1 INTRODUCTION

Albacore, Thunnus alalunga Bonnaterre, is a widely distributed tuna species (Yeh et al., 1990; Sun et al., 2002; Viñas et al., 2004), which constitutes a commercially important stock with recruitment, growth, and mortality rates as all varying continuously over time. T. alalunga is highly migratory, cosmopolitan, and inhabits tropical and temperate pelagic oceans waters (Yeh et al., 1990; ICCAT, 1999; Viñas et al., 2004). The International Commission for the Conservation of Atlantic Tunas (ICCAT) has delimited three groups of albacore stock in the Atlantic: the northern and southern Atlantic stocks (Separated at 5°N) and the Mediterranean stock (ICCAT, 1999). Since the early 2000s the southern Atlantic stock has been considered to a have big development potential. Its catch was stabilized at about 30 000 t from 1988 to 2001, with a peak of 40 630 t, but the average has since declined to 21 000 t for the past five years (ICCAT, 2013). International Seafood Sustainability Foundation (ISSF) and ICCAT (2012) reported that the T. alalunga stock is in a mildly over-fished state, and that measures should be taken for the sustainable utilization of this stock in future.

Many different methods have already been used to assess the albacore (T. alalunga) stock, such as the surplus production model (SPM), the age structured production model (ASPM), and the delay-difference model (D-DM) (Yeh et al., 1990; Sun et al., 2002; Viñas et al., 2004; ICCAT, 2012; Zhang et al., 2015a). In the ASPM, detailed biological information is required, and the SPM lacks biological realism (Quinn and Deriso, 1999; Haddon, 2011). The delaydifference model (D-DM) considers biological information in a simple way (Deriso, 1980; Musick and Bonfil, 2005; Collette et al., 2006), in which the fishing mortality rate is used independently of the age of the fish at recruitment age and older (Quinn and Deriso, 1999; Su and Peterman, 2012). It has already been successfully used to estimate stock size in sharklike fishes (Musick and Bonfil, 2005), fishes with long life cycles (Pallare and Restrepo, 2003; Jensen et al., 2009), lobster (Hall, 1997), and prawn (Dichmont et al., 2003). The delay-difference equation is a biomass dynamic model, but with biologically meaningful (directly measurable) parameters and also accounts for basic time delays due to growth and recruitment (Hilborn and Walters, 1992). The discrete time formulation does not work well for populations with the consideration of continuously varying time delays of biological processes (Quinn and Deriso, 1999; Walters and Martell, 2004; Musick and Bonfil, 2005). Walters and Martell (2004) modified this delaydifference model, then Walters (2011), proposed a continuous time delay-difference model (CTDDM), in which recruitment, growth and mortality rates are treated as varying continuously over time. This study provides an application of CTDDM for this stock assessment in the southern Atlantic.

CTDDM treats recruitment, growth and mortality rates as all varying continuously over time, which is considered as a theoretical bridge between simple surplus-production models and complicated agestructured models (Walters and Martell, 2004). The dynamics of CTDDM characterize some commercially important stocks, such as shrimps and smaller tunas, and treats recruitment, growth, and mortality rates all as varying continuously over time (Walters and Martell, 2004; Musick and Bonfil, 2005; Walters, 2011). Being a bridge between ASPM and SPM, CTDDM is an alternative model necessary for appropriate albacore stock assessment (Walters and Martell, 2004). CTDDM provides an extremely compact performance of the exact dynamics of total numbers and biomass for the highly migratory cosmopolitan fish populations, such as albacore (T. alalunga). The CTDDM performs better than the discrete models for stocks that reproduce continuously over time (Walters, 2011). The CTDDM has not yet been applied in this albacore (T. alalunga) stock, and it may turn out to be a useful alternative model necessary to provide further information on this important international fishery.

In this study, the Beverton-Holt stock-recruitment relationship (SRR) model is used to explain the relationship between spawning biomass and recruitment. The maximum sustainable yield (MSY) and other biological reference points (e.g., B_MSY, E_MSY, and F_MSY) are estimated. The MSY will change through time as the mixture of fishing mortalities by class changes. CTDDM were modified from the discrete-time delay-difference model, in which recruitment, growth, and mortality rates are treated as varying continuously over time. It is the first time that CTDDM has been used to assess the Atlantic albacore (T. alalunga) fishery. By using it, we hope to provide an alternative approach to the stock assessment of albacore (T. alalunga) and fill the gaps in simple surplus production models and complex fully age-structured models. The estimated results have been analyzed to verify the proposed methodology and provide reference information for sustainable management of the southern Atlantic albacore stock.

