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Chilean fishing law, maximum sustainable yield and the stock-recruitment relationship.

Ley de pesca chilena, rendimiento maximo sostenible y la relacion stock-recluta

INTRODUCTION

Chile lands around 2.5 million ton of fish every year, the eighth largest catch of any country in the world (FAO, 2014). Fishing activity became important in the early 1920s when mostly European investors developed the small pelagic fishery in the northern zone, and afterwards, during the 1940s, the demersal fishery in the central and southern zones of Chile. Despite this relatively long history of fishing exploitation, only in the preceding three decades fishing monitoring, stock assessment and fishery management became well-established programs. Fishing monitoring program collects information regarding operational and biological attributes of the main target species, while stock assessment in Chile is based on integrate models that combine all available information to produce estimates of abundance and exploitation status. Fishery management takes the advice from stock assessment to propose an annual total allowed catch (TAC) of each stock, usually based on biological reference points (RPs) and the associated risk of incompliance with management objectives. Since 1991, fisheries management in Chile has been framed by "the General Law of Fishing and Aquaculture" including a system for quota allocation based on individual transferable quotas (ITQs) from 2001 to 2012. This system, however, lacked a specific procedure to establish a TAC using RPs. This often resulted in managers setting TACs along political and/or social criteria, instead of emphasizing the associated risk of not fulfilling the conservation objective associated to the RPs (Leal et al., 2010). Such bad and incorrect management practice contributed to the current overfishing and depletion of most Chilean fisheries managed by TAC (SUBPESCA, 2013). During December 2012, several amendments to the general fishing law were introduced. One of the most important amendments is concerned with the ownership of the fishing licenses in those fisheries subjected to TAC. In this case, ITQs were adjudicated to a small group of industrial fishermen for 20 years with prorogation. Legislators also wanted to make sustainability the core of the new legal framework and indicated that management must explicitly consider the guidelines of the precautionary and ecosystem approaches. In this framework, the maximum sustainable yield (MSY) became the cornerstone by playing two main roles: it is a target RP for fishing management, and it also defines the threshold upon which the remaining surplus quota may be auctioned, allowing new actors into the fishery market. The latter was one of the most controversial issues in the legislative discussion during 2012. Several members of parliament and fisheries scientists exposed the shortcomings of a focus on MSY. Nevertheless, expert opinion had little impact on the final changes introduced to the general fishing law. The mandate of using MSY in the new fishing law was introduced to align fishery management, imposing greater specificity and less flexibility in the way TACs are set every year. During 2013, these modifications to the general fishing law came into effect, which has triggered a demand for estimating MSY-based RPs in each fishery resource managed by TAC.

Since the popularization of the concept of MSY by Schaefer (1954), fisheries science has seen a controversial debate about the usefulness (or lack thereof) of MSY in managing fisheries (e.g., Larkin, 1977; Mace, 2001; Punt & Smith, 2001). During the last few years, several studies have noticed that MSY estimates and MSY-based RPs are deeply connected to one single parameter: the steepness of the stock-recruitment relationship (see Brooks et al., 2010; Mangel et al., 2010; Hart, 2013). Steepness (h) is a common measurement of stock resilience and is defined as the recruitment produced by a spawning biomass at 20% of unfished spawning biomass, relative to the recruitment produced by unfished spawning biomass (Mace & Doonan, 1988). Steepness is firmly rooted in evolutionary ecology (Enberg et al., 2012) and it integrates several different aspects of the reproductive biology of fish stocks (Mangel et al., 2010). RPs are based on the reproductive biology of fishes and thus MSY and MSY-based RPs have a strong connection with steepness. Williams (2002) and Punt et al. (2005) showed that MSY and MSY-based RPs can be accurately predicted using only steepness. Recently, Mangel et al. (2013) derived an analytical function connecting MSY-based RPs and steepness using a production model. In such cases, MSY-based RPs are the only function of the steepness and other life history parameters have no influence. On the hand, complex age-structured models produced MSY-based RPs merging several biological and fishery aspects to compute MSY-based RPs. Thus, evaluation on MSY-based RPs estimates using production and age-structured models became relevant. This comparison will shed light on the influence steepness has on the estimates MSY-based RPs when using a complex age-structured model. In this paper, we compare estimates of MSY-based RPs from production models which only depend on steepness, with age-structured models for eleven fish demersal species harvested in Chile.

