Extinction risk depends strongly on factors contributing to stochasticity

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1 co rs dpds srogly o facors corbug o sochascy r A. Mlbour & Ala Hasgs 2 parm of cology ad voluoary ology Uvrsy of Colorado ouldr CO 839 USA 2 parm of vromal Scc ad Polcy Uvrsy of Calfora avs CA 9566 USA 8 Posso vromal dmographc gamma Possobomal bomalv bomaldm bomalgamma Supplmary gur Sochasc ralsaos of h produco fucos for a famly of sochasc cr modls. or ach lvl of al abudac ralsaos of h abudac h grao wr smulad. Pos ar jrd alog h -as for clary. Th curv shows h horcal ma of h dsrbuo. rror bars show h horcal sadard dvao. Modl paramrs: Th sochasc paramrs ( ) wr s so ha h oal varac du o dmographc hrogy was qual o h oal varac du o vromal sochascy; hs also corrspods o qual varac a h saoary po h produco fuco. Abbrvaos dfy h modls lsd g. of h ma. 2 3 Supplmary gur 2 aa from h Trbolum prm showg h bs fg sochasc cr modl. Th bs fg modl was h gav bomalbomal-gamma cr modl whch s a modl for h combd ffcs of all sochasc sourcs: dmographc sochascy sochasc s drmao vromal sochascy ad dmographc hrogy. Th curv shows h horcal ma of h dsrbuo. rror bars show h horcal sadard dvao. als of h modl f ar gv Tabl of h ma

2 Supplmary Mhods rvao of sochasc cr modls. Th orgal drvao of h drmsc cr modl was for fsh populaos udrgog cabalsm of ggs by aduls 26. cr assumd frs ha ggs wr lad a shor dscr v a h bgg of h yar. or h rs of h yar aduls wr fr o cabals ggs ad juvls. Hr w d cr's modl o drv a famly of sochasc modls ha clud sochascy varous aspcs of h lfcycl (s g. h ma ). Th foudao of h sochasc modls s h Posso cr modl whch w drv frs. I cluds hr sourcs of dmographc sochascy h lfcycl: brhs dsy-dpd moraly ad dsy-dpd moraly. To hs basc modl w h add hr vromal hrogy (varao h brh ra m or spac) dmographc hrogy (varao h brh ra bw dvduals wh h populao) or boh o drv rspcvly h gav bomal-vromal () h gav bomal-dmographc (d) ad h gav bomal-gamma (g) cr modls. W h drv modls whr s s drmd sochascally. W frs add sochasc s drmao o h Posso cr modl whch lads o h Posso-bomal (P) cr modl. ally w add vromal hrogy dmographc hrogy or boh o h Possobomal cr modl o drv rspcvly h gav bomal-bomal-vromal () h gav bomal-bomal-dmographc (d) ad h gav bomal-bomal-gamma (g) cr modls. Ths modls form a sd famly of sochasc cr modls whr h g modl s h full modl. Posso cr modl. Th Posso cr modl s a basc modl of dmographc sochascy. Thr ar aduls h populao a m h bgg of h lfcycl. L dvdual aduls gv brh radomly accordg o a Posso procss a a cosa ra β a shor dfd prod a h bgg of h lfcycl. Th h umbr of ggs or youg producd by adul a h bgg of h lfcycl s a Posso radom varabl (.g. rf. 3): ~ Posso β (S) whr β s h ma umbr of brhs pr adul. To bcom a adul ach dvdual offsprg mus ow survv bg cabalsd or dyg from dsy-dpd causs. cr assumd ha a dvdual adul cours ad as ggs or youg radomly wh cosa probably ad o hadlg m 26. Wh hs assumpos abou h sochasc sarch procss h probably c ha a dvdual offsprg s o a by adul by h d of h prod of posur o prdao s c (S2) whr s h adul sarch ra (s.g. p 53 rf. 32). Th probably c ha a dvdual offsprg s o a by ay aduls s hus c c. (S3) Th probably ha a dvdual survvs all forms of moraly durg h lfcycl s h s ( m)c (S4) whr m s h probably of dsy-dpd moraly. Summg up survval of all offsprg from adul gvs a bomal dsrbuo for S h umbr survvg o h adul sag gv ha wr producd by ha adul. Tha s S ~ omal( s). (S5) Sc s Posso (q. S) S has a compoud bomal-posso dsrbuo. y h law of oal probably hs compoud dsrbuo rducs o a Posso dsrbuo (s.g. rf. 3): ( ) S ~ Posso (S6) whr β (-m) ad s mmdaly dfabl as h f ra of populao cras of h drmsc cr modl. ally w add up h survvg offsprg producd by all of h aduls. Sc h sum of dpd Posso radom varabls s also Posso (s.g. rf. 33) h oal offsprg survvg o bcom aduls s: ( ) S ~ Posso. (S7) Thus wh cr's assumpos w fd ha has a Posso dsrbuo wh ma qual o h drmsc cr modl. Th probably mass fuco (pmf) for h Posso cr modl s gv Supplmary Tabl. s al. also drvