Estimating the population parameter, r, q and K based on surplus production model. Wang, Chien-Hsiung

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1 SCTB15 Working Pper ALB 7 Esiming he populion prmeer, r, q nd K bsed on surplus producion model Wng, Chien-Hsiung Biologicl nd Fishery Division Insiue of Ocenogrphy Nionl Tiwn Universiy Tipei, Tiwn

2 Tile: Esiming he populion prmeers r, q nd K bsed on surplus producion model Auhor: Wng, Chien-Hsiung Biologicl nd Fishery Division Insiue of Ocenogrphy Nionl Tiwn Universiy No. 1, Sec. 4, Roosevel Rod Tipei, Tiwn, R.O.C. Tel: , or ex. 415 Fx: e-mil: chwng@ccms.nu.edu.w 1

3 Esiming he populion prmeers r, q nd K bsed on surplus producion model ABSTRACT A heoreicl cch curve similr o he Schnue's mehod ws derived from he Schefer s surplus producion model. Theoreiclly, i shows he relionships beween he index of he relive biomss nd he prmeers of he inrinsic growh re r nd cchbiliy q. If he environmenl condiions re comprively sble, hen he minimum CV of he index of relive biomss migh be considered s he mos possible cse of he consn environmenl condiions. Hence, he crrying cpciy K cn be esimed by he iniil biomss B0 dividing by he index of he relive biomss s K=Bo/. As numericl exmple, for he cse of he Souh Pcific lbcore socks, hey re r= , q=9.2925e-09, nd K=161,786 m. Key words: producion model, populion prmeers, relive biomss 2

4 Esiming he populion prmeers r, q nd K bsed on surplus producion model by Wng, Chien-Hsiung INTRODUCTION Schefer s surplus producion model ws exensively used o ssess he fish socks due o simple, convenien, esy of his model nd requiring he cch nd effor d only. Generlly, he ssumpion of wheher or no cch equilibrium ws necessry. A equilibrium, MSY (mximum susinble yield) ws esimed wihou he esimions of he inrinsic growh re r, cchbiliy q nd crrying cpciy K. A non-equilibrium, hey ofen obined he negive prmeers. This is biologiclly impossible (Hilborn nd Wlers, 1992). This pper ries o derive heoreicl cch curve similr o he Schnue's mehod (Schnue, 1977). Nex, o show h heoreiclly his curve shows he relionships beween he index of he relive biomss nd he prmeers of r nd q, only. Third, o sugges possible mehod for esiming K. Forh, o evlue he populion prmeers of he Souh Pcific lbcore s numericl exmple. MATERIALS Long erms of cch nd effor d of Tiwnese un longline fishery opering in he 3

5 Souh Pcific Ocen were used o esime he effecive fishing effor by Honm s mehod (Honm 1974). The effecive cch per uni of fishing effor, ECPUE, ws evlued. Overll of he effecive fishing effors were esimed s follows. (Tol effecive effors) =(ol cch) /(ECPUE) Tol nd Tiwnese un longline cches of he Souh Pcific lbcore socks were doped direcly from he Tble 64, p118~119 of Tun Fishery Yerbook, 2000 published by SPC (Souh Pcific Commission). The logbooks of Tiwnese un longliners were provided by OFDC of Tiwn. METHODS 1954). Under exploiion, Schefer s surplus producion model expressed s follows (Schefer db d B ( F B (1) K = rb 1 ) Here, F= insnneous fishing morliy re, B= biomss, r= inrinsic growh re, K= crrying cpciy, = ime. Se α = r F nd β = r / K nd F=consn during he uni ime inervl ~+1, hen db d 2 = rb (1 B / K) FB =αb βb (2) Inegred equion (2) implied he biomss s follows. α αb e B + 1 = (3) α α + βb ( e 1) Expressing by he iniil biomss, hen equion (4) cn be obined. 4

6 B αb e α = 0 + ( α α βb 0 e 1) (4) Here, Bo= iniil biomss. On he oher hnd, he nnul cch Y cn be defined s follows. Y = + 1 FB d = F + 1 αb0e α + βb e α 0( α d 1) (5) for F=consn during he uni ime inervl. Following cch curve cn be obined by inegring equion (5) nd subsiuing U = Y / in i. X U = qk + qk r B ln( B q K ) r X (6) Here, q= cchbiliy, X= fishing effors, U=E CPUE by cch per effor, nd =ime. This is similr o he Schnue's mehod (1977). I is liner funcion, n index of he chnge of he relive bundnce. The differenil equion model of Wlers nd Hilborn (1976) is nlogous o he esimion procedure derived by Schnue. Equion (6) seems useful for esiming he prmeers of K, q, nd r by he mehod of he les squres. However, equion (6) implied follows. U qk 1 B = 1+ ln( r B + 1 ) q r X (7) Se U / q = B nd = B K hen / 1 B = 1 + ln( r B + 1 q ) r X (8) I shows he relionships beween he index of he relive biomss nd he prmeers of 5

7 he inrinsic growh re r nd he cchbiliy q, only. K becomes n implici fcor in = B K. Theoreiclly, K cn no be esimed direcly from equion (7). / Equion (8) shows h i is depending on he fishing inensy X nd he biomss sring B nd ending B+1 of his yer. If environmenl condiions re sble, or he iniil biomss of ech yer is he sme, i.e., B = B + 1, hen i implies h = 1 q X / r. Th is he index of he relive biomss decreses from 1 nd depending on he fishing inensiy X, only. If X0=0, i.e., wihou fishing, hen = 1. This mens h = B / K = 1 or B = K only if in consn environmenl condiions nd excluding he influence of he fishery. Generlly, 1, hence i implies h K cn no be obined direcly from equion (7). As shown in Figure 1, cch curve is heoreiclly depending on he iniil biomss Bo nd he consn environmenl condiions. As poined ou by Hilborn nd Wlers (1992), if ll of d re vilble righ from he sr of he fishery, hen i my be resonble o ssume h Bo =K. Such d re generlly unvilble. Environmenl condiions included he chnges of bioic nd bioic condiions, hisoric exploiion of his fishery, nd oher ype of fishery. Generlly, hey re no sble. Assuming h he consn environmenl condiions re vilble, hen B = B0 implies = due o he definiion of = B / K. Therefore, i need o esime B 0 nd before evluing K = B /. As shown in Figure 2, numericl exmple of he Souh 0 Pcific lbcore socks, hey flucued yer by yer. Here, he mos possible of consn environmenl condiions could be considered s follows. If he flucuion of biomss in nure cn express s B = m +1 B e, hen under fishing i 6

8 becomes equion (9). B = m F + 1 Be (9) Here, m =insnneous ne producion re. Under fishing, biomss is incresing if m > F. F m < implies he decresing of biomss under fishing. m = F mens he cch equilibrium. Subsiuing equion (9) in (8), equion (10) cn be obined. m = 1 (10) r I reveles h r m 0 if nd only if 0 1 due o he definiion of = B K. > < Theoreiclly, B = 0 implies = 0 nd hen m = r. / If m is n index of he populion chnges corresponding o he goodness of he environmenl condiions, hen migh be noher well defined index. However, hey hve lile difference, including he fishing s shown in equion (6), bu m excluding he fishing s defined bove. From equion (10), consn implies consn m, i.e., consn environmenl condiions, nd vice sers. Consn re generlly unvilble, bu pproxmely consn could be obined by he verge of wih he minimum coefficien of vrinces. On he oher hnd, he iniil biomss B 0 could be evlued s follows. B o = Y exp( F ) (11) 1s / 1s Here, Y 1 s = nnul ch of he firs yer fiing o he equion (6), F 1 s = fishing morliy of 7

9 he firs yer fiing o he equion (6), F = qx. As sed bove, if cch nd effor d re vilble, hen he esimion of r nd q, nd hen K, becomes possible. NUMERICAL EXAMPLE The Souh Pcific lbcore socks were minly exploied by un longline fishery, especilly by Tiwn (Wng 1984, 1988; Wng e l. 1988). Tol cch ws bou 33353m in 1999 (Tble 1). Numerous sudies ried o ssess he socks (Skillmn 1975; Weherll e l. 1979; Weherll nd Yong 1984, 1987; Wng 1988; Yeh nd Wng 1996). However, cchbiliy (q), inrinsic growh re (r) nd crrying cpciy (K) of his sock re sill unknown. Tble 1 lised he ol cch, effecive fishing effor ( X ) nd cch per uni effecive fishing effor ( U ) of he Souh Pcific lbcore socks. U represens he bundnce index ' ' of he verge biomss -yer. U = U + U ) / 2 nd U ( U U ) 1 / 2 represen ( = + + he bundnce index he beginning nd ending of h yer, respecively. Replcing ' ' ' ' ln( B / B + 1) ln( U / U + 1) nd fiing U, X nd ln( U / U + 1) o he equion (6), he prmeers, r nd q cn be obined s q = E 09, r = In order o esime K, ll of were evlued by equion (8). As shown in Figure 2, vried yer by yer. I reveled he flucuions of he relive biomss yer by yer. Before 1974, hey showed cler decresing rend. All of hem re lrger hn 1/2. I is he generl cse of erly developmen of he fishery. Afer 1974, vried round 1/2. 8

10 As sed bove, he mos possible of he sble environmenl condiions implies he verge of wih he minimum CV. I ws found by compring ll of CV which ws clculed by deleing firs some from he sr of he fishery. As shown in Figure 3 nd 3b, minimum CV ws obined by deleing ll of before The verge of during 1974~1999 is bou This is = On he oher hnd, F 1967 = qx = implies B 1967 = Y / exp( F) = m. This is he iniil biomss. Therefore, K cn be evlued s K = B / = m. Bsed on he populion prmeers, he curren sus of he fish socks under exploiion cn be deeced s follows. From equion (10), m ws evlued by m = r 1 ). As shown in Figure 4 nd ( 5, he difference beween m nd F provides us he curren sus of he fish socks under exploiion. Figure 4 reveled h fishing morliy depends on he ne producion re, i.e., fishery reflecing he curren sus sufficienly. Generlly, his is he cse of sufficienly developed fishery. Figure 5 reveled h before 1974 he fishing morliy ws lwys lrger hn he ne producion re. I is ofen he erly developmen of fishery. ws sill lrger hn 1/2. Afer 1974, he difference vried round zero, i.e., ner he cch equilibrium. Rising he efficiency of he fishery mngemen, i is possible o mke he difference ner zero, i.e., cch equilibrium. Due o he difficuly for predicing he environmenl condiions nd hndling he chnges of he biomss, generlly i is impossible o mke he difference jus o be zero. However, o mke he difference s smll s possible is one of he 9

11 gol of fishery mngemen. Therefore, he difference of heses wo indices migh be n index of he efficiency of he fishery mngemen. Under fishing, exp( F ) = B+ 1 m / B vried in he rnges of ~ wih he men , lower bu ner o 1. I implies h he hrves of he Souh Pcific lbcore socks ws closing o cch equilibrium. DISCUSSIONS Bsed on Schefer s surplus producion model, heoreicl cch curve cn be obined. However, i cn esime he r, q nd he relive biomss only. Generlly, i cn no esime K, direcly. As sed bove, esimion of K becomes possible if he informion of he iniil biomss he mos possible of he consn environmenl condiions re vilble. However, some poins should be noiced. 1. Minimum CV of could be found by choosing priculr ime inervl. However, long ime endency nd he recen sus s suggesed in his pper migh be more pprecible. 2. K could be he lrges one of K = B /, =1 n, or verge of hem, or he iniil biomss of However, i is difficul o exclude he influence of he fishery. The iniil biomss s suggesed in his pper migh be hving he les disurbnce cused by fishery. 3. Consn fishing morliy during he uni ime inervl, consn cchbiliy nd sble environmenl condiions re he minimum requiremens for heoreicl developmen. 4. During 1967~1999, he flucuions of he biomss, exp( m ) = B+ 1 / B, vried in he 10

12 rnges of 1.35~5.82. The rio of mx/min is bou imes. Observed ECPUE vried in he rnges of 17.38~75.12 kg per 100 hooks. The rio of he mx/min is bou imes. Correspondingly, r=1.90, i.e., exp( 1.89) = seems resonble. 5. Under fishing, exp( m F ) = B+ 1 / B vried in he rnges of ~ wih he men , lower bu closing 1. I is consisen wih he comprively sble of he fishery. SUMMARY 1. Theoreicl cch curve derived from Schefer s surplus producion model cn esime he relive biomss nd he prmeers, r nd q only. 2. The esimion of K is possible if he informion of he iniil biomss nd he sble environmenl condiions re vilble. 3. In order o evlue K, new index m of he ne producion re in nure ws inroduced. 4. The ssumpion of he consn environmenl condiions implies he consn m, nd hence consn. The consn ws esimed by he verge of wih he minimum CV. 5. Iniil biomss ws esimed by he biomss sr of he fishery. 6. The populion prmeers of he Souh Pcific lbcore socks were esimed s r= , q=9.2925e-09, K= m, respecively. 7. The difference beween he index of nd m is useful for deecing he curren sus under fishing. Zero mens he cch equilibrium. 11

