Institute, Guangzhou , China. Technology, China. Chinese Academy of Fishery Sciences, Qingdao , China. *[

Transcription

1 Indian Journal of Geo Marine Sciences Vol. 46 (05) May 2017 pp Applicaion of Bayesian surplus producion model and radiional surplus producion model on sock assessmen of he souhern Alanic albacore (Thunnus alalunga) Baochao Liao 135 Kui Zhang 2 Xiujuan Shan 34 Xiao Chen 1 Abdul Base 5 Khadim Hussain Memon 5 & Qun Liu 5* 1 Deparmen of Mahemaics Shandong Universiy (Weihai) Weihai China 2 Key Laboraory of Souh China Sea Fishery Resources Exploiaion & Uilizaion Souh China Sea Fisheries December Research Insiue Guangzhou China 3 Laboraory for Marine Fisheries Science and Food Producion Processes Qingdao Naional Laboraory for Marine Science and Technology China 4 Key Laboraory for Susainable Uilizaion of Marine Fisheries Minisry of Agriculure Yellow Sea Fisheries Research Insiue Chinese Academy of Fishery Sciences Qingdao China 5 College of Fisheries Ocean Universiy of China 5 Yushan Road Qingdao China *[ qunliu@ouc.edu.cn] Received 20 March 2015; revised 29 May 2015 Bayesian surplus Producion model (BSP) and radiional surplus Producion models (TSP) were used o evaluae he souhern Alanic albacore (Thunnusalalunga) sock. Populaion parameerswere esimaed using CEDA (cach-effor daa analysis) and ASPIC (a surplus-producion model incorporae covariaes) compuer sofware packages. Performance of he BSP model and TSP model were compared by a Bayesian informaion crierion (BIC). Maximum susainable yield (MSY) from he TSP model and BSP model were used o verify he MSY esimaions by Inernaional Commission for he Conservaion of Alanic Tunas (ICCAT). Cach of 2011 (24122 ) was higher han he MSY from BSP ( ) and he relaive fishing moraliy raio (F 2011 /F MSY ) of he sock was higher han 1.0 which shows hahis sock overexploied. Differen harves sraegies were se o assess he risk for his sock and hese esimaes were used opredic he biomass and cach in 2025 (B 2025 C 2025 ) and oher five indexes (B 2025 /B MSY B 2025 /K P (B 2025 > B 2012 ) P (B 2025 > B MSY ) P (B 2025 < B MSY/4 )). Evaluaed biological reference poins (BRPs) from Bayesian model were compared wih he resuls from radiional modeling mehod on he souhern Alanic albacore (T. alalunga) sock and resuls showed ha he measures should be aken for he susainable uilizaion of his fish sock and he harves rae of 0.15 seemed obe he bes managemen measures [Keywords: Bayesian model Fox model Thunnus alalunga Souhern Alanic Risk assessmen] Inroducion Global opimizaion and Bayesian approaches were used increasingly in assessing and managing fisheries socks because of heir flexibiliy in incorporaing daa from differen sources 1 2. The goodness-of-fi and saisical performance of he Bayesian surplus producion model was compared wih ha of he classical Fox surplus producion model 3. Bayesian models were used o address he emporal variaion in populaion growh rae caused by changes in species composiion 45. Variaions in environmenal variables and measuremen errors ofen creaed problems for fish sock assessmen scieniss 6. Fisheries scieniss involved recovery planning in a conservaion siuaion in which a populaion was currenly depleed i was criical o obain good qualiy esimaes o esimae he maximum percenage of he harves rae ha could be oleraed 7. There have been Bayesian sock assessmens analyses using populaion dynamics models 89 bu none of hose analyzes have been compared wih classical approach for saisical inference. Bayesian surplus producion model was referred as mulilevel priors o represen a siuaion wihou hierarchical daa 2. Bayesian models have been increasingly used in ecological applicaions o quanify muliple sources of uncerainy 9. How- due o heir complexiy Bayesian models were compuaionally very inensive and difficul o fi 10. The Markov chain Mone Carlo (MCMC) mehod uses each of he one-dimensional full condiional poserior disribuion in urn o generae a sample from he join poserior disribuion of all he unknowns Mos of hese models have no ye been subjec o exensive evaluaion by means

