We propose a novel Bayesian hierarchical structure of state-space surplus production models that accommodate multiple catch per unit effort (CPUE) data of various fisheries exploiting the same stock. The advantage of this approach in data-limited stock assessment is the possibility of borrowing strength among different data sources to estimate reference points useful for management decisions. The model is applied to thirteen years of data from seven fisheries of the lebranche mullet (

Proponemos una nueva estructura jerárquica bayesiana para modelos de producción excedente espacio-temporales que permite incorporar datos de captura por unidad de esfuerzo (CPUE) de diversas fuentes para varias pesquerías que explotan el mismo stock. La ventaja de este enfoque en la evaluación de stocks con datos limitados es la posibilidad de reforzar las estimaciones a partir de diferentes fuentes de datos para estimar puntos de referencia útiles para las decisiones de gestión. El modelo se aplica a trece años de datos de siete pesquerías de la población meridional de lebranche (

Stochastic versions of biomass dynamic models are called state-space models. These models have a hierarchical structure which simultaneously accounts for uncertainties in the time and space dynamics of biomass production and for errors in the observational process of some abundance indices (e.g. Catch per Unit Effort, CPUE) that relate data to the (unknown or latent) biomass.

Surplus Production (SP) models are simple but robust non-linear models for stock assessment that are widely used to model biomass dynamics in state-space models for exploited fish populations (

The search for improvements in SP models has grown over the last few years. Revisions have pointed towards limitations related mostly to nonconformities in utilized data rather than the predictive capability of the underlying model (

It is an essential premise in the observational component of state-space models that CPUE has a known proportionality relation to stock biomass. However, stock concentration profile, changes in fishing power, gear type, season and fishing ground can all affect this relation over time and space and seriously bias biomass dynamic predictions if not properly accounted for (

Another path has been to ignore fishing effort all together and rely on catch-only data (

One motivation for this study is to try a new alternative for dealing with fisheries assessment in data-limited situations. It consists in retaining CPUEs of various fishing fleets (characterized by differences in gear type, fishing operation and fishing ground) and model them simultaneously as multiple but integrated observation models. The only requirement is that all fleets exploit the same stock biomass, even if time-windows within a year or over time do not coincide.

The Bayesian approach to perform inference in fishery assessment and management is very appealing because, conditional on the proposed model, it provides direct estimates of biological reference points while automatically retaining and integrating all sources of (data and process) uncertainties (

The lebranche mullet

This is also one of the most frequent and abundant fish species in the south and southeast regions of Brazil, representing an important cultural and historical artisanal fishery for this region (

Since 2004 the species has been classified as overexploited by the Brazilian Government (

This paper assesses the current status of the lebranche mullet southern population distributed along the southern and southeastern shelf regions of Brazil, using hierarchically structured Bayesian state-space models. Given the need to integrate various data sources within a single structure, we divide this goal into three steps: (i) develop a robust hierarchical Bayesian state-space biomass model capable of integrating multiple CPUEs; (ii) use this model to estimate the reference points together with the associated uncertainties; and (iii) provide a modelling structure to make predictions that can help integrate management actions among all fisheries targeting the mullet (

The proposed state-space models were based on the Bayesian approach, considering the integration of uncertainties involved in the latent process dynamic (biomass) and errors associated with the observational component consisting of various CPUE data collected from a multitude of fishery statistics (see

The deterministic component of the process dynamic was defined in discrete time (annual) variation, where the biomass (B_{t}) at the start of year t, is a known function of the previous year’s biomass (B_{t−1}) and total catch (C_{t−1}), parameterized by the intrinsic population growth rate (r), the average unfished stock size or support capacity (K) and a shape parameter (z).

$${\text{B}}_{\text{t}}={\text{B}}_{\text{t\u20131}}+{\text{rB}}_{\text{t\u20131}}\left(1-{\left(\frac{{\text{B}}_{\text{t\u20131}}}{\text{K}}\right)}^{\text{z}}\right)\u2013{\text{C}}_{\text{t\u20131}}$$ |

This equation is known as the Pella and Tomlinson SP model (

For computational convenience and to reduce parameter confounding, the model was reparametrized in terms of relative abundance (B/K=P) (_{t} were further included as independent, identically distributed (iid) Gaussian random variables with mean zero and process variance σ_{µ}^{2}.

$${\text{P}}_{\text{t}}=\left({\text{P}}_{\text{t\u20131}}+{\text{rP}}_{\text{t\u20131}}\left(1-{\text{B}}_{\text{t\u20131}}^{\text{z}}\right)\u2013\frac{{\text{C}}_{\text{t\u20131}}}{\text{K}}\right){\text{e}}^{{\mu}_{\text{t}}}$$ |