MATERIALS AND METHODS

Although, the most commonly used stock-recruitment functions (SRF) can be parameterized in terms of steepness (Hillary et al., 2012), we use the Beverton-Holt SRF (Beverton & Holt, 1957) because of its wide use in the demersal stocks in Chile. The Beverton-Holt SRF (BH-SRF) parameterized in terms of steepness (h) is as follows:

[R.sub.t+[DELTA]] = 0.8[R.sub.0]h[S.sub.t]/0.2[S.sub.0](1-h)+[S.sub.t](H-0.2), (1)

where R is the recruitment produced by the spawning biomass, S, after [DELTA] time (t). Likewise, [S.sub.0] and [R.sub.0] are the unfished levels for spawning biomass and recruitment, respectively. Steepness is a dimensionless parameter that takes values in the range [0.2-1] and defines the shape of the BH-SRF (see Fig. 1). Thus, h is useful in comparing resilience across fish populations. Stocks with low values of steepness have low recruitment compensation, and therefore the rebuilding of those stocks after overfishing will take longer and TAC should be kept small (Mace & Doonan, 1988).

Here, we are interested in three MSY-based RPs proposed in Mangel et al. (2013). They are the ratio between the biomass leading to MSY and the unfished biomass ([B.sub.msy]/[B.sub.0]), the ratio between the fishing mortality that gives the MSY and natural mortality ([F.sub.msy]/M) and the ratio of spawning biomass per recruit (SR) when the population is fished at [F.sub.msy] to the spawning biomass per recruit (SR) of an unfished population, SR([F.sub.msy])/[SR.sub.0] = [SPR.sub.msy]. According to Mangel et al. (2013) these MSY-based RPs derived from a production model can be computed as follows, using only steepness:

[B.sub.msy]/[B.sub.0] = [square root of (4h/1-h)]-1/4h/1-h-1, (2)

[F.sub.msy]/M = [square root of (4h/1-h)]-1 (3)

SR([F.sub.msy])[SR.sub.0] = [SPR.sub.msy] = [square root of (1-h/4h)]. (4)

For reasons of brevity, estimates from Eq. (2) to (4) will be called hereafter "production model". When assuming BH-SRF and the population dynamic is modeled using a production model, these three MSY- based RP are deterministic functions depending only on the steepness parameter. Mangel et al. (2013) also explored the connection between MSY-based RPs using an age-structured model, although no analytical solution is found in such cases. In age-structured models, life history parameters and fishing selectivity will have some influence in determining the MSY- based RPs. However, as discussed in Mangel et al. (2013), even in those cases estimates of MSY-based RPs from highly sophisticated age-structured models can be closely approximated using only the deter- ministic models in Eq. (2) to (4). Therefore, our goal in this paper was to compare the prediction of MSY-based RPs from a production model described by Mangel et al. (2013), with estimates from the age-structured models described below and implemented for eleven demersal stocks managed by TAC in Chile.

An age-structured per recruit model was implemented, using life history and fishing selectivity parameters reported in DER (2012) (Table 1) to estimate [B.sub.msy]/[B.sub.0],[F.sub.msy]/M and [SPR.sub.msy] in the following demersal stocks managed by TAC in Chile: 1) Patagonian toothfish (Dissostichus eleginoides); 2) yellow-nose skate (Zearaja chilensis); 3) alfonsino (Beryx splendens); 4) orange roughy (Hoplostethus atlanticus); 5) cardinalfish (Epigonus crassicaudus); 6) Chilean hake (Merluccius gayi gayi); 7) southern blue whiting (Micromesistius australis); 8) pink cusk-eel (Genypterus blacodes), northern stock; 9) pink cusk-eel, southern stock; 10) southern hake (Merluccius australis), and 11) Patagonian grenadier (Macruronus magellanicus). These stocks were selected because they are the most important demersal species harvested in Chile in terms of total catch and economic importance (Paya et al., 2014). Pelagic stocks were not considered here for the following reason. Small pelagic fish in Chile have a particular biology including fast growing species making difficult a straightforward comparison of results with demersal species. Although comparison of h between pelagic and demersal species is interesting, is beyond the scope of this paper.

The relative size for each stock, as well as the yield curve under variable fishing mortality levels, was modeled using an age-structured production model that incorporates the BH-SRF and the most feasible life- history parameters. The survival per recruit for unfished (na) and fished (Na) populations are defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where M is the instantaneous natural mortality rate assumed age-invariant, [f.sub.e] is the equilibrium fishing mortality rate, and Ta is the fishing selectivity at age a. Table (1) contains the data used to implement the age-structured model.

