, then the old equilibrium biomass was greater than the new B e. and we want to determine how long it takes for B(t) to reach the value B e.

Transcription

1 SURPLUS PRODUCTION (coiud) Trasiio o a Nw Equilibrium Th followig marials ar adapd from lchr (978), o h Rcommdd Radig lis caus () approachs h w quilibrium valu asympoically, i aks a ifii amou of im o acually rach h w quilibrium Howvr, w ca drmi how log i will ak o g o wihi ay fixd proporio of h w quilibrium followig a sudd chag i h ra of fishig moraliy If h iiial ra of fishig moraliy is lss ha h w ra, h h old quilibrium biomass was grar ha h w ad w wa o drmi how log i aks for () o rach h valu +ε W wa o fid lag such ha + ε lag + ε > ( lag ) + ε > + C xp r lag + ε whr C w_ old_ ad r old_ 4 Trasiio Afr a Icras i Hr is a picur of wha w ar doig lag ε { Solv h followig quaio for lag + ε l ε + ε + C xp ( r ) lag + ε C xp r lag > C r l lag > lag + ε + ε + ε ε xp ( r ) lag C w_ r old_ old_ W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 34

2 Suppos is lss ha (i, h w is grar ha h old ) I his cas h approach o h w quilibrium is from blow ad ε lag Th soluio for lag is ε > ( lag ) ε lag l ε ε w_ r old_ old_ Trasiio Afr a Dcras i lag ε { I ihr cas, h yild ha accumulas durig h rasiio priod (, lag ) is lag Y lag + C xp[ ( r ) ] lag Y Th igral is of h form + C xp[ ( r ) ] + a xp( b u) du xp( b u) xp( b u) + a du l( xp( b u) + a) b + Arb X + a xp( b u) du l( xp( b X) + a) b l( + a) b xp b X l ( ) + a b + a W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 35

3 W ca us his rsul wih b r - ad a C ad wri h quaio for Y as C Y l xp ( r ) lag + wih C r + C xp ( r ) lag + Now subsiu for C o g Y l r + Y l xp ( r ) lag + r Y l xp ( r ) lag r + Th firs rm ca b wri as r r r r r r r + r r So, Y ca b wri as Y ( ) l + xp r lag Plla ad Tomliso's Gralizd Surplus-Producio Modl O problm wih h Graham-Schafr modl is ha h maximum susaiabl yild always occurs wh h biomass is half h carryig capaciy This is a dirc cosquc of h parabolic rlaioship bw d/ ad, which i ur follows from h liar rlaioship bw pr capia produciviy ad populaio siz Plla ad Tomliso (969), o h Supplmal Radig lis, proposd a alraio o h modl for la produciviy, which ucoupls msy from d a b b for < < a for < This is o a covi formulaio for his modl Th valus for,, ad msy all dpd o paramr, ad you hav o rvrs h sigs of paramrs a ad b dpdig o whhr is grar or lss ha o W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 36

4 lchr's Paramrizaio of h Plla & Tomliso Modl lchr (978), o h Supplmal Radig lis, proposd h followig alraiv paramrizaio o avoid h problms miod abov d γ γ wih γ Paramr γ (gamma), which is a pur umbr, auomaically chags sig as icrass hrough o Paramr corols h locaio of msy To s his, diffria h quaio for d/ wih rspc o d d d γ γ If w s his o zro ad solv for, w ca drmi msy γ msy γ > msy msy > msy > msy Wh 2, h modl is quival o h Graham-Schafr modl msy 2 2 Wh > 2, msy > ad maximum produciviy is closr o 2 or xampl, 5 > 5 msy Wh < 2, msy < ad maximum produciviy is closr o zro 2 or xampl, 5 > 5 msy 5 25 W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 37

5 5 Produciviy, d/ <2 >2 Pr Capia Producio, (d/)/ >2 <2 Wh, h sysm rducs o h so calld "xpoial" surplus-producio modl of ox (97), o h Supplmal Radig lis d l This is dscribd as a xpoial modl bcaus h graph of quilibrium CPUE vrsus ffor dclis xpoially Hr is a drivaio of ox's modl from lchr's vrsio of h Plla ad Tomliso modl d γ γ acor ou / ad wri ou γ i full d Now ak h limi as gos o o lim lim X l( X) > lim d l Wih h ox surplus producio modl msy W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 38

6 Explor h ifluc of paramrs,, ad (<2, >2, ad ) usig h Excl dmosraio I pracic rsarchrs hav of had difficuly i fiig h Plla-Tomliso modl o ral daa caus of h ihr rlaioship bw h curvaur i h modl ad h valu of, his paramr is of difficul o drmi Vry diffr valus for ca giv almos h sam fi o may daa ss O approach o his problm is o choos o h basis of ohr aalyss or cological argums ad h drmi h rmaiig paramrs lchr (978), o h Supplmal Radig lis, discusss h problm of applyig h Plla-Tomliso modl lchr (974), o h Supplmal Radig lis, dvlops a alraiv mhod for corollig h locaio of msy, by roaig h mai axis of h parabolic curv Ths surplus producio modls do o spcify h biological procsss rsposibl for h curvaur i h graph of d/ vrsus, bu w hav alrady sudid a modl ha dos iclud h biological dails; h modl for quilibrium yild, which w dvlopd by combiig a yild-pr-rcrui modl wih a sock-rcrui rlaioship Surplus Producio Modl Equilibrium Yild Modl Equil Yild Equil Yild Th Equilibrium Yild Surfac Th horizoal axis is, h axis goig io h pag is, ad h vrical axis is Y Y W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 39

7 Wih ay of h surplus producio modls h populaio is driv o xicio if h ra of fishig moraliy is grar ha or qual o h slop of h graph of d/ vrsus a h origi (quival o paramr r i h Graham-Schafr modl) Wih h quilibrium yild modl, xicio oly occurs if h ag-a-ry is oo small This discrpacy ariss bcaus h wo modls mak diffr assumpios abou whhr all h rproduciv aimals ar suscpibl o capur I h quilibrium yild modl maur aimals yougr ha h ag-a-ry provid a buffr agais h ffcs of fishig I h surplus producio modls all h produciv aimals ar vulrabl Surplus Producio Modls ad ishris Maagm Alhough surplus-producio modls ar iuiivly appalig ad ivolv rasoably simpl mahmaics, usig hm as guids for fishris maagm ca lad o srious problms Durig h '6s ad '7s h rsarch focus of may fishris agcis was o drmi for all h xploid socks, ad h mai maagm objciv was o maiai h socks a msy Som rsarchrs cauiod agais such a arrow viw Larki (977), o h Rcommdd Radig lis, dscribs som of h problms wih usig as h objciv for maagig a fishry Hr ar som of h problms: I mulispcis fishris, h socks wih lowr produciviy may d up big limiad if h ra of fishig moraliy is maiaid a msy Th valus for diffr socks may o b idpd Spcis ar o cologically isolad, ad as a cosquc chags i h biomass of o may affc h valus of r ad for ohr spcis ishig a h msy ra, as opposd o ay lowr ra, implis a yougr avrag ag ad a rducd umbr of ag classs, which may lad o grar variabiliy i rcruim ishig may rduc h gic variabiliy of a fish sock ad lad o rducd produciviy i h log ru ishig a is o cos ffciv Th coomically opimum yild is grally lss ha W43/53 Copyrigh 28 by David Sampso Surplus3 - Pag 4