Abdul-Aziz Yakubu. Department of Mathematics Howard University Washington, D.C

Transcription

1 Impc of Periodic nd Consn Proporion Hrvesing Policies On TAC-Reguled Fisheries Sysems Abdul-Aziz Ykubu Deprmen of Mhemics Howrd Universiy Wshingon, D.C Collborors Jon Conrd, Ninpeng Li nd Mry Lou Zeemn

2 Emerging Ocen Diseses Disese is incresing mong mos mrine orgnisms Wrd nd Lffery, Emples: Recen epizooics epidemics in nimls of Alnic Ocen bolenose dolphins nd endngered Florid mnees. Conribuing Fcors include globl wrming hbi desrucion humn overfishing ec

3 Overfishing Impliced In Se Urchin Epidemics Se urchin epidemics hve risen over he ls 30 yers, nd diseses hve decimed urchin populions in mny prs of he world. In he erly 980s, n epidemic killed more hn 95 percen of he long-spined se urchins Didem nillrum in he Cribben. Afer he urchins died, prevlence of seweeds incresed drmiclly; ody, mny corl reefs here re ded. Biologiss hve suggesed h overfishing urchin predors such s odfish Opsnus sp. nd queen riggerfish Blises veul my hve plyed role in his epidemic.

4 World's Fish Supply Running Ou, Reserchers Wrn Journl of Science By Julie Eilperin Wshingon Pos Wrier, November 3, 2006 Economiss nd ecologiss wrning: No more sefood s of 2048 Bsed on 4-yer sudy of Cch d Effecs of fisheries collpses Cuses Overfishing Polluion Oher Environmenl Cuses Loss of Species ffecs ocens biliy o Produce sefood Filer nuriens Resis he spred of disese Sore CO 2

5 Tol Allowble Cch TAC Mny fisheries re reguled using TAC. A TAC wihin sysem of individul rnsferble quos ITQs is currenly used o mnge he Alskn hlibu fishery. The Alskn hlibu is one of he few success sories in he book on US fisheries mngemen. The TAC did resonble good job of prevening overfishing, bu creed noher se of problems. Reguled open ccess: If TAC is imposed on fishery where ccess o he resource is free or of miniml cos, fishers hve n incenive o rce for he fish, rying o cpure s lrge shre of he TAC for hemselves before he cumulive hrves reches he TAC nd he seson is ended. Reguled open ccess my resul in severely compressed fishing seson where vs mouns of fishing effor re epended in few dy hlibu derby Prior o 995 one or wo dy seson. -fishers si idle or re-ger nd cuse overfishing in oher fisheries.

6 Periodic Proporion Policy PPP A sr of yer, esimed fish sock biomss y ol llowble cch TAC y PPP F F me m + F F m fishing morliy nurl morliy Under Pulse Fishing, fishing morliy is F + p F. Therefore, + p. periodic nd

7 Consn Proporion Policy CPP y, CPP where F e m m + F F. CPP is rnspren, esy o implemen nd ccepble o fishers.

8 Hrvesed Fish Sock Model Escpemen Model y S g m or S g S S m S f

9 Compensory Dynmics nd CPP Wihou Allee Effec f m + g. When he Allee effec is missing, g :[0, [0, is sricly decresing smooh funcion, nd F me m + F F.

10 Compensory Dynmics nd CPP Coninued g0 m If >, hen he + g0 m sock size pproches zero for ny iniil sock level. g0 m If < nd he dynmics + g0 m is compensory, hen he sedy se biomss is he fied poin g m.

11 Emple: Beveron-Hol Model nd Consn Hrvesing 0.5 m Hlibu Alskn. whenever poin fied rcing globlly on persiss sock The. when depleed is sock The. where, + < + + > > α α β α α α α β α m m m m m m m m f

12 Allee Effec Criicl Depension in Rel Populions Soner nd Ry-Culp showed evidence of he Allee effec in nurl populions of he Cribben queen conch Srombus gigs, lrge moile gsropod h suppors one of he mos imporn mrine fisheries in he Cribben region. There is eperimenl evidence of he Allee effec in urchins. In fisheries sysems, he Allee mechnism is relevn o issues of species eincion, conservion, fisheries mngemen nd sock rehbiliion.

13 Srong Allee Effec The eploied sock hs srong Allee effec if here eiss criicl posiive sock level min, such h lim f 0 for ll in [0,min, nd he sock persiss uniformly on subse of min,.

14 Compensory Dynmics nd CPP Wih Srong Allee Effec criiclly depensory ne growh funcion When he Allee effec is we ssume h is smooh one - hump mp h increses from zero o posiive vlue h is nd hen decreses so h lim g <. g :[0, presen, [0, mimum bigger hn,

15 Compensory Dynmics nd CPP Wih Srong Allee Effec Coninued Modified Beveron-Hol Model: β β α α β β α α β α β α min. 2, 2 2 m m nd m m Then m where m f + > + +

16 Modified Beveron-Hol Model nd CPP. 2, : bifurcion fold ehibis he m f Theorem + + β α

17 Compensory Dynmics nd CPP Coninued Under compensory dynmics nd CPP, he sock size ehibis disconinuiy cr when he srong Allee effec is presen. The sock size suddenly jumps o zero s eceeds cr.

18 Overcompensory Dynmics nd CPP Ricker Model: + m + r e, m 0.2 slmon

19 Ricker Model nd CPP Wihou Allee Effec Under overcompensory dynmics vi he Ricker model no Allee effec nd CPP, he sock size decreses smoohly o zero wih incresing levels of hrvesing. Period-doubling reversls L. Sone, Nure 993.