Individual length at age ([l.sub.a]) was modeled using the von Bertalanffy growth function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [l.sub.[infinity]] is the asymptotic length, k is the growth parameter and [a.sub.0] is the theoretical age at length zero. The weight at age ([w.sub.a]) was computed using the allometric function [w.sub.a] = [phi][([l.sub.a]).sup.b], were [phi] and b are parameters. Selectivity ([[tau].sub.l]) and maturity ([m.sub.l]) at length (l) were modeled independently using a logistic function such as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which was parametrized in terms of length at 50% (l50) and the length range between 0.5 and 0.95 probability ([DELTA][l.sub.50]). Selectivity and maturity at age (Ta,ma) were computed using a deterministic conversion between lengths and age. The calculation was conducted using the inverse von Bertalanffy growth function.

The spawning biomass per recruit in unfished (n) and fished (N) populations is given by

[spr.sub.n] = [[summation].sup.[infinity].sub.a=1][n.sub.a][w.sub.a][m.sub.a], [spr.sub.N] = [[summation].sup.[infinity].sub.a=1][N.sub.a][w.sub.a][m.sub.a], (7)

and the vulnerable biomass per recruit is expressed as a function of [[tau].sub.a] using:

[bpr.sub.n] = [[summation].sup.[infinity].sub.a=1][n.sub.a][w.sub.a][[tau].sub.a], [bpr.sub.N] = [[summation].sup.[infinity].sub.a=1][N.sub.a][w.sub.a][[tau].sub.a]. (8)

Equilibrium recruitment (Re) was modelled using the Beverton-Holt stock-recruitment function of equation (1),

[R.sub.e] = [alpha][S.sub.e]/1+[beta][S.sub.e] = [alpha][R.sub.e][spr.sub.N]/1+[beta][R.sub.e][spr.sub.N], (9)

where [S.sub.e] = [R.sub.e][spr.sub.N] represents the equilibrium spawning biomass, [alpha] = 4h/(1-h)[spr.sub.n] is the maximum juvenile survival rate, and [beta] =5h-1/(1-h)[R.sub.0][spr.sub.n] represents the density -dependent term defined as the compensation rate of juvenile survival (Myers et al., 1999). Replacing the expressions [alpha], [beta] and [S.sub.e] into equation (9) and solving for Re, the equilibrium recruitment ([R.sub.e]) for a given fishing mortality rate ([f.sub.e]), is given by:

[R.sub.e] = [R.sub.0]4h-(1-h)[THETA]/5h-1, (10)

where [THETA] = [spr.sub.n]/[spr.sub.N], is the ratio of spawners per recruit, and [R.sub.0] = [B.sub.0]/[spr.sub.n] represents the unfished recruitment as function of unfished spawning biomass ([B.sub.0]).

MSY-based RPs from Eq. (2) to (4) can be obtained from maximum equilibrium yield [Y.sub.e] for a plausible range of steady-state fishing mortality rate [f.sub.e]. Under specific steady-state recruitment, the equili-brium yield is given by,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where the total mortality at age [Z.sub.a] was computed as [Z.sub.a] = M + [f.sub.e][[tau].sub.a]. To calculate MSY-based RPs from associated life-history parameters, the most common approach is to plot [Y.sub.e] versus [F.sub.e] and find the fishing mortality value that maximizes [Y.sub.e]. Using a sequence of 10,000 equally spaced points falling in a range between 0 to 2 values of [F.sub.e], the maximum equilibrium yield was identified and subsequently mapped onto the [F.sub.e] range. Relative differences between production and age-structured models were computed across the range of h but only formally compared in each species at h = 0.7 (see Table 2). We choose this value of h for comparing methods, because its represent a standard or "average" stock recruitment relationship in demersal fish stocks.

RESULTS

Differences between production model and age-structured estimates increases with steepness in all the MSY-based RPs evaluated (Figs. 2-4). Estimates of [SPR.sub.msy] presented the smallest differences between production model and age-structured estimates across h, while [F.sub.msy]/M shows the largest differences between estimates. Predictions from the production model for [B.sub.msy]/[B.sub.0] and [SPR.sub.msy] are smaller than those estimates of the age-structured model across the range of h analyzed. On the other hand, production model predictions for [F.sub.msy]/M produced larger values than those predicted by the age-structured models, except alfonsino and pink cusk-eel (southern stock) where the production model produced smaller estimates (Figs. 2-4).