2 MATERIAL

Data from 1975 to 2011 on the southern Atlantic albacore (T. alalunga) stock (ICCAT, 2012) have been used in this study. According to ICCAT, the southern Atlantic albacore stock is one of the biggest tuna populations in the world; the total catches of this stock in the previous 30 years ranged from 15 000 t to 40 000 t, which was mostly attributed to longline fisheries (ICCAT, 2011). The standardized catch per unit effort (CPUE) based on the Chinese Taipei longline fishery significantly indicates the relative abundance index of the southern Atlantic albacore (T. alalunga) stock (ICCAT, 2012). The length-weight relationship is from Penney (1994); the estimates of Von Bertalanffy growth parameter K, W_∞, mean body weight at age, and a recruitment age of 5 years are from Lee and Yeh (2007); an instantaneous natural mortality is assumed to be constant at 0.3 per year for all age classes for this Atlantic albacore (ICCAT, 2012).

3 METHOD 3.1 Models

Continuous time delay-difference model (CTDDM) provides an extremely compact performance of the exact dynamics of total numbers and biomass for agestructured populations.

This model differs from previous such models in that the continuous-time rather than the discrete-time form of the biomass dynamics equation is considered. Summary statistics for the “known” (Y) or “estimated” (N) parameters and Biological Reference Points (BRPs) are shown in Table 1.

In the discrete time delay-difference model (DDM), the mean body weight of the fish follows a difference relationship for age a≥k that leads to the Ford-Brody version of the von Bertalanffy growth model (Hilborn and Walters, 1992). Schnute (1985) proposed that the point estimates of δ, ρ, and mean body weight at age k were obtained from von Bertalanffy growth model.

where s_t is the overall survival for year t, a is age, w_a is body weight at aged a, and δ and ρ are parameters. The Stock-Recruitment Relationship (SRR) suggested by Schnute (1985) and Fournier and Doonan (1987), has three-parameters. Hilborn and Walters (1992) suggested the Beverton-Holt SRR models to explain the relationship between spawning biomass and recruitment. The discrete time formulation of D-DM results in simple predictions of equilibrium biomass and numbers (Musick and Bonfil, 2005). CTDDM provides an extremely compact performance of the exact dynamics of total numbers and biomass for age structured populations. CTDDM is expressed as follows:

where B_t is the stock biomass for year t, and R_t recruitment for the year t in the Beverton-Holt SRR model, and N_{a, t} is stock number, N_a, t=s_t–1 N_{a–1, t–1}; K and W_∞ are von Bertalanffy growth coefficients and asymptotic weight, respectively; biomass B_∞ and the recruitment R_∞ are the asymptotic values. The total instantaneous mortality rate Z(t)=F(t)+M, assumed to vary over time with changes in fishing mortality rate F(t). Any analytical solution of CTDDM was difficult to evaluate because of the integral form of the model expression. Under the constant R(t) and F(t) conditions, and taking differential Eqs.8 and 9 to equal zero, then C_∞=FB_∞, and N_∞=R/Z, B_∞=BPR×R, where BPR is biomass per recruit; α, β, and κ are the parameters of the Beverton-Holt SRR model. Under the conditions as the recruitment R(t) and fishing mortality rate F(t) are treated as piece-wise constant over short time intervals Δt, the N(t+Δt) is the exact predictions of population numbers at the end of each interval given starting values, and B(t+Δt) is the solution (analytical solution) of continuous time delay-difference population model.

where B_∞ and N_∞ are the asymptotic values, H^* and s^* are transitional parameters, while ρ^* and α^* do not change except in case where the growth curve varies over time.

3.2 Predicted catch values from CTDDM compared with those from D-DM

In this study, the predicted results from CTDDM will be compared with those from the discrete time delay-difference model (D-DM) using three types of error (i.e., normal, log-normal and Gamma distribution). The error assumptions will be constructed in the predicted catch values from CTDDM and D-DM.

3.3 Biomasses and numbers from CTDDM compared with those from ASPM

This study constructed the Stock-Recruitment Relationships (SRR) model with very small age-time increments Δa, Δt < 0.1, and forced this model with the recruitment R(t) varying in an arbitrary pattern over time, and then the biomass and the numbers predicted from CTDDM do indeed track the age-structured production model (ASPM) predictions more and more precisely as the age-time increment becomes smaller.

