USP Surplus-Producion Models
2 Overview Purpose of slides: Inroducion o he producion model Overview of differen mehods of fiing Go over some criique of he mehod Source: Haddon 2001, Chaper 10 Hilborn and Walers 1992, Chaper 8
3 Inroducion Nomenclaure: surplus producion models = biomass dynamic models Simples analyical models available Pool recruimen, moraliy and growh ino a single producion funcion The sock is hus an undiffereniaed mass Minimum daa requiremens: Time series of index of relaive abundance Survey index Cach per uni of effor from he commercial fisheries Time series of cach
4 General form of surplus producion Are relaed direcly o Russels mass-balance formulaion: B+1 = B + R + G - M C B+1 = B + P C B+1 = B + f(b) C B+1: Biomass in he beginning of year +1 (or end of ) B: Biomass in he beginning of year P: Surplus producion he difference beween producion (recruimen + growh) and naural moraliy f(b): Surplus producion as a funcion of biomass in he sar of he year C: Biomass (yield) caugh during year
5 Funcional forms of surplus producions Classic Schaefer (logisic) form: f B rb B K 1 The more general Pella & Tomlinson form: f B r p B 1 B K p Noe: when p=1 he wo funcional forms are he same
Sock biomass B 6 Populaion rajecory according o Schaefer model 100 90 80 70 60 50 40 30 20 10 0 K: Asympoic carrying capaciy B 1 B rb 1 B K 0 5 10 15 20 25 Time Unharvesed curve
Surplus producion 7 Surplus producion as a funcion of sock size rb B 14 K 12 1 Maximum producion 10 8 6 4 2 K /2 K 0 0 10 20 30 40 50 60 70 80 90 100 Sock biomass B
8 Reference poins Once he parameers have been esimaed, fishery performance indicaors useful o fisheries managemen can be calculaed. Biomass giving maximum susainable yield: BMSY K 2 Maximum susainable yield: MSY rk 4 Effor ha should lead o MSY: E r 2q MSY Fishing moraliy rae a MSY: FMSY r 2
USP Mehods of fiing
10 Esimaion procedures All mehods rely on he assumpion ha: Normally assume: Equilibrium mehods CPUE Never use equilibrium mehods! C qb CPUE Ofen use: U f B CPUE E Regression (linear ransformaion) mehods Compuaionally quick Ofen make odd assumpion abou error srucure Time series fiing Currenly considered o be bes mehod available
11 The equilibrium model 1 The model: B B rb B K C 1 1 The equilibrium assumpion Cach = Producion CPUE B 1 B and hus C rb 1 B K Each years cach and effor daa represen a an equilibrium siuaion where he cach is equal o he surplus producion a ha level of effor If he fishing regime is changed (effor or harves rae) he sock is assumed o move insananeously o a differen sable equilibrium biomass wih associaed surplus producion. The ime series naure of he daa is hus ignored. THIS IS PATENTLY WRONG, SO NEVER FIT THE DATA USING THE EQUILIBRIUM ASSUMPTION C E qb
12 The equilibrium model 2 Given his assumpion: I can be shown ha: B 1 B rb 1 B K Y Can be simplified o C rb 1 B K CPUE C E qb Where C a be E E MSY = a/(2b) MSY = (a/2) 2 /b 2 C ae be
13 The equilibrium model 3
Yield 14 Equilibrium model 4 20 18 16 14 12 10 8 6 4 2 0 If effor increases as he fisheries develops we may expec o ge daa like he blue rajecory Observaions and he fi The rue equilibrium 0 10 20 30 40 50 60 70 80 90 100 Effor
15 The regression model 1 Nex biomass = his biomass + surplus producion - cach 1 2 B B B 1 B rb (1 ) qe B K CPUE q U Subsiuing 2 in 1 gives: U 1 U U U r (1 ) q q q qb q EU U ru U r 1 r E q 1 r U qe U qk U qk 1 1.and dividing by Y F B qe B U q
16 The regression model 2 rae of change in B U U 1 1 inrinsic growh rae r r U qe qk densiy dependen reducion fishing his means: moraliy r M F 0 Regress: U U 1 1 on U and E which is a muliple regression of he form: Y b b X b X where X U and X E 0 1 1 2 2 1 2 r b0 r, b1, b2 q K qk b0 b b 1 2
17 Time series fiing 1 The populaion model: B B B rb Y K 1 1 The observaion model: Uˆ qb or Uˆ qb e i i The saisical model min 2 2 SS U Uˆ or SS lnu lnuˆ min 0 0
18 Time series fiing 2 The populaion model: Given some iniial guesses of he parameers r and K and an iniial saring value B0, and given he observed yield (Y) a series of expeced B s, can be produced. The observaion model: The expeced value of B s is used o produce a prediced series of Û (CPUE) by muliplying B wih a cachabiliy coefficien (q) The saisical model: The prediced series of Û are compared wih hose observed (U ) and he bes se of parameer r,k,b0 and q obained by minimizing he sums of squares
19 Example: N-Ausralian iger prawn fishery 1 40000 35000 30000 25000 20000 15000 10000 5000 0 Effor 1970 1975 1980 1985 1990 1995 0.30 0.25 0.20 0.15 0.10 0.05 0.00 CPUE 1970 1975 1980 1985 1990 1995 7000 6000 5000 4000 3000 2000 1000 0 Cach 1970 1975 1980 1985 1990 1995 Hisory of fishery: 1970-76: Effor and cach low, cpue high 1976-83: Effor increased 5-7fold and cach 5fold, cpue declined by 3/4 1985-: Effor declining, cach inermediae, cpue gradually increasing. Objecive: Fi a sock producion model o he daa o deermine sae of sock and fisheries in relaion o reference values.
20 Example: N-Ausralian iger prawn fishery 2 0.30 0.25 0.20 0.15 0.10 0.05 0.00 CPUE 1970 1975 1980 1985 1990 1995 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 B/Bmsy 1970 1975 1980 1985 1990 1995 1.2 1.0 0.8 0.6 0.4 0.2 0.0 E/Emsy 1970 1975 1980 1985 1990 1995 The observe cach raes (red) and model fi (blue). Parameers: r= 0.32, K=49275, Bo=37050 Reference poins: MSY: 3900 Emsy: 32700 Bcurren/Bmsy: 1.4 The analysis indicaes ha he curren saus of he sock is above Bmsy and ha effor is below Emsy. Quesion is how informaive are he daa and how sensiive is he fi.
21 Adding auxiliary informaion In addiion o yield and biomass indices daa may have informaion on: Sock biomass may have been un-fished a he beginning of he ime series. I.e. B0 = K. Thus possible o assume ha U0 = qb0=qk However daa mos ofen no available a he beginning of he fisheries Absolue biomass esimaes from direc couns, acousic or rawl surveys for one or more years. Those years may be used o obain esimaes of q. Prior esimaes of r, K or q q: agging sudies r: basic biology or oher similar populaions In he laer case make sure ha he esimaes are sound K: area or habia available basis
USP Some word of cauion
23 One way rip One way rip Increase in effor and decline in CPUE wih ime A lo of cach and effor series fall under his caegory. Effor CPUE Time Time Cach Time
24 One way rip 2 Fi # r K q MSY Emsy SS R 1.24 3 10 5 10 10-5 100,000 0.6 10 6 ---- T1 0.18 2 10 6 10 10-7 100,000 1.9 10 6 3.82 T2 0.15 4 10 6 5 10-7 150,000 4.5 10 6 3.83 T3 0.13 8 10 6 2 10-7 250,000 9.1 10 6 3.83 Idenical fi o he CPUE series, however: K doubles from fi 1 o 2 o 3 Thus no informaion abou K from his daa se Thus esimaes of MSY and Emsy is very unreliable
25 The imporance of perurbaion hisories 1 Principle: You can no undersand how a sock will respond o exploiaion unless he sock has been exploied. (Walers and Hilborn 1992). Ideally, o ge a good fi we need hree ypes of siuaions: Sock size Effor ge parameer low low r high ( K) low K (given we know q) high/low high q (given we know r) Due o ime series naure of sock and fishery developmen i is virually impossible o ge hree such divergen & informaive siuaions
26 The imporance of perurbaion hisories 2 db d U U 1 1 r r U qe qk r = max when B is low and E is low This means overfishing When C/E = Kq, hen r = 0 E = rq, r = 0 Biomass Effor Usual siuaion, CPUE (C/E) declines as effor (E) increases. Very lile info on r
27 One more word of cauion Mos abused sock-assessmen echnique! Published applicaions based on he assumpion of equilibrium should be ignored Problem is ha equilibrium based models almos always produce workable managemen advice. Non-equilibrium model fiing may however reveal ha here is no informaion in he daa. Laer no useful for managemen, bu i is scienific Efficiency is likely o increase wih ime, hus may have: q q q Commercial CPUE indices may no be proporional o abundance. I.e. Is he relaionship is ruly proporional? CPUE 0 incr qb
CPUE 28 CPUE = f(b): Wha is he rue relaionship?? 10 9 8 7 6 5 4 3 2 1 0 0 5 10 Biomass Hypersabiliy Proporional Hyperdepleion
29 Accuracy = Precision + Bias No accurae and no precise Accurae bu no precise (Vaguely righ) Precise bu no accurae (Precisely wrong) Accurae and precise I is beer o be vaguely righ han precisely wrong!