The relative differencce between production and agestructured model predictions at h = 0.7, can be found in Table 2. Cardinalfish and Chilean hake showed the biggest differences between estimates for [B.sub.msy]/[B.sub.0] while Patagonian toothfish and yellownose skate showed smaller ones. When we analyzed [F.sub.msy]/M at h = 0.7, most of the stocks show large differences between production and age-structured estimators. Only in the cases of orange roughy and cardinalfish, differences between estimates of [F.sub.msy]/M are negligible. In general, estimates of [SPR.sub.msy] from production model produces the best prediction for the age-structured estimates especially for Patagonian toothfish and yellownose skate, while the biggest differences between estimates were found in alfonsino and Chilean hake.

DISCUSSION

Equations (2) to (4) were analytically derived from a production model. Thus, models based on these equations will only approximate those estimates from an age-structured model as shown in our results. MSY-based RPs from age-structured models are determined mostly by the value of h, but also by the life history parameters and fishing selectivity. Here, [SPR.sub.msy] estimated from the age-structured model were accurately predicted by the production models, in most of the stocks analyzed and across the range of steepness. This means that this RP is mostly determined with information on h alone. For the selected demersal fish stocks in Chile, production models produced the best predictions in [SPR.sub.msy] and the worst in [F.sub.msy]/M. This is in line with the theory in Mangel et al. (2013) in which [SPR.sub.msy] is independent of the life history parameters or selectivity function and is strongly determined by h. The same authors added that [F.sub.msy]/M is highly influenced by h, but also by the fishing selectivity. MSY-based RPs estimated here for the demersal stocks in Chile confirm the theory in Mangel et al. (2013) in which [B.sub.msy]/[B.sub.0] and [SPR.sub.msy] are predicted almost exclusively using h, whereas for [F.sub.msy]/M the selectivity pattern became also important. In this context, [F.sub.msy]/M is the only RP assessed here that may incorporate fishing dynamics on the demersal stocks in Chile and the parameters defining the selectivity function in each stock may explain the reported differences between the production model predictions and those from the age-structured model.

In all demersal stocks managed with TAC in Chile, a common practice is to fix M and h in the stock assessment and the shape of the SRF is selected beforehand. M values are estimated using empirical models from meta-analysis, while h values usually lack a clear rationale and no further considerations (e.g., sensitivity analysis) are given to this parameter. As we discussed above, BH-SRF is commonly used in Chilean fisheries, although, in some cases, such as Chilean hake and Patagonian grenadier, the Ricker-SRF has also been implemented. Fixing M, h and pre-determining the shape of the SRF will certainly either fix MSY-based RPs when using a production model or highly influence them when using an age-structured model. For the latter case, fixing h, M and the shape of the SRF will only guarantee that observed data used in the stock assessment will have at best little influence on the estimates of MSY-based RPs of stock analyzed, as shown above.

Following Mangel et al. (2013), there are two main avenues we could follow in order to surpass these shortcomings. First, we could explicitly incorporate an extra parameter in the SRF, such as the compensation parameter used in the general SRF proposed by Shepherd (1982). This will allow enough flexibility for another parameter to be estimated directly from the data (Martell et al., 2008), then avoiding the specification of SRF beforehand. A good example of how to compare steepness parameters across SRFs is found in Hillary et al. (2012). A second approach is to avoid the assumption of fixed h. We are aware that the problems associated with fixing h in the stock assessment are not exclusive to Chilean fisheries. Fishery management systems based on MSY, like those in the USA, are also facing this problem. Although in the USA for some data-rich fisheries, h is estimated within the stock assessment model, still an important proportion of their stock assessments consider a fix value of h. Yet, when h is fixed in those assessments, a clear and explicit rationale is given for this parameter, and it is usually approximated by using the meta-analysis described in Rose et al. (2001) and Dorn (2002). In addition, a sensitivity analysis of h is commonly conducted in those stocks.