3.4 Catch and effort data analysis (version 3.0.1, Hoggarth et al., 2006) (CEDA) and A surplus production model incorporating covariates (version 5.0, Prager, 2005) (ASPIC)

The CEDA (Catch-effort data analysis, Hoggarth et al., 2006) and ASPIC (A surplus-production model incorporating covariates, Prager, 2005) software packages were used to compare the accuracy of the parameter estimates, notably B₁/K, MSY, K, q, E_MSY. The initial proportion of B₁/K is assumed to be the ratio of the catch in 1975 (the initial year of data) over the catch in 1987 (the year of the largest catch) in CEDA and ASPIC, which gives a value of approximately 0.5.

4 RESULT

The CTDDM was used to evaluate the growth parameters and Biological Reference Points (BRPs) of the southern Atlantic albacore (T. alalunga) stock, including r (the intrinsic growth rate), B_∞ (the carrying capacity), B_MSY (the biomass at MSY) (Table 2).

We obtained von Bertalanffy’s growth equation, which is L_t=147.5(1–e^{-0.126(t+1.89)}), and the lengthweight relationship which is W_t=1.3718×10^-5L_t^3.093. The estimated growth parameters are listed as follows: W_∞=71.56 kg, W_k=16.59 kg, and W_k−1=12.09 kg. The results of CEDA and ASPIC have been compared with the results from CTDDM (Table 3).

The catch in 2011 (24 100t) is higher than the values of MSY from CTDDM and the relative fishing mortality ratio in 2011 (F₂₀₁₁/F_MSY) is higher than 1.0. The South Atlantic albacore (T. alalunga) in 1975– 2011 was assessed by discrete time delay-difference models and continuous time delay-difference models. The error assumptions of normal, lognormal, and gamma distribution were considered in the predicted catch values of the albacore (T. alalunga). Predicted catch values of CTDDM and discrete time D-DM with three types of error (i.e., normal, log-normal and Gamma distribution) are compared in Fig. 1.

Comparing the coefficient of variation (CV) between the discrete model (D-DM) and the continuous model (CTDDM) shows that the CV values from normal error distribution were higher than those in the lognormal and the Gamma distributions. The relationship between the average weight and the survival rate was obtained. This relationship can be used to explain the effect of variations in survival rate on the mean body weight and to estimate the survival rate for a particular observed mean body weight. With increasing growth parameter ρ, the average weight also increased (Fig. 2).

The performance of CTDDM was compared to that of the age structured production model (ASPM). The total biomass and numbers predicted from CTDDM do indeed track the ASPM predictions more and more precisely as the age-time increment becomes smaller. Because we have little detailed age-structure information about the southern Atlantic albacore (T. alalunga) stock, the trajectory of the CTDDM tracks the biomass or numbers predicted from ASPM will not be considered in this study, instead of in the further study. The annual fluctuations in observed and predicted CPUE in the CTDDM are discussed here. The observed and predicted CPUEs (catch per unit effort) of CTDDM are compared in Fig. 3.

The annual fluctuations in observed and predicted CPUE in the CTDDM are very close. Harvest rates (HR) between 0.05–0.40 were set to estimate the indicators such as biomass, catch, B₂₀₂₅/B_MSY, B₂₀₂₅/K, and the probability of P (B₂₀₂₅ < B₂₀₁₂). Risk assessment results of the surplus production model (SPM) and CTDDM are shown in Tables 4 and 5.

The SPM and CTDDM risk assessment outputs have a similar trend: with the increased harvest rates, biomass decreases, and the catch rises before the first half of the curve (harvest rate of 0.2) then decreasing afterwards. The results of CTDDM suggested a harvest rate of 0.15–0.20 may be the best option (Table 5).

4 DISCUSSION

The main objective of this study is to provide an alternative method (a continuous time type model) to the southern Atlantic albacore (T. alalunga) stock assessment. Many methods have been already used to assess the albacore (T. alalunga) stock (Yeh et al., 1990; Sun et al., 2002; Viñas et al., 2004; ICCAT, 2012; Zhang et al., 2015b). The results of fish stock assessment can be different using different model approaches or using different data (Chen et al., 1992; Musick and Bonfil, 2005; Wang and Liu, 2006; Liu et al., 2012). Thus, the model and the data must be used carefully to ensure that the conclusions can be consistent with the current fisheries both management philosophy and realistic situations. The utility of a stock assessment method depends on the nature of the fisheries taking the stock. ICCAT (2011) used ASPIC that obtained B₂₀₀₉/B_MSY at about 0.624–1.204, and F₂₀₀₉/F_MSY at about 0.795–1.342; and used ASPM which obtained the two ratios at 1.063 and 0.748. In this study, CTDDM obtained the relative biomass ratio (B/B_MSY) and the relative fishing mortality ratio (F/F_MSY) of the stock in 2011 at about 1.178 and 1.386, respectively. They are higher than the values obtained by ICCAT using the traditional production models. Lee and Yeh (2008) suggested the MSY at about 28 771 t using ASPM; ICCAT (2011) used Fox non-equilibrium production model to obtain the MSY with 23 630 t to 27 390 t. CTDDM gets an 80% confidence interval of MSY of South Atlantic albacore population at about (21 510 t–23 118 t). Because the MSY will change through time as the mixture of fishing mortalities by age class changes, the risk assessment results showed that the catch initially increased until the harvest rate reached approximately 0.15 and then decreased. The equilibrium biomass decreased consistently in the whole process, the biomass decreased slower when catch declined.

