Review of Salmon Management Model Migration Algorithms

Transcription

1 Review of Salmo Maagemet Model Migatio Algoithms by James G. Nois Columbia Basi Reseah Shool of Fisheies Uivesity of Washigto Box Seattle WA 9895 (360) Pepaed Fo: NMFS Salmo Model Review Committee Jue 9 997

2 Itodutio The Natioal Maie Fisheies Sevie (NMFS) has esposibility fo admiisteig the US Edageed Speies At (ESA) fo aadomous fish ad shaed esposibility fo maagig oea salmo fisheies. Beause of this dual esposibility NMFS has a eed to evaluate the ostaits of ESA osideatios o havest maagemet ad the impats of havest ad habitat maagemet o the eovey of depleted stoks. These evaluatios eed to be made o a oastwide basis ad should be iteally osistet ad osistet with models used i othe maagemet aeas. I 997 NMFS otated with the Uivesity of Washigto Shool of fisheies to begi developig a sigle boad modelig famewok that will assist NMFS i meetig its salmo maagemet esposibilities. A Review Committee omposed of State Tibal ad Fedeal salmo biologists ad modeles was fomed to help develop model speifiatio idetify eseah pioities esue osistey with existig models ad etify eseah models fo maagemet use. The oveall pojet has the followig objetives:. Povide a ommo famewok fo both osevatio isk assessmet ad havest maagemet aalysis; 2. Expad the geogaphi sope of uet havest models; 3. Lik oho ad hiook models; 4. Iopoate life yle (podutio) models fo both speies to evaluate log-tem havest ad osevatio stategies; 5. Allow flexibility to aommodate ew methods ad model desigs; 6. Povide a itefae with a lage subset of oea ad feshwate databases maitaied by the Paifi States Maie Fisheies Commissio. The two salmo maagemet models uetly used most extesively fo fisheies that impat stoks listed ude the ESA ae the Paifi Salmo Commissio (PSC) Chiook Model ad the Fishey Regulatio Aalysis Model (FRAM). Neithe of these models ilude speifi migatio algoithms to simulate the movemet of fish though a gautlet fishey. Thee ew models have bee poposed to simulate salmo migatio moe auately: a State Spae Model (Newma 995) the PSC Seletive Fishey Model (PSC 995) ad the Popotioal Migatio (PM) Model (Mooe et al. 996). The pupose of this epot is to desibe the migatio algoithms used i these five salmo maagemet models. The State Spae Model is the most geeal of the five models ad has bee poposed fo use i a ew salmo life yle model. Thus this epot desibes the existig algoithms i tems ommo with the State Spae Model.

3 Notatio The otatio used i this epot follows that ommoly used by the State Spae Model. I desibe the migatio algoithms with efeee to a sigle ohot of fish. A ohot may be a sigle age lass fom a sigle stok o it may be a maked/tagged sub-ompoet of that age lass. No stok o age subsipts ae used to make the otatio easie to ead. All of the models use disete time steps ad the otatio is defied with espet to time iteval [t- t). Table lists the otatio ad Fig. illustates the otatio gaphially. Table. Commo otatio used i this epot (also see Fig. ). Vaiable Defiitio Idex Vaiables t R Time Geogaphi egio. Fo models i whih thee is a oe-to-oe oespodee betwee geogaphi egio ad fishey also idexes fishey. The total umbe of geogaphi egios i the model. State Vaiables Abudae of fish i egio at time t Obseved ath i egio duig the time iteval [t-t). s Abudae of fish i egio that do ot suffe atual motality ad/o fishig motality duig the time iteval [t-t). Natual Motality ad Suvival Paametes M Istataeous atual motality ate i egio duig the time iteval [t- t). υ Fatio of the ohot i egio killed by atual motality duig the time iteval [t-t). s Total suvival ate i egio duig the time iteval [t-t). Fishig Motality Paametes E Amout of fishig effot i egio duig the time iteval [t-t). h Regioal havest ate defied as the fatio of the ohot loated i egio havested as legal ath i egio duig the time iteval [t-t). F Istataeous total fishig motality ate i egio duig the time iteval [t-t). µ Fatio of the available ohot i egio killed by all types of fishig motality (iludig iidetal motalities) duig the time iteval [t-t). Migatio Paametes D A set of doo egios that otibute fish to egio at the ed of time peiod t- (e.g. used i the PM Model). m i j Fo a give ohot ad time the fatio of the abudae i egio j movig to egio i. 2