2 INDIAN J. MAR. SCI. VOL. 46 NO. 05 MAY of he simulaions across he wide ranges of he scenarios/siuaions o deermine heir performance relaive o ha of simpler sandard regression models ha only are considered as one source of uncerainy. In his sudy he Bayesian model and classical sur- -plus producion models were used o assess populaion dynamics on he souhern Alanic albacore (T. alalunga) sock. Albacore (T. alalunga) is a highly migraory cosmopolian and emperae fish species of una inhabiing ropical and emperae pelagic waers of all oceans from abou 45-50N o 30-40S Since he early 2000s he souhern Alanic sock is considered o have big poenial for developmen by ICCAT (Inernaional Commission for Conservaion of Alanic Tunas) 15. Bu he ISSF (Inernaional Seafood Susainabiliy Foundaion) repored ha he souhern Alanic albacore (T. alalunga) sock is in a mild over fishing sae 16 he measures should be aken for he susainable uilizaion of his sock in he fuure. In his sudy he esimaed maximum susainable yield (MSY) from Bayesian model and Fox model was used o verify MSY esimaes used by ICCAT. The evaluaed biological reference poins (BRPs) from Bayesian model were compared wih he resuls from radiional modeling mehod on he souhern Alanic albacore (T. alalunga) sock. Differen harves sraegies were se o assess he risk for his sock and hese esimaes were used o predic he biomass and cach of his albacore (T. alalunga) sock. A cenral advanage of Bayesian modeling over he classificaion rules of he cach-based mehods is he calculaion of F/F MSY and B/B MSY in which he probabiliy of differen saes can also be calculaed. Maerial and Mehods The daa of he souhern Alanic albacore (T. alalunga) sock ( ) were used (From Inernaional Commission for he Conservaion of Alanic Tunas) in his sudy 17. Global populaion disribuion of albacore (T. alalunga) and he souhern Alanic sock (area No 12#) are showed (Fig.1). According o ICCAT a sandardized cach per uni effor (CPUE) based on he Chinese Taipei longline fishery ( ) was used as a relaive abundance index of he souhern Alanic albacore (T. alalunga) sock 17. For his albacore sock he oal caches in he pas 30 years ranged from approximaely o which was mosly aribued o he longline fisheries 15. Bayesian surplus producion models (BSP) wih mulilevel priors are called he hierarchical models even when he daa are no hierarchically srucured 3 5. Bayesian sae-space model was an exension of he sae-space model proposed by Walers and Marell 18. Walers and Marell demonsraed ha sae-space models may be no helpful for correcing ime series bias for individual socks 18. The daa available on Alanic albacore (T. alalunga) sock was no sagesrucured and he sae-space surplus producion models were used here as he basic model srucure 1 2 : E( B 1 B G C (1) E G ) ( I i ) q i B rb (ln( K ) ln( B 2 r ~ N ( r ) )) (2) (3) (4) where B G and C are he populaion abundance producion funcion and he oal cach in year and qi is he cachabiliy coefficien for he i-h ype of relaive abundance index I i. Here we used he Fox model as he producion funcion and i was used widely in fisheries and ecology In he Fox model r is he populaion inrinsic growh rae and K is he carrying capaciy. Insead of assuming a consan populaion growh rae in he Fox model a hierarchically srucured prior was used o model he populaion growh rae. The hierarchical populaion srucure was implied in he model hrough a mulilevel prior of r. Tradiional surplus producion models (SPM) are among he major academic models in assessmen and managemen of modern fishery resources due o heir simpliciy and relaively undemanding daa needs 20. From surplus producion models (SPM) we could predic he MSY (maximum susainable yield) and F MSY which are sill he major goals of fisheries managemen There are several forms of surplus producion models (SPM) in he lieraures. The dynamic equaions used in his sudy were lised below. U U 1 1 r ln( qk U ) qe 1 MSY rke Fig. 1 Map showing caches localiies of albacore (T. alalunga)

3 924 LIAO e al.: APPLICATION OF A BAYESIAN MODEL ON STOCK ASSESSMENT OF THUNNUS ALALUNGA f msy r / q (5). where B is he sock biomass r is he inrinsic populaion growh rae K is he carrying capaciy G e is he cach in equilibrium f is he effor q is he cachabiliy coefficien U is he cach per uni effor or abundance index in year. The CEDA (Cach-effor daa analysis Hoggarh e al. 2006) sofware package was used o evaluae he value of parameers 22. ASPIC (A surplus producion model incorporaing covariaes Prager 2005) sofware package was also used o compare he accuracy of he parameer esimaes 23 such as B 1 /K Table 1 Populaion parameers (r K q) of Fox surplus producion models from CEDA and ASPIC Sofware package (CEDA / ASPIC) K q r R yield R 2 CEDA Fox (non-equilibrium) E ASPIC Fox (non-equilibrium) E Fig. 2 Observed and prediced cach per uni effors (CPUE) of he souhern Alanic albacore (T. alalunga) sock. MSY K q F MSY. Tradiional surplus producion analyses used sandard regression and reaed B as an independen variable G as a dependen variable were fraugh wih esimaion problems 20. The observaion error of CPUE was considered as a lognormal disribuion and expressed as follows (where U and U^ were he observed and prediced CPUEs respecively): Log U Log ( Uˆ ) (6) Ĉ Uˆ qb (7) E Parameer esimaes for such relaionships provided he basis for seing key values of variables used in fish sock managemen such as classes were analyzed Bayesian inferenial mehods for he sae-space model were based on he poserior disribuion of parameers σ and saes r given he daa 10. To make inferences for θ one needs o consruc a marginal prior disribuion of θ by inegraing ou he saes r in p(y θ). For he above Bayesian surplus producion model he likelihood of any se of observaion can be represened by he following equaion: 2 Ln I ˆ i Ln I i n pdaa e i1 I i 2 (8) 2 I i pi N(0 ) (9) where p (I θ) is he likelihood of he daa given he parameer vecor θ θ denoes (k rσ 2 ) and I i and I i^ are he I is observed and prediced recruimen value of he daa respecively. The basic idea behind maximum likelihood is o find he values of he parameers for which he observed daa is mos likely o occur. In recen years mos reliable fisheries parameer esimaions were currenly being achieved by using he maximum likelihood (ML) mehod 20. However i was difficul o evaluae p(θ r Y) direcly by inegraion. MCMC (Markov chain Mone Carlo) simulaion echniques bypass he need o evaluae he high dimensional inegral in p(θ r Y) by generaing dependen draws from he poserior disribuion p(θ r Y) via Markov chains 24. For he fiing process we develop a Bayesian approach via MCMC sampling o make inferences abou he Bayesian models Resuls For he albacore sock we used radiional surplus producion (TSP) model (non-equilibrium) o obain MSY a abou ( ) and used he ASPIC model o obain B 2009 / B MSY raio a abou1.18 and F 2009 /F MSY raio a abou 1.42 (Fig. 2). The observed and prediced cach per uni effors (CPUE) of he souhern Alanic albacore (T. alalunga) sock are compared (Fig. 2). We applied CEDA and ASPIC o he cach and sandardized CPUE daa of souhern Alanic albacore o ge informaion of carrying capaciy K inrinsic growh rae r and he cachabiliy coefficien q (Table 1). The inrinsic populaion growh rae r is ( ) carrying capaciy K is ( ) and cachabiliy coefficien q is ( ).

4 INDIAN J. MAR. SCI. VOL. 46 NO. 05 MAY Prior disribuion of hree parameers were N ( ) N ( ) and N ( ( ) 2 ) respecively. We supposed ha N (equal 3) chains of MCMC differen iniial condiions were esimaed and he lengh (equal 10000) number of parameers (p) and he average variance of he chains were obained. Some sequence of parameer converged and ohers were no converged. If he sequences have srong correlaion i was acceped ha he Markov chain could be converged (Fig. 3). Frequency The Relaive index R^2 Frequency The Key parameer r Frequency The Key parameer K Fig. 3 Plo he convergence of he Markov chains of parameers in he Bayesian esimaor. Table 2 Summary saisics for parameers and biological reference poins (BRPs) from Bayesian Model Bayesian Model Mean Median 2.5%quanile 97.5%quanile r 0.372(0.126) K (10 4 ) 30.55(0.116) q(10-10 ) (0.18) B MSY (10 4 ) 15.28(0.116) F MSY 0.186(0.126) F (0.126) MCMC algorihm was used o compue he poserior disribuions of he hree parameers. In some cases he models did no give converge o a sable mixing disribuion for a leas runs. We used a burn-in period of runs which reduces he effec of saring values on he MCMC esimaes. The poserior disribuions values and densiy disribuions of r K and q were showed (Fig. 4). Bayesian surplus producion model was used o esimae he MSY of Souh Alanic albacore populaion (Table 2). The Fox model and he ASPIC esimaed he MSY and he raios of he B 2009 /B MSY and he F 2009 /F MSY and he sensiiviy of he model oucomes o he specified priors were esed (Priors were r ~ N ( ) K ~ N ( ) q~ N [ ( ) 2 ]). Inrinsic rae of increase Densiy disribuion r Ieraions Inrinsic rae of increase Carrying capaciy ( 10000) Densiy disribuion 3e+08 5e+08 7e+08 9e+08 0e+00 1e-09 2e-09 3e K Ieraions 2e+08 6e+08 Carrying capaciy ( 10000) Cachabiliy coefficien Densiy disribuion 2e-10 4e-10 6e-10 8e e e e e q Ieraions 2e-10 6e-10 Cachabiliy coefficien Fig. 4 Plo of poserior disribuion values and densiy disribuion for parameers r K and q.

5 926 LIAO e al.: APPLICATION OF A BAYESIAN MODEL ON STOCK ASSESSMENT OF THUNNUS ALALUNGA The esimaed maximum susainable yield (MSY) from Bayesian surplus producion (BSP) model and Fox surplus producion model (SPM) were used o verify MSY esimaes by ICCAT on he Souh Alanic albacore sock and he comparison beween MSY and he acual cach in 2011 were presened in his sudy (Table 3). Table 3 Summary saisics for MSY and he acual cach in 2011 MSY 2011 ( ) ICCAT Fox Model BSP Model ( ) ( ) C Bayesian models were compared wih he Fox surplus producion model based on he esimaes of parameers. We applied hose models o fi he daa and used Bayesian informaion crierion (BIC) o compare he predicions of parameers (Table 4) and hen he smaller BIC value indicaes he beer fi. Table 4 Model selecion crierion (BIC) resuls of he Bayesian surplus producion model and he Fox surplus producion model. Normal errors Lognormal errors Gamma errors Bayesian surplus producion model Fox surplus producion model Table 5 Risk analysis of Bayesian surplus producion model and he Fox surplus producion model using harves coefficien. Mehods and parameers Differen harves coefficien (from 0.05 o 0.4) Fox model H=0.05 H=0.1 H=0.15 H=0.2 H=0.25 H=0.3 H=0.35 H=0.4 B 2025 ( 10 4 ) C 2025 ( 10 4 ) B 2025 /B MSY B 2025 /K BSP H=0.05 H=0.1 H=0.15 H=0.2 H=0.25 H=0.3 H=0.35 H=0.4 B 2025 ( 10 4 ) C 2025 ( 10 4 ) B 2025 /B MSY B 2025 /K P(B 2025 > B 2012 ) P(B 2025 > B MSY ) Risk analysis of Bayesian model and he Fox model were compared under differen harves sraegies. We se differen harves sraegies o conduc risk assessmen for his sock and we used hose wo kinds of surplus producion models o predic he biomass and cach in 2025 (B 2025 C 2025 ) and oher five indexes (B 2025 /B MSY B 2025 /K P (B 2025 > B 2012 ) P (B 2025 > B MSY ) P (B 2025 < B MSY/4 )) (Table 5) Discussion Differen mehods have been used o esimae MSY of he souhern Alanic albacore (T. alalunga) sock for example Lee and Yeh (2007) used he age srucured producion model (ASPM) wih an esimaed MSY in base case 25 ; ICCAT (2012) used ASPIC o give a range of esimaed MSY wih o We esimaed he MSY based on TSP model o ge he 80% confidence inerval of MSY a abou We used Bayesian surplus producion model o ge an 80% confidence inerval of MSY of Souh Alanic albacore populaion a abou ICCAT (2012) used non-equilibrium producion model o obain he MSY wih o The resul of Bayesian model was lower han ha repored by Lee and Yeh (2007) which was obained using ASPIC primarily 25. The esimaed MSY from he TSP model was similar wih MSY esimaes used by ICCAT 17 and he esimaed MSY from he Bayesian model was lower han MSY esimaes used by ICCAT on he Souh Alanic albacore (T. alalunga) sock. Bayesian modeling has been increasingly used in ecological applicaions o quanify muliple sources of uncerainy 26. There have been Bayesian sock assessmens analyses using populaion dynamics models 11 bu none of hose analyzes have been compared wih classical approach for saisical inference. Bayesian surplus producion model was referred as mulilevel priors o represen a siuaion wihou hierarchical daa 2. A cenral advanage of Bayesian modeling over he classificaion rules of he cach-based mehods is he calculaion of F/F MSY and B/B MSY in which he probabiliy of differen saes can also be calculaed. Due o heir complexiy Bayesian models were compuaionally very inensive and difficul o fi 10. Mos of hese models have no ye been subjec o exensive evaluaion by means of simulaions across wide ranges of scenarios/siuaions