ICCAT (2011) assessment report showed that the spawning stock of species in this study from 1997 to 2007 dropped by about 32%, which may result in overfishing. Reports showed that the southern Atlantic albacore stock had experienced overfishing from 1985 to 2005 (ICCAT, 2012). After that, this stock rebuilds gradually and the current situation is not critical, but it must be taken care of in the future. The International Foundation for Sustainable Seafood (ISSF) believes that this group was in a mild overfishing state in the early 2000s (ISSF, 2011). The yield of 2011 was 24 100 t, and the relative fishing mortality ratio (F₂₀₁₁/F_MSY) of the stock was higher than 1.0 using CTDDM. Statistical properties of model parameters and other derived quantities used for setting management targets are relevant for conservation and management of fisheries resources. Most of delay-difference models that only considered one source of uncertainty had not yet been subject to extensive evaluations (Walters and Martell, 2004; Jiao et al., 2010). The uncertainties of CTDDM were considered in this study. The performance of CTDDM was in comparison with that of the age structured production model (ASPM) are discussed in this study. Annual fluctuations in the total biomass and the numbers predicted from CTDDM do indeed track the ASPM predictions more and more precisely as the age-time increment becomes smaller. We evaluated the accuracy of estimates of population parameters and biological reference points (BRPs) of the South Atlantic albacore population, and compared the results of traditional modeling method with those from CTDDM. The CTDDM has not been applied in this albacore (T. alalunga) stock, and it would be an alternative model necessary to provide some further information for this important international fishery filled the gaps in simple surplus production models and complex fully age-structured models.

Few publications about the continuous time type delay-difference model are available on the south Atlantic albacore fisheries (Su and Liu, 1998; Walters and Martell, 2004). The CTDDM provides a good performance of the dynamics of total numbers and biomass for age-structured populations. Standard D-DM has already been applied to fishes with long life cycles, shark-like fishes, and lobster. However, the discrete time formulation does not work well for populations with the consideration of continuous time delays of biological processes (Quinn and Deriso, 1999; Walters and Martell, 2004; Musick and Bonfil, 2005). The dynamics of CTDDM characterize some highly migratory cosmopolitan fish stocks, such as albacore (T. alalunga), which treat recruitment, growth, and mortality rates as all varying continuously over time (Walters and Martell, 2004, Walters, 2011). For such species, the continuous rate of recruitment, growth, and mortality rates as all varying continuously over time may be better to be treated by CTDDM. The CTDDM performs better than the discrete time models for stocks when consideration is given continuous time delays of biological processes (Walters, 2011). Thus, we apply a continuous time delay-difference type model for assessing the Atlantic albacore (T. alalunga) fishery.

In conclusion, the main objective of this study has been to provide an alternative method (a continuous time type model) to the southern Atlantic albacore (T. alalunga) stock assessment. The CTDDM was used to evaluate this albacore (T. alalunga) stock for the population parameters and biological reference points (BRPs), and provided a good fit to the catch data. The CTDDM performs better than the discrete time models for the stocks when consideration is given to continuous time delays of biological processes. Being a bridge between ASPM and SPM, it is an alternative model that is necessary for the appropriate stock assessments of the southern Atlantic albacore. A major value of this study is that it is the first time CTDDM be provided to assess the Atlantic albacore (T. alalunga) fishery. The method verified the MSY estimates used by ICCAT and diversified the models on this stock assessment. Estimated results were analyzed to verify the proposed methodology and provide reference information for the sustainable management of the southern Atlantic albacore (T. alalunga) stock, and we plan to conduct further study on the uncertainty of MSY with the perturbation histories of the data in the future.

5 ACKNOWLEDGEMENT

We thank Drs. JIAO Yan, SU Zhenming and two anonymous reviewers for comments on the manuscript. The first author would like to acknowledge the Chinese Scholarship Council (CSC) for the funding of his study in Virginia Polytechnic Institute and State University (USA).