In the demersal stocks managed by TAC in Chile, fishing monitoring and stock assessment have com- parable standards to those implemented in developed countries (Paya et al., 2014). Fishing monitoring is a lengthy process and well established. Stock assessments use sophisticated models with an appropriate treatment of uncertainty in the state variables (e.g., spawning biomass). In addition, the assessments are regularly scrutinized through an international peer-review process. This means that, if we are following the same MSY-based RPs route as the USA, we should also follow a similar procedure in dealing with h. This means, we should estimate h as best as possible in those fisheries with adequate information. In other cases, an approximation for h is desirable when meaningful estimates are impossible to obtain. Although this may seem simple, h is one of the most difficult parameters to estimate within the stock assessment models, even in data-rich stocks. This is because the usual lack of information in the stock-recruitment relationship and the confounding effect between h with other life history parameters (e.g.,M). However, h can be estimated in stocks where a good contrast between high and low spawning abundance is available (Lee et al., 2012).

Meta-analysis is a useful tool for proposing informative priors of h (Shertzer & Conn, 2012). Meta-analysis has been widely applied in fisheries science to approximate parameters that in other cases are difficult to obtain (Minte-Vera et al., 2005). The use of meta-analysis rests upon the assumption that certain parameters such as h, can be interchangeable between stocks with similar life history attributes. Mixed effect models, such as in Myers et al. (2002) and hierarchical Bayesian analysis described in Dorn (2002), are promising methods for estimating priors of h for fish stocks in Chile. The main drawback of such meta-analysis methods is that available estimates of h coming almost exclusively from highly valuable species inhabiting the northern hemisphere (see compilation of species in Shertzer & Conn, 2012). This may challenge the use of meta-analysis for h in Chilean fisheries and thus careful selection of available data for estimating h is recommended. In applying such meta-analysis, special attention needs to be paid to the double use of data explained in Minte-Vera et al. (2005) and the manner to select the group of species to be included in the analysis (Myers et al., 2002). In addition to the meta-analysis, Mangel et al. (2013) indicated that possible values for h should be constrained with the used value of M and observed mean age T using the following expression exp(TM)/M [approximately equal to] 4h/1-h. This approximation will be useful in delimiting a solution for h to have, at least, consistency with M and T used in the stock assessment. The procedure described using meta-analysis and constraining of h values can be useful in the context of proposing priors for estimating h and to approximate this parameter in those cases where it cannot be estimated within the stock assessment model.

In Chile, stocks with extensive data available such as southern hake, Chilean hake and Patagonian grenadier are good candidates to attempt an estimation of h within the stock assessment model. This could be done without using extra funding or human resources. For the rest of the species listed above, we may not be able to get meaningful estimates of h in the short-term. For those cases, the approximation using meta-analysis described above can be implemented, although we recognize that in extreme cases it will be impossible to have an approximation for h. The endemic and/or unique biology of some fish species fished in Chile, such as orange roughy, alfonsino, Patagonian toothfish and cardinalfish, violate the assumption of interchan-geability of h between stocks on which meta-analysis rests. In such species, additional alternatives to approximate h need to be explored. In the meantime, fishery management need to be fully aware of the limitations of MSY-based RPs in each stock managed with TAC in Chile. We recommend to develop workable protocols to deal with those stocks in which meaningful estimates of h are difficult (if not impossible) to obtain with the current level of knowledge. Fully considering the uncertainty in mana-gement actions is also desirable in the light of the precautionary approach in FAO guidelines. The Chilean MSY-based management system in which h and M are fixed, gives a false sense of certainty about the specific productivity at which each stock can be harvested. This is a very undesirable situation because incorrect assumptions about h will lead to inaccurate estimates of MSY-based RPs. When fixing h in the stock assessment, the rest of the data contributes little to the estimation of MSY-based RPs, and thus alternatives to estimate or approximate h in Chilean stocks need to be explored.

During 2014, the Instituto de Fomento Pesquero (IFOP) conducted a project on biological reference points in Chilean fisheries subjected to annual quotas (for further details see Paya et al., 2014). The project included extensive work to first classify stocks in tiers (groups) according to quantity and quality of the data available (poor, medium and rich data), and then selecting the best method to estimate MSY-based RPs in each tier. Most of the stocks illustrated here, except common hake, were classified in such tier in which proxy quantities are used as reference points. This means that these stocks contain enough information to conduct an age-structured stock assessment, but none of them has reliable information to estimate h within the stock assessment model (except common hake). Extensive simulation analyses were carried out to assess the robustness of such proxy measurements of MSY-based RPs. Proxies seem an appropriate way to deal with in these fisheries, although they need to be interpreted cautiously. The proxies implicitly assume a value of steepness that may either under-or overestimate the stock-specific productivity, which could distort the population status and total allowable catch estimates. We do not offer a solution for this problem; rather we emphasize the high dependence of MSY-based RPs and steepness on most of selected Chilean stocks. Thus, results obtained here should be seen as complementary to the simulations analyses presented in Paya et al. (2014).