4 Note that the total fishig motality ate iludes both legal ad iidetal motalities. Thus the legal havest ate is ot eessaily equal to the total fishig motality ate (i.e. µ ). Also ote the followig elatioships: h s ( Mt Ft e ) = + υ = e Mt F µ t e = s = s The vaiable s a be thought of as the umbe of fish i egio at the ed of time iteval [t- t) just befoe they ae edistibuted amog the geogaphi egios fo the stat of the ext time iteval [t t+). State Spae Model I matix otatio the detemiisti State Spae Model osists of two equatios: = M S = H t t t t t t t The abudae vetos t ad t ae omposed of R elemets (oe abudae fo eah egio). Eah migatio matix M t is a R x R squae matix of m i j elemets ad the elemets i eah olum must sum to oe. Eah suvival matix S t is a diagoal matix with R elemets (e.g. s ). I expaded fom equatios () ad (2) look like this: t m = Rt mr m m R t h = Rt hr R R h h R R R s t Rt s t Rt t Rt Note that we a defie a ew veto s = S t t that epesets the fish that do ot suffe atual o fishig motality duig the iteval [t-t). Thus the migatio matix a be thought of as beig applied to the suvivig ohot at the ed of the time iteval ad eah elemet of the ew abudae veto a be witte 3

5 R = m s. j jt j= I Ke Newma s appliatio of the State spae Model he sets M =0 fo all ad t. Thus i his appliatio we have s = e F υ =0 F t µ t = e = s t. PSC Chiook Model The PSC Chiook Model defies o speifi geogaphi egios ad theefoe has o speifi fish migatio algoithm. Howeve by sepaatig the 25 model fisheies ito petemial ad temial ategoies the model assumes a de fato oept of fish migatio fom the petemial aea to the temial aea. The petemial fisheies ae geeally offshoe oea fisheies (e.g. oea toll fisheies) while temial fisheies ae geeally eashoe ad ive fisheies (e.g. oea et ad spot fisheies). Oe the petemial fishey havests ad iidetal motalities have bee take the model sepaates eah ohot ito a oea u (that stays i the oea ad is available fo havest the followig yea) ad a temial u (that is available fo havest by temial fisheies). Thus what ae temed matuatio ates i the PSC Chiook Model a be thought of as migatio ates fom the oea (petemial aea) to the eashoe ad ive aeas (olletively alled the temial aea). Some oea et fisheies ae osideed to be temial fisheies fo olde age lasses eve though these fisheies ae ot physially loated ea the atal steam. The idea is that at some poit duig the yea the olde (matue) membes of eah ohot deide to stat migatig dow the oast. It is assumed that the olde fish aptued by the eashoe et fisheies ae pat of the matue migatig potio of the stok. Usig the mathematial otatio of the PSC Chiook Model the matuatio ates ae applied to eah ohot as follows: whee TemRu = ( ORu PeTemMot) MatRt TemRu ORu = the abudae of fish i the temial aea at the stat of the temial time peiod; = the abudae of fish i the petemial aea duig the petemial time peiod afte atual motalities have bee emoved; 4