13 8. To mke he difference of nd m s smll s possible is one of he gol of fishery mngemen. ACKNOWEDGMENTS This reserch is suppored by Fisheries Adminisrion, Council of Agriculure, Execuive Yun, he Republic of Chin. The logbook d of he un longline fishery of Tiwn ws provided by OFDC (Overses Fisheries Developmen Council of he Republic of Chin). 12

14 CITED PAPERS Hilborn, R. nd C. J. Wlers Quniive fisheries sock ssessmen. Chpmn & Hll. New York, London. pp570. Honm, M., Esimion of overll effecive fishing inensiy of un longline fishery. Bull. Fr. Ses Fish. Lb., Rep. 10: Schefer, M. B Some specs of he dynmics of populions imporn o he mngemen of commercil mrine fisheries. Bull. Iner-Am. Trop. Tun Comm. Bull. 1: Schnue, J Improved esimes from he Schefer producion model. heoreicl considerions. J. Fish. Res. Bd. Cn. 34: Skillmn, R.A., An ssessmen of he souh Pcific lbcore, Thynnus llung, fishery, Mr. Fish. Rev. 37(3):9-17 Wlers, C. J. nd R. Hilborn Adpive conrol of fishing sysem. J. Fish. Res. Bd. Cn. 33: Wng, C. H., Review of he developmen of Tiwnese fr ses un longline fisheries. Chin Fisheries Monhly, 375: (in Chinese). Wng, C. H., Sesonl chnges of he disribuion of souh Pcific lbcore bsed on Tiwn s un longline fisheries, ACTA Ocenogrphic Tiwnic, 20: Wng, C. H., Chng, M. S., Lin, M. C., Esiming he mximum susinble yield of he souh Pcific lbcore, ACTA Ocenogrphic Tiwnic, 21: Wng, C. H., Reconsiderion of ssessing fish socks wih he surplus producion model. ACTA Ocenogrphic Tiwnic, 35(4): (in Chinese). Wng, C. H., Reconsiderion of ssessing souh Pcific lbcore socks (Thunnus 13

15 llung). ACTA Ocenogrphic Tiwnic, 37(3): Wng, C. H., 1999b. Flucuion of he souh Pcific lbcore socks (Thunnus llung) relive o he se surfce emperure. TAO, 10(2): Wng, C. H., Applied he improved surplus producion mehod o ssess he Souh Pcific lbcore socks, (Thunnus llung), ALB-2/SCTB13, July 5~12, 2000, Noume, New Cledoni. 19pp. Weherll, J. A., Riggs, F. V., Yong, M. Y. Y., Assessmen of he souh Pcific lbcore socks. U.S. N. Mr. Fish. Serv. Souhwes Fish. Cener Admin. Rep. H-79-6, 17pp. Weherll, J. A., Yong, M. Y. Y., Assessmen of he souh Pcific lbcore socks bsed on chnges in cch res of Tiwnese longliners nd esimes of ol nnul yield from 1964 hrough U.S. N. Mr. Fish. Serv. Souhwes Fish. Cener Admin. NPALB/87. 14pp. Weherll, J. A., Yong, M.Y.Y., Souh Pcific lbcore sock ssessmen nd reled issues. U.S. N. Mr. Fish. Serv. Souhwes Fish. Cener Admin. Rep. H pp. Yeh, Y. M., Wng, C. H., Sock ssessmen of he souh Pcific lbcore by using he generlized producion model, ACTA Ocenogrphic Tiwnic. 35(2):

16 Tble 1. Cch, effor nd effecive CPUE of he Souh Pcific lbcore socks, ol cch effecive effors ECPUE U'= Yer (meric ons) (million hooks) U=kg/100H (U+U+1)/ men mx min noe: Tiwn nd ol LL cches doped from Tble 64, p118~119, Tun Fishery Yerbook, 2000, SPC 15

17 Figure 1. Theoreicl cch curve bsed on he relionships mong he ne producion, biomss nd fishing morliy. Figure 2. Flucuions of he index of relive biomss,. Figure 3. Minimum coefficien of vrince, Figure 3b. Minimum coefficien of vrince,, (by smll scle)., (by lrge scle). Figure 4. Comprisons of he ne producion re, m, nd fishing morliy re, F. Figure 5. Comprisons of he differences: d = m F. 16

18 ne producion rk/4 cch curve F=F' 0 K/2 Bo biomss F=0 F=F" fishing morliy vlue YEAR 17

19 CV-vlues YEAR CV-vlues * minimum CV YEAR 18

20 m F m-vlue YEAR Differences YEAR 19