Our analysis reveals the complexity of implementing the Chilean Fishing and Aquaculture Law into real situations where data are incomplete and/or absent. We also show that even in data rich fisheries, the uncertainty surrounding the estimation of h and, by extension, MSY and MSY-based RPs indicates that MSY should have been considered as limit rather than objective RP. The above is the recommendation of the international and Chilean fishery science community. Insisting in using MSY as target RP is clearly against the precautionary and ecosystem approaches that should guide the decision making process, and does not secure sustainability and conservation. In addition to our advice on how to deal with h values in Chilean fisheries, we posit that more research is needed to include a full ecosystem approach into the RPs, since the current efforts are based only in the traditional single-species approach. For example, fisheries of low trophic levels should be managed as to fulfil the requirements of other fishing resources located in higher trophic levels as well as other predators such mammals and birds.

DOI:10.3856/vol44-issue2-fulltext-19

Received: 10 December 2015; Accepted: 21 March 2016

ACKNOWLEDGMENTS

We are grateful to two anonymous reviewers for their valuable comments and suggestions. This manuscript was funded by CONICYT/FONDECYT Post-doctoral project number 3130425 and CONICYT project CAPES FB 0002. J.C. Quiroz was awarded with the CONICYT BECAS-CHILE scholarship from the Chilean Government and the Top-Up University of Tasmania Flagship Postgraduate Scholarship.

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Rodrigo Wiff (1), J.C. Quiroz (2,3), Sergio Neira (4), Santiago Gacitua (1) & Mauricio A. Barrientos (5)

(1) Center of Applied Ecology and Sustainability (CAPES) Pontificia Universidad Catolica de Chile, Santiago, Chile

(2) Institute for Marine and Antarctic Studies (IMAS), University of Tasmania, Hobart, Australia

(3) Division de Investigacion Pesquera, Instituto de Fomento Pesquero (IFOP), Valparaiso, Chile

(4) Copas Sur-Austral, Departamento de Oceanografia, Universidad de Concepcion, Concepcion, Chile

(5) Instituto de Matematicas, Pontificia Universidad Catolica de Valparaiso, Valparaiso, Chile

Corresponding author: Rodrigo Wiff (rodrigo.wiff@gmail.com)

Corresponding editor: Patricio Arana

Caption: Figure 1. Characterization of Beverton-Holt stock recruitment relationship, using different steepness values.

Caption: Figure 2. Estimates of [B.sub.msy]/[B.sub.0] across steepness from production (grey lines) and age-structured models (black lines).

Caption: Figure 3. Estimates of [F.sub.msy]/M across steepness from production (grey lines) and age-structured models (black lines).

Caption: Figure 4. Estimates of [SPR.sub.msy] across steepness from production (grey lines) and age-structured models (black lines).
Table 1. Life history and fishery parameters used to implement the
age-structured per recruit model in each demersal stock assessed.
Growth parameters are asymptotic length ([l.sub.[infinity]]),
intrinsic growth rate (K) and age at length zero ([a.sub.0]). Max.
Age, represents the lifespan in years and logistic model for maturity
ogive is described by the slope and shape parameters ([l.sub.50] and
[DELTA][l.sub.50]). The length at weight relationship (L-W) is
represented by two parameters ([phi] and b) and [[tau].sub.50] is the
length at 50% of selectivity.

                                                Growth

                [l.sub.[infinity]]          K          [a.sub.0]
                       (cm)          ([year.sup.-1])    (year)

Patagonian            176.2               0.06           -1.65
  toothfish
Yellownose            128.3               0.11           -0.51
  skate
Alfonsino             63.69               0.09           -2.57
Orange roughy         52.13               0.03           -1.72
Cardinalfish          35.25               0.08           -3.39
Chilean hake          80.40               0.14           0.92
Southern blue         59.64               0.16           -2.51
  whiting
Pink cusk-eel         117.40              0.19           -0.84
  (north)
Pink cusk-eel         123.45              0.15           -1.78
  (south)
Southern hake         115.00              0.12           -1.60
Patagonian            105.50              0.17           -0.81
  grenadier