The observational component in the state-space model was kept explicit for the CPUE (I_{ti}) information provided by each fishery i in year t. These fisheries, each with a specific gear and fishing strategy but all exploiting the same stock biomass, were classified in terms of gear (purse seine and bottom gill nets), area of operation (estuaries and sea), and type of labour relation (artisanal and industrial). Hence, it was assumed that all observed CPUEs relate to the same overall stock, B_{t}, but each with a fishery-specific catchability coefficient, q_{i}, for i = 1, 2, ..., F, where

$${\text{I}}_{\text{ti}}=\left({\text{q}}_{\text{i}}{\text{B}}_{\text{t}}\right){\text{e}}^{{v}_{\text{ti}}}$$ |

A stochastic component was further included with the random quantities v_{ti} which, conditional on B_{t}, are assumed iid Gaussian with mean zero and process variance σ^{2}. The lognormal multiplicative structure used in both process and observation models has been used in this form by other authors as well (

All model parameters were estimated using MCMC implemented in BUGS code, using the

All information considered is composed of data gathered from official and unofficial bulletins published by governmental and private research agencies, as follows: the Research and Management Centre of Fishing Resources in South and Southeast Coastline (CEPSUL/ICMBio), the Federation of Fishermen of Santa Catarina (FEPESC), the Research and Management Centre of Estuary and Lagoon Fishery Resources (CEPERG), the Fishery Institute of São Paulo (IP/SP), the Fishery Studies Group from the University of Vale do Itajaí (UNIVALI/GEP) and the IBAMA Regional Office of Rio de Janeiro (IBAMA/RJ).

The time window for this study covers an overall period of 13 years, from 2000 to 2012. Annual total catches come from different fishing methods in industrial (purse seine and bottom gill net) and artisanal (bottom gill net) fisheries. Based on reliability considerations, we only used data from the industrial and artisanal fishery landings monitored in the states of São Paulo and Santa Catarina as follows: A, industrial purse seine fleet from Santa Catarina State; B, Industrial fishery fleets from Santa Catarina State (except purse seine); C, all industrial and artisanal fishery fleet from São Paulo State (except the fleet classified as D); D, all the industrial and artisanal fishery fleet from São Paulo State that operates only off the southeastern and southern coast of Brazil; E, the artisanal fishery fleet from São Paulo State that operates only in estuarine waters (except the fleet classified as G); F, the industrial and artisanal gill net fleet from São Paulo State that operates only off the southeastern and southern coast of Brazil; and G, the Artisanal gill net fleet from São Paulo State that operates only in estuarine waters.

A key component in a Bayesian analysis is the inclusion of previous knowledge about model parameters in the form of informed prior distributions. Regarding the SP model parameters, the required priors consist of the probability distribution for the support capacity (K), the maximum intrinsic growth rate (r) and the shape parameter (z). Based on recommendations by

Model | Parameter | Description | Prior |
---|---|---|---|

Schaeffer | K | Carrying capacity | Lognormal (10, 0.5)I(15000,) |

r | Intrinsic growth rate | Lognormal (0.4, 0.5) | |

q_{i} |
Catchability | Uniform (0, 0.00001) | |

Process error variance | Inverse-Uniform (0, 5) | ||

Observation error variance | Inverse-Uniform (0, 5) | ||

Pella-Tomlinson | K | Carrying capacity | Lognormal (10, 0.5)I(15000,) |

r | Intrinsic growth rate | Lognormal (0.4, 0.5) | |

z | Shape parameter | Uniform (1, 5) | |

q_{i } |
Catchability | Uniform (0, 0.00001) | |

Process error variance | Inverse-Uniform (0, 5) | ||

Observation error variance | Inverse-Uniform (0, 5) |

The prior distribution for K has its lower limit fixed at 15000 t, since this is close to the historical maximum reported landings of 13375 t. A prior 95% credibility interval ranging from about 15500 to 60400 t (and a mean at 28600 t) covers the range of most plausible support capacity from a biological standpoint.

Similarly, for r, the proposed prior 95% credibility intervals ranges from about 0.6 to 3.9 (with a mean about r=1.7) and covers all reasonable growth rates for this species. The uniform priors for q_{i} only establish the order of magnitude for the relation between CPUEs (measured in tonnes per unit effort) and biomass (measured in thousands of tonnes). Finally, uniform prior distributions are defined for the precision parameters rather than variances (i.e. precision = 1/variance) because this is the parameterization used in JAGS. In all cases, posterior estimates remained well within the prior windows, away from the extremes, suggesting that these prior specifications did not conflict with information provided by the data likelihood.