6 INDIAN J. MAR. SCI. VOL. 46 NO. 05 MAY o deermine heir performance relaive o ha of simpler sandard regression models ha only are considered as one source of uncerainy. Maximum Susainable Yield (MSY) had been acceped as one of he arge biological reference poins (BRPs) and i consiued he foundaion of federal fishery managemen policy in he USA This sudy esimaed he souhern Alanic albacore (T. alalunga) sock based on he CPUE daa of Chinese Taipei longline fishery. For his albacore sock he oal caches in he pas 30 years ranged from approximaely o which was mosly aribued o he Chinese Taipei longline fishery 17. We evaluaed he BRPs by radiional modeling mehod and he Bayesian model and risk assessmens of hose models were compared under differen harves sraegies. The Bayesian approach was he mos general mehod for fiing non-linear sae-space models and he Bayesian analysis gave suiable poserior disribuion of he esimaed parameers 28. We used BIC o selec he bes model in his sudy and he smalles value of BIC indicaes he bes fi. Resuls showed ha he value of BIC for Bayesian model were lower han he BIC values of he Fox model. A cenral advanage of Bayesian modeling over he classificaion rules of he cachbased mehods is he calculaion of F/F MSY and B/B MSY in which he probabiliy of differen saes can also be calculaed. In his sudy he evaluaed biological reference poins (BRPs) from Bayesian model were compared wih he resuls from radiional modeling mehod on he souhern Alanic albacore (T. alalunga) sock and he comparison beween MSY 2011 and C 2011 were presened in his sudy. Those performances make us believe ha he Bayesian mehods are likely o be appropriae for he assessmen of he souhern Alanic albacore (T. alalunga) sock. In his sudy he esimaed MSY from Bayesian model and Fox model were used o verify MSY esimaes by ICCAT on he Souh Alanic albacore sock. Differen harves sraegies were se o assess he risk for his sock and hese esimaes were used o predic he biomass and cach in 2025 (B 2025 C 2025 ) and oher five indexes (B 2025 /B MSY B 2025 /K P (B 2025 > B 2012 ) P (B 2025 > B MSY ) P (B 2025 < B MSY/4 )). The Bayesian modeling provides an exremely compac performance of he exac dynamics populaion. The mehods migh presen a useful choice for he souhern Alanic albacore (T. alalunga) sock assessmen. Conclusion The sudy verifies he MSY esimaes of albacore (T. alalunga) sock o suppor he managemen (i.e. seing of MSY) for he souh Alanic albacore sock. The biological reference poins (BRPs) of he albacore sock were evaluaed and verified by Bayesian surplus producion model and Fox surplus producion model. The MSY esimaes from BSP was lower han hose from convenional esimaes; he relaive biomass raio or relaive fishing moraliy raio from BSP were higher han hose from radiional modeling mehod and he resuls showed ha he measures for he caches should be aken for he susainable uilizaion of his fish sock. This sudy was based on he CPUE daa of Chinese Taipei longline fishery of he souhern Alanic albacore sock. In he fuure we plan o conduc furher sudy on quanify muliple sources of uncerainy for his sock. Acknowledgemen This research was suppored in par by he Naional Naural Science Foundaion of China (Gran No ) Fundamenal Research Funds for he Cenral Universiies of China (Gran No ) Public Science and Technology Research Funds Projecs of Ocean (Gran No ) and Laboraory for Marine Fisheries Science and Food Producion Processes Qingdao Naional Laboraory for Marine Science and Technology China (Gran No. 2016LMFS-B14). We hank Drs Yan Jiao Muhsan Ali Kalhoro and wo anonymous reviewers for commens on he manuscrip. References 1 Buckland S. T. Newman K. B. and Thomas L. Saespace models for he dynamics of wild animal populaions. Ecol. Model. 171 (2004) Jiao Y. Hayes C. and Core s E. Hierarchical Bayesian approach for populaion dynamics modelling of fish complexes wihou species-specific daa. ICES. J. Mar. Sci. 66 (2009) Robers G. O. and Rosenhal J. S. Infinie hierarchies and prior disribuions. Bernoulli. 7 (2001) Clark J. S. Uncerainy and variabiliy in demography and populaion growh: a hierarchical approach. Ecology 84 (2003) Gelman A. Carlin J. B. and Sern H. S. Bayesian Daa Analysis (Chapman and Hall London) 2004 p Quinn T. J. Deriso R. B. Quaniaive fish dynamics. (Oxford Universiy Press) 1999 pp Su Z. Randall M.P. Performance of a Bayesian sae-space

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