20 Modified Ricker Model Wih Allee Effec nd CPP Theorem : r f, m + e ehibis he fold bifurcion.

21 Allee Effec nd CPP Under CPP, he Allee mechnism generes sudden disconinuiy cr, wih he sock size suddenly jumping o zero s pproches he criicl vlue fold bifurcion, when he sock dynmics is eiher compensory or overcompensory.

22 Sock Dynmics nd Periodic Proporion PolicyPPP We ssume k periodic fishing morliy F + k F, so h f, m + g, where + k.

23 Compensory Dynmics nd Theorem : For ech j { 0,, 2,...,k- }, le f j be n j incresing PPP m + g j compensory dynmics in 0,, where j + k j. Then he sock under period-k hrvesing ehibis globlly sympoiclly sble r-cycle, concve down mp under populion where r divides k. Proof : Use he generl resul of Elydi-Scker JDEA' 05, period-k eension of he resul of Cushing-Henson JDEA' 0.

24 Beveron-Hol Model Wihou Allee effec nd PPP orollry : For ech j { 0,, 2,...k }, le f j m + j + where j m + α >, β > 0 α, β j sock populion under period-k hrvesing ehibis globlly sympoiclly sble k-cycle. nd j + k j. Then, he

25 Compensory Dynmics, Srong Allee Effec nd PPP Theorem: For ech j {, 2,...,k }, le f j j + k j hrvesing ehibis wo m + g j ehibi he Allee effec, where f incresing concve down mp under compensory dynmics in [ min,, nd j. Then, he sock under period k nd globlly sympoiclly sble posiive r-cycle in min,,where r divides k. j coeising is n rcors; zero

26 Modified Beveron-Hol Model, Srong Allee Effec nd PPP orollry: For ech j { 0,, 2,...,k }, le α j f j m +, j β j 2 + α -j where > 2 β nd j + k j. - -j -m hen, he sock populion und er period-k hrvesing s wo co eising rcors; sympoiclly sble posiive zero nd n k-cycle Allee effec.

27 Overcompensory Dynmics nd PPP Ricker Model nd PPP As in he cse of CPP, under PPP nd no Allee effecs he sock size ehibis he bubble bifurcion s i decreses smoohly o zero. *Arcors in periodic environmens S. M. Henson, J. M. Cushing e l., Bull. Mh. Biol. 999, nd J. Frnke nd J. Selgrde, JMAA 2003.

28 Overcompensory Dynmics, Allee Effec nd PPP Under PPP nd overcompensory dynmics, low populion sizes led o he eincion of he sock, whenever he srong Allee effec occurs during ech pulse fishing seson.

29 Modified Ricker Model Wih Allee Effec nd PPP

30 Hlibu D

31 Pcific Hlibu

32 Prmeer Esimion Minimize MSE subjec o s 2007 { + s y nd m 0.5. m + g s } 2

33 Prmeer Esimion: Coninued

34 Pcific Hlibu & Modified Ricker Model

35 Fuure Of Pcific Hlibu 0.277

36 Hlibu Under Period-2 Hrves

37 Hlibu Under High Consn Fishing Pressure

38 Quesion Wh re he inercions beween clime chnge, Allee effec nd persisence of eploied species?

39 Sochsic Model Rndom Environmen nd Fisheries Le ζ~u σ, + σ be " men preserving spred" uniformly disribued rndom vrible. Sochsic Model : + m + ζ g

40 Unsrucured Populions In Rndom Environmens + where G ζ, m + ζ * Lewinon nd Cohen 969 * Birkhoff Le γ Εpeced G ζ, Ergodic Theorem { ln G ζ,0}. low bundnces in he long - erm. * Chesson 982, Ellner 984, Hrdin e l If γ < g 0 he populion goes einc wih probbiliy. If γ > 0 he populion hs low probbiliy of 988 ec reching

41 Unceriny nd Allee Effec

42 Unceriny nd Allee Effec

43 Sochsic Model Predicions

44 Sochsic Model Eincions

45 Cod D From Georges Bnk

46 Modified Ricker Cod Model

47 Cod s Fuure & Fishing Pressure

48 Opiml CPP nd PPP A ime, indusry revenue is R E By cos C Cobb moun equion py ce is p of - Dougls fishing effor producion d c q b funcion, where p>0 is he dockside or e-vessel price per uni for y, c>0 is he uni cos for fishing effor, q cchbiliy coefficien, nd b, d >0 re he elsiciies of hrves.

49 Opiml PPP nd CPP is discoun erm. δ nd is discoun fcor is given nd where g m o subjec d b q c p Mimize PPP nd CPP wi ll opiml The d b q c p is funcion revenue Ne } { > + > < + + Π Π δ ρ ρ

50 D Tble. Imporn species lnded or rised in he Norhes, heir lndings, L housnd m, e-vessel revenue, R $, millions, nd prices,p $ per lb, Yer L R P L R P L R P L R P L R P Americn lobser Se scllops Blue crb Alnic slmon2 Goosefish Hrd Clm Surf clm Menhdin Squid Loligo Cod

51 Conclusion Consn eploiions diminish socks while preserving compensory dynmics. Periodic nd consn eploiions simplify comple overcompensory sock dynmics wih or wihou he Allee effec. In he bsence of he Allee effec, sock size decreses smoohly o zero wih incresing levels of consn or periodic fishing pressure. Consn nd periodic eploiions force sudden decline in fisheries sysems h show evidence of he Allee mechnism. The probbiliy of eincion increses wih lrge enough vrince in he clime vrible. Opiml conrol echniques for discree-ime models wih eploiion.

52 Thnk You.