The

Model | DIC | pD |
---|---|---|

Schaeffer | –150.815 | 25.437 |

Pella-Tomlinson | –150.158 | 25.870 |

The documented time series of lebranche mullet catches is summarized in

In

Posterior predictions of CPUE (

Posterior distributions for model parameters (

Model | Parameter | Mean | Sth. deviation | 2.5% | 50% | 97.5% |
---|---|---|---|---|---|---|

Schaeffer | K | 30139.53 | 10932.14 | 18222.95 | 27328.04 | 58910.14 |

r | 1.009 | 0.552 | 0.198 | 0.917 | 2.328 | |

B_{MSY} |
15069.77 | 5466.07 | 9111.48 | 13664.02 | 29455.07 | |

F_{MSY} |
0.505 | 0.276 | 0.099 | 0.459 | 1.164 | |

MSY | 6903.21 | 3114.61 | 1640.01 | 6594.10 | 13889.35 | |

Pella-Tomlinson | K | 29765.57 | 10588.31 | 18081.71 | 27088.99 | 57586.30 |

r | 0.634 | 0.357 | 0.143 | 0.565 | 1.532 | |

z | 2.724 | 1.156 | 1.064 | 2.586 | 4.859 | |

B_{MSY} |
18059.22 | 6694.32 | 10392.98 | 16440.87 | 35483.47 | |

F_{MSY} |
0.432 | 0.215 | 0.102 | 0.400 | 0.936 | |

MSY | 7148.15 | 3087.48 | 2037.94 | 6829.71 | 14194.20 |

To assess the impact of all fisheries combined on the historic evolution of the stock, the time dynamics of the ratios B_{t}/B_{MSY} (_{t}/F_{MSY} (_{MSY} from 2010 for the Pella and Tomlinson model (posterior means as point estimates), the threshold is estimated to have been crossed only in the last year of the observed series for the Schaeffer model. However, there are wide credibility intervals covering the threshold line in both cases. This indicates high uncertainty and weak statistical evidence to support this conclusion in either of them. Regarding the estimated exploitation rate, both models suggest that in recent years these ratios are lower than one. However, these conclusions are also weak since marginal credibility intervals are wide and do not support conclusive statistical evidence.

Analysing the plots of CPUE versus effort for the seven fisheries (not shown), we might identify weak one-way-trip–type behaviour in fisheries E, F and G, while in fisheries A and B no such pattern is apparent. This evaluation is, of course, quite subjective. It would further be hard to justify among stakeholders why data from some particular fisheries should by ignored in any assessment exercise.

To partially circumvent limitations due to the relatively short time window (13 years) and also to minimize unreliability in estimation due to the one-way-trip phenomenon, we chose to incorporate seven different fisheries simultaneously into a single model. Although all these fisheries explore the same stock, each has its own characteristics, so the impact of some of the unwanted features is expected to be diminished.

For illustrative purposes, let us suppose for a moment that the requirements for a regression-based CPUE standardization among fisheries are not met. Furthermore, let us suppose that a preliminary visualization of all available CPUE series shows a decreasing pattern over time for many, while some are stable and a few others show a reversed increasing pattern. Which CPUE series should we select for assessment? This would be a difficult call to make and to justify politically.

By simply including all CPUEs, as we did in our proposed model, the dominant pattern will eventually drive the fitting process and biomass dynamic predictions. Discrepancies among different CPUE patterns will affect the uncertainties (i.e. posterior variance) for estimated parameters and derived reference points such as MSY. The more discrepancies, the larger posterior variances will become, and vice-versa. All very much in line with what common sense would dictate.

The historic development of lebranche mullet exploitation is summarized with the phase plot in _{t}>B_{MSY}) and a low exploitation rate (F_{t}<F_{MSY}) in year 2000. With the exception of 2001 and 2002, when biomass showed a sudden (unexpected) decrease, the stock continued to display a large biomass until 2009, while the exploitation rate increased steadily. From 2009 to 2012, biomass stayed below B_{MSY}, with the exploitation rate remaining high (F_{t}>F_{MSY}) until 2012, when it fell below F_{MSY}. This suggests that this reduction in exploitation rate was possibly linked mostly to economic considerations rather than to low stock density. Otherwise, since stock density has been falling, one would expect a still further increase in the exploitation rate in order to sustain catches. The phase plot suggests that care must be taken to guarantee sustainability for the lebranche mullet stock. If low exploitation rates could be maintained for a longer period, one might expect stock biomass to gradually move again towards the region in the graph were it was in the year 2000.

Since mullet performs reproductive aggregations at the time of its fishing season, CPUE can remain artificially stable while the stock is in fact being fished down. This phenomenon, known as hyperstability, can misinform about the stock status. It becomes a risk for stock collapse in cases where the density of stock aggregation remains above the economic threshold (

Once again, our attempt to prevent hyperstability from contaminating the estimates reinforces the argument for including CPUEs for different fisheries simultaneously into the model. After all, if we combine seven mullet fisheries from various regions, some using different gears, fishing practices and fishing grounds but all exploring the same stock, a Bayesian hierarchical model becomes a very handy tool for integrating them all into a single, consistent structure for estimating biomass time dynamics. By doing so, we expect to be able to produce more robust and reliable reference point estimates to guide fisheries management. In this specific case study, the wide posterior credibility intervals on key reference points indicate that conclusive evidence cannot be drawn yet, although the estimated trends suggest reasons for concern.

We are indebted to all technician involved in different fisheries monitoring programs in Southeast and South of Brazil, their hard work and assistance in collecting and compiling data made this work possible. This research was funded by the National Research Council (CNPq) and Ministry of Fisheries and Aquaculture (MPA) (CNPq/MPA n. 42/2012).