629 30.4127 37.2746 cod-2532 5.99 5.16 6.50 9.8338 7.7396 11.5713 cod-347d 4.61 4.46 4.71 58.1481 51.047 63.9161 cod-7e-k 2.58 2.30 3.10 5.6092 4.5086 8.5127 cod-arct 2.67 2.37 2.74 14.3659 10.6912 15.6806 cod-coas 0.88 0.66 1.16 1.6554 1.4509 1.9883 cod-farp 9.94 8.41 10.80 19.3995 15.43 22.1903 cod-iceg 22.23 20.82 22.78 70.791 57.2075 76.8256 cod-kat 17.08 16.04 17.39 27.8056 24.6851 29.2819

cod2j3kl 5.54 3.50 7.23 12.243 6.3397 20.7054 cod3m 28.17 23.73 29.18 71.5356 51.2905 77.8295 cod3no 2.53 2.02 2.71 4.1983 3.4975 4.476

cod3pn4rs 3.65 2.59 5.03 5.1054 3.9038 7.6282 cod3ps 2.58 2.29 2.97 5.1416 4.3399 6.4477 cod4tvn 7.52 7.31 7.60 13.4299 12.6464 13.9036 cod4vsw 4.34 1.97 5.10 9.9713 3.9213 13.1582

cod4x 4.32 4.31 4.33 8.5027 8.253 8.5857 codgb 5.36 4.86 5.62 12.1927 10.0968 13.5046

codgom 5.78 5.38 5.92 18.2678 15.893 19.1094 codvia 4.31 4.12 4.64 10.9883 10.0422 12.9231 codviia 9.12 7.04 9.88 16.263 10.8312 19.0351

67

Table 7: Mean, maximum and minimum CCmax estimates obtained by the Ricker and the BH multi-level models.

Ricker BH mean min max mean min max Change

cod-2224 23.46 20.17 25.18 33.2335 28.149 35.0759 -0.11 cod-2532 7.72 6.01 9.24 9.7757 7.5814 11.3214 -0.14 cod-347d 34.72 30.27 38.25 57.2598 49.9756 62.9622 -0.22 cod-7e-k 3.34 2.74 5.02 5.2726 4.2222 7.7243 -0.42 cod-arct 9.47 6.50 10.30 14.1944 10.3213 15.4283 0.24 cod-coas 1.00 0.86 1.22 1.6183 1.412 1.9137 -0.29 cod-farp 14.23 10.49 17.01 18.8836 14.6731 21.7152 -0.34 cod-iceg 63.95 52.81 69.13 71.0584 57.0194 76.9682 -0.25 cod-kat 22.26 20.02 23.11 25.0635 21.7785 26.3884 -0.15

cod2j3kl 11.29 4.27 23.29 12.1212 6.0949 20.135 0.83 cod3m 59.26 46.49 63.63 59.8493 35.1879 66.5998 -0.11 cod3no 2.65 2.19 2.84 3.9795 3.2006 4.286 0.03

cod3pn4rs 3.75 2.81 5.74 4.9848 3.7328 7.2006 0.42 cod3ps 3.38 2.78 4.37 5.0215 4.2269 6.1995 0.44 cod4tvn 10.49 10.03 10.80 13.3776 12.413 13.7873 0.05 cod4vsw 6.39 2.29 8.53 9.4306 3.4177 12.2188 -0.35

cod4x 5.22 5.18 5.34 8.0868 7.6983 8.1989 -0.02 codgb 7.63 6.42 8.46 11.7744 9.629 12.9459 -0.21

codgom 10.77 9.25 11.50 15.8557 13.5382 16.5026 -0.09 codvia 6.76 6.15 8.06 9.7847 8.8936 11.431 -0.34 codviia 12.82 8.50 15.20 14.0301 8.993 16.3536 -0.28