6 PeTemMot = total fishig motalities (i.e. legal athes plus iidetal motalities) i the petemial aea duig the petemial time peiod. The tem ORu - PeTemMot is the abudae of fish ot suffeig atual motality ad suvivig the fishig poess duig the petemial time peiod. I the otatio of the State Spae Model this tem is equivalet to the fist elemet of the s t veto. The seod elemet of the s t veto is zeo beause it is assumed that o fish ae loated i the temial aea duig the petemial time peiod. The tem TemRu is equivalet to the seod elemet of the t veto (i.e. abudae i egio 2 at the stat of time peiod 2). The fist elemet of the t veto is the umbe of fish fom the ohot that emai i the petemial (o oea) aea. I matix otatio the aual migatio patte fo eah ohot is expessed as follows (whee = petemial aea ad time step; 2 = temial aea ad time step): whee 2 22 m = m m m s s 2 m s + m s = m s + m s = petemial aea (i.e. oea) abudae at the stat of the temial time step 22 = temial aea abudae at the stat of the temial time step m = - MatRt m 2 = 0 (i.e. o fish move fom the temial aea to the petemial aea) m 2 = MatRt m 22 = 0 (i.e. thee ae o fish i the temial aea to keep i the temial aea) s = petemial aea (i.e. oea) abudae at the ed of the petemial time step s 2 = 0 (i.e. o fish i the temial aea at the ed of the petemial time step) These equatios ae ow i the same fom as Equatio (3) fo the State Spae Model. I tems of the matuatio ate we have = ( MatRt) s = MatRt s PSC Seletive Fisheies Model This model a have ay umbe of maie geogaphi aeas but i its iitial appliatio otais just five: 5

7 West Coast Vaouve Islad (OCNN); Washigto/Oego Oea (OCNS); Stait of Geogia (GEOS); Stait of Jua de Fua ad Sa Jua Islads (SJDF); South Puget Soud (SSND). Some stoks also have temial aeas (e.g. CRTM = Columbia Rive temial aea) ad all stoks have a geei esapemet o spawig aea (ESCP). This model was desiged pimaily as a oho model i whih all membes of a ohot (o bood) matue ad etu to the atal steam i the same yea. The model is geeally used to examie the potetial effets of seletive fishey maagemet o a limited umbe of stoks duig a sigle yea. Thus this model is ot desiged to simulate the effets of oastwide maagemet egulatios o all stok ad fisheies. This model a be u i eithe detemiisti o stohasti mode. The model flow is as follows:. Compute iitial abudae of eah stok; 2. Distibute iitial abudae to the five fishig aeas; 3. Time loop with: a. Natual motality; b. Fishig motality;. Redistibute the fish (iludig esapemet). Seveal simplifyig assumptios wee made to estimate the popotio of eah stok i eah aea migatig i eah time step:. No migatio oued betwee geogaphi egios duig statistial weeks though 32; 2. All migatio was dieted towad the ive of oigi; 3. Hathey ad wild fish had the same migatio timig ad pathway; 4. 33% of IStk i the OCNN egio migated aoud the oth ed of Vaouve Islad to the GEOS egio; ad 5. Cath pe uit effot i a fishey povided a ubiased measue of stok abudae. Mathematially the migatio ompoet of this model is vitually idetial i stutue to the State Spae Model. Duig eah time step atual ad fishig motalities (iludig iidetal motalities) ae emoved fom the ohot fist. The emaiig fish i a give aea ae the edistibuted amog the aeas by assigig dispesio ates to eah aea. I the otatio of the PSC Seletive Fishey Model the ew abudaes ae omputed by the followig equatio: 6

8 N A * = I sa t saa t a= whee N sat = the abudae of stok s i aea a at time t; I sabt = the umbe of fish fom stok s that move fom aea a to aea b at time t. I stohasti mode the elemets of the migatio matix ae seleted adomly fom a multiomial distibutio with paametes: γ sabt = pobability that a fish fom stok s moves fom aea a to aea b at time t These pobabilities ae estimated fom ath ad effot data usig a solve outie i Miosoft Exel. Table 2 lists the dispesio paametes. I detemiisti mode the pobabilities ae eplaed by the fatios movig fom oe egio to aothe. Thus i the PSC Seletive Fishey Model the γ sabt ae equivalet to the m i j elemets of the migatio matix of the state spae model. Table 2. Dispesio paametes by week fo the PSC Seletive Fishey Model. Stok Doo Reeive WCVI OCNN ESCP WCVI SJDF OCNN LFGS OCNN SJDF LFGS OCNS SJDF LFGS SJDF GEOS LFGS GEOS ESCP NPSD OCNN SJDF NPSD OCNS SJDF NPSD GEOS SJDF NPSD SJDF NSND NPSD NSND ESCP SPSD OCNN SJDF SPSD OCNS SJDF SPSD GEOS SJDF SPSD SJDF SSND SPSD SSND ESCP NWAC OCNN OCNS NWAC OCNS WCTM NWAC SJDF OCNS NWAC WCTM ESCP CRIV OCNS CRTM CRIV SJDF OCNS CRIV CRTM ESCP

9 Table 2. Coluded. Stok Doo Reeive WCVI OCNN ESCP WCVI SJDF OCNN LFGS OCNN SJDF LFGS OCNS SJDF LFGS SJDF GEOS LFGS GEOS ESCP NPSD OCNN SJDF NPSD OCNS SJDF.00 NPSD GEOS SJDF NPSD SJDF NSND NPSD NSND ESCP SPSD OCNN SJDF SPSD OCNS SJDF SPSD GEOS SJDF SPSD SJDF SSND SPSD SSND ESCP NWAC OCNN OCNS NWAC OCNS WCTM NWAC SJDF OCNS NWAC WCTM ESCP CRIV OCNS CRTM CRIV SJDF OCNS CRIV CRTM ESCP The followig example illustates the elatioship betwee the γ sabt ad the m i j. Table 3 lists the dispesio ate (ad No-Dispesio Rate ) paametes fo the South Puget Soud stok duig week 40. Table 3. Dispesio ad o-dispesio (= - dispesio) ate paametes fo the South Puget Soud stok duig week 40 used i the PSC Seletive Fishey Model. Doo Regio Reeivig Regio Dispesio Rate No-Dispesio Rate Oea Noth Stait Jua de Fua Oea South Stait Jua de Fua Geogia Stait Stait Jua de Fua Stait Jua de Fua South Puget Soud South Puget Soud Esapemet It is assumed that fish that do ot dispese fom a doo egio emai i the doo egio. Thee ae six aeas outig the Esapemet aea ( = OCNN; 2 = OCNS; 3 = SJDF; 4 = GEOS; 5 = SSND; ad 6 = ESCP). The oespodig State Spae Model migatio matix epesetig this movemet patte is: 8

10 M = Note that values alog the diagoal epeset the o-dispesio ates (i.e. the fatio of the ohot that emais i the egio). Popotioal Migatio (PM) Model The PM model does ot have speifi geogaphi objets. Howeve fishey defiitios at as de fato geogaphi objets. Fo example the spot fisheies ae geeally defied by statistial epotig aea suh as Aea 7 Aea 8 Buoy 0 et. Thee ae 45 fisheies/geogaphi aeas i this model. Thee ae thee gea types used to defie fisheies ad eah gea type patitios the geogaphi age diffeetly. Fo example withi the toll gea goup thee is a Nothwest Vaouve Islad ad a Southwest Vaouve Islad ad withi the spot gea goup thee is oly oe fishey fo all of the West Coast of Vaouve Islad. This methodology assumes that fish available fo the WCVI spot fishey ae ot the same fish that ae also available fo the two WCVI toll fisheies. I othe wods the model taitly assumes that thee ae thee distit geogaphi egios off WCVI. I tems of the otatio used i this epot the subsipt idexes both geogaphi egios ad thei assoiated fisheies. Thee is o expliit movemet of fish betwee fishig egios. Howeve the fisheies ae assumed to have a geogaphi ad tempoal odeig suh that the fish available i a give fishey ad time ae eeived oly fom desigated doo fishig egios (desigated by the set D ). Note that seveal eeivig egios may shae some of the doo egios. Fo example egio 2 may eeive fish fom egios ad 2 ad egio 3 may eeive fish fom egios 2 ad 3. The model expesses hages i fishey motalities elative to a base yea. Fo a give base yea ad stok the model begis with iput data fo thee paametes (obtaied fom u eostutios): h ad E. The havest ates ( h ) ae assumed to be fishey ad time speifi but ot stok speifi. That is the same h tem is applied to all stoks havested i egio (fishey) at time t. The egio ad time speifi abudaes at the stat of time iteval [t-t) ae omputed by 9

11 t = EV h whee EV is a stok ad yea speifi Eviometal Vaiability sala. Fo the base yea EV =. Fo yeas othe tha the base yea these iitial abudaes ae adjusted by the EV salas to eflet the bood yea stegth elative to the base yea. These salas have the same effet as EV salas i the PSC hiook model. Note that the egioal abudaes ae detemied solely fom the iput data fo that egio ad time step. Thus it is possible fo the total abudae fo a stok to be geate at time t tha at time t-. That is? >=<. The suvivos afte the fishig poess duig the peiod [t-t) ae omputed by s = The model simulates the impats of hages i egio speifi havest ates as follows (pimes i the otatio idiate vaiables ude the ew havest maagemet seaio). Let E be the ew effot level i egio at time t. Let the assoiated ew havest ate be some futio of the base peiod effot base peiod havest ate ad the ew effot level. That is ( ) h = f h E E t t t This futio may be a simple liea salig suh as h E = ht E I the fist time step the ew havest ate is applied to the oigial abudae ( 0 ) to get the ew ath: = h 0 New suvivos i the fist time step ae omputed by s = 0 Thus duig the fist time step we have 0

12 = h 0 = h 0 ad we a wite h =. h Solvig fo the ew ath we get = h h Thus the ew egioal athes i the fist time step ae just the old egioal athes saled up o dow by the atio of the ew ad old havest ates. Note that the salig is based o the elative havest ates ot the effot levels. If a o-liea elatioship is used to elate effot ad havest ate doublig the effot may ot eessaily double the havest ate ad the ath. I subsequet time steps the ew havest ate is ot applied to the oigial loal abudae. Istead the oigial loal abudae is saled up o dow to eflet the total hages i fishig motality i the doo egios duig the pevious time step. The salig fato is the atio of the ew suvivos to oigial suvivos (fom the doo aeas) duig the pevious time step ad the ew abudae is omputed as follows: = Dt Dt s s whee the ew suvivos at time t- ae omputed by s =. If we sepaate the oigial (base) vaiables fom the adjusted vaiables we a wite t = s D D s Note that this equatio is quite simila to the equatio fo the elemets of the ew abudae veto i the State Spae Model fomulatio amely:

13 R = m s j jt j= Reall that m j is the fatio of the fish loated i doo egio j movig ito egio. If oe assumes a ostat migatio ate duig the iteval [t-t) fom all doo egios j ito egio (all the ostat Kt = m j fo all j) the migatio ate tems a be moved outside the summatio sig: R = jt j= K s Fo the PM model we a set K = t. st D I the PM model the tem K must be thought of as a migatio idex athe tha a migatio ate. It is the atio of the fish loated i egio at the stat of time peiod t to the total potetial doo fish fo egio at the ed of time peiod t-. Sie the abudaes fo diffeet time peiods ae omputed idepedetly this atio a be geate tha oe ad does ot epeset a movemet of fish fom oe aea to aothe. Regadless of the biologial itepetatio of the K tems the impotat poit is that the mathematial omputatios of the PM Model a be oduted usig the matix fomulatio of the State Spae Model by substitutig the K tems i the appopiate positio of the migatio matix. Fo time t eah K tem is plaed i eah olum of the th ow that oespods to a doo egio fo egio. The State Spae Model fomulatio of the PM model is best explaied with a example. Coside a ase of thee fisheies (egios) ad two time steps. Assume that fo the seod time step the doo egios ae as follows: Reeivig Regio Doo Regios 2 ad ad 3 The fom the iput data we have: K 2 = s 2 2

14 K K = = s s 22 + s s 2 3 I tems of the matix omputatios we have K = K K 0 K K s s s 2 3 whee the s abudae veto epesets the fish suvivig the ew havest egime duig the fist time step. I patie s veto would be omputed by multiplyig a ew suvival matix (with the diagoal elemets epesetig the ew suvivals detemied fom the oigial ad ew effot levels) times the oigial abudae veto. The speifi elemet fomulas fo the 2 veto ae: s = s s = s s = s s + s s + s Thus it is possible to ast the PM model i the geeal matix famewok as follows: Use the egioal abudaes at the stat of the fist time step fo the iitial abudae veto; Use the iput ath ad havest ate data to ompute the K tems; Use the iput ad adjusted effot data to eate ew suvival maties; Plae the K tems i the appopiate loatio i the migatio maties; Pefom the matix omputatios i hoologial ode. 3

15 It is lea fom the above fomulae that the PM model makes the tait assumptio that fo a give ohot ad time step fish migate fom all doo egios at the same ate. This is vey diffeet fom the PSC Seletive Fishey Model that estimates sepaate migatio ates fo eah doo aea. Reall fom the PSC example fo week 40 fo the SPSD stok that fish migate ito the SJDF aea fom fou othe aeas at vey diffeet ates: OCCN = 0.67; OCNS = 0.79; GEOS = 0.03; ad SJDF = The PM model would equie that fish would ete the SJDF aea at the same ate fom eah of the fou aeas. We also ote that if the set of doo egios fo eah aea ad time step ( D ) is the etie set of possible egios the PM model is vitually idetial to a sigle pool model i whih the oigial egioal havest ates ae defied with espet to the total abudae of the ohot at ay give time (istead of loal abudae withi a egio). That is ude a sigle pool model the pooled havest ate ( ph ) is defied as: ph = t. t The ath equatio is: = ph t. The base egioal athes ae the same ude both fomulatios of the havest ate (PM ad Sigle Pool) so we a wite: ad pht t = ht ph = h Thus the base havest ate i a sigle pool model is equal to the base loal havest ate i the PM model saled by the fatio of the total ohot abudae loated i the egio. Reall that duig the fist time step of the PM model the loal abudaes ae ot adjusted befoe applyig the adjusted havest ates ad the adjusted athes ae just saled by the atio of the ew to the old havest ate: = h h 4

16 Fo the sigle pool model the same is tue. Hee we have = ph 0 = ph 0 Thus we a wite ph = ph ad = ph ph Thus at the fist time step the egioal athes ae saled up ad dow to eflet the hages i the havest ates. If the set of doo egios fo eah aea ad time step is the etie set of possible egios the the same will be tue fo all time steps. Fishey Regulatio Aalysis Model (FRAM) This model is simila to the PSC Chiook Model i that it has o speifi geogaphi aeas but does patitio the fisheies ito petemial ad temial ategoies ad also iludes sepaate exteme temial aeas. This model uses mothly time steps. At eah time step the matuatio algoithm is alled but it is ulea whethe o ot a matuatio alulatio atually ous at eah time step. Need futhe laifiatio o this. The Temial Aea Maagemet Modules (TAMMs) used i FRAM do ot have ay migatio ompoets. The havest ates i the exteme temial aeas ae all defied with espet to the abudae of fish eteig the exteme temial aea. 5