                                  Maturity

                  Max     [l.sub.50]       [DELTA]
                Age (A)      (cm)      [l.sub.50] (cm)

Patagonian        36         85.2           14.0
  toothfish
Yellownose        34        103.9           11.1
  skate
Alfonsino         19         33.1            1.9
Orange roughy     90         38.0            2.2
Cardinalfish      42         23.2            4.0
Chilean hake      12         38.5            5.1
Southern blue     24         38.2            2.6
  whiting
Pink cusk-eel     14         82.2            2.0
  (north)
Pink cusk-eel     14         82.2            2.0
  (south)
Southern hake     24         72.0           15.0
Patagonian        14         54.0           28.0
  grenadier

                   Mortality             L-W

                       M          [phi]([g.sup.*]    b
                ([year.sup.-1])    [cm.sup.-1])

Patagonian           0.15             4.9E-03       3.17
  toothfish
Yellownose           0.12             2.6E-03       3.22
  skate
Alfonsino            0.28             2.4E-02       2.89
Orange roughy        0.05             8.8E-02       2.65
Cardinalfish         0.08             2.0E-02       2.90
Chilean hake         0.30             1.3E-02       2.83
Southern blue        0.20             8.3E-04       3.50
  whiting
Pink cusk-eel        0.27             2.5E-03       3.15
  (north)
Pink cusk-eel        0.23             2.3E-03       3.16
  (south)
Southern hake        0.21             2.4E-03       3.25
Patagonian           0.35             3.4E-03       2.96
  grenadier

                 Selectivity

                [[tau].sub.50]
                     (cm)

Patagonian          86.53
  toothfish
Yellownose          73.83
  skate
Alfonsino           36.66
Orange roughy       39.90
Cardinalfish        23.65
Chilean hake        25.01
Southern blue       49.57
  whiting
Pink cusk-eel       89.60
  (north)
Pink cusk-eel       84.79
  (south)
Southern hake       84.65
Patagonian          54.23
  grenadier

Table 2. Estimates of MSY-based reference points from production
and age-structured model (Age-S) predicted at h = 0.7 across
species analyzed. DEV is the relative difference (in percentage)
between production and age-structured model estimates. [B.sub.msy]
/[B.sub.0] is the ratio between the biomass leading to MSY to the
unfished biomass. [F.sub.msy]/M is the ratio between the fishing
mortality that gives the MSY and natural mortality. [SPR.sub.msy]
is the ratio of spawning biomass per recruit (SR) when the population
is fished at [F.sub.msy] to the spawning biomass per recruit of an
unfished population.

                [B.sub.msy]/[B.sub.0]   [F.sub.msy]/M

Production      0.247   DEV %   2.055       DEV %
(h = 0.7)       Age-S           Age-S

Patagonian      0.273   10.7    0.880       -57.2
  toothfish
Yellownose      0.262    6.3    0.900       -56.2
  skate
Alfonsino       0.347   51.7    9.389       356.9
Orange roughy   0.288   16.8    2.040       -0.7
Cardinalfish    0.346   40.3    1.900       -7.5
Chilean hake    0.357   44.9    0.773       -64.3
Southern blue   0.317   28.5    1.815       -11.7
  whiting
Pink cusk-eel   0.289   17.3    1.641       -20.2
  (north)
Pink cusk-eel   0.322   30.7    3.022       47.0
  (south)
Southern hake   0.314   27.5    1.581       -23.1
Patagonian      0.306   24.0    0.966       -53.0
  grenadier

                        [SPR.sub.msy]

Production      0.327
(h = 0.7)       Age-S       DEV %

Patagonian      0.351        7.2
  toothfish
Yellownose      0.341        4.2
  skate
Alfonsino       0.441       34.8
Orange roughy   0.364       11.3
Cardinalfish    0.416       27.1
Chilean hake    0.426       30.2
Southern blue   0.390       19.1
  whiting
Pink cusk-eel   0.365       11.6
  (north)
Pink cusk-eel   0.395       20.6
  (south)
Southern hake   0.388       18.5
Patagonian      0.380       16.1
  grenadier
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Title Annotation:Research Article
Author:Wiff, Rodrigo; Quiroz, J.C.; Neira, Sergio; Gacitua, Santiago; Barrientos, Mauricio A.
Publication:Latin American Journal of Aquatic Research
Article Type:Ensayo
Date:May 1, 2016
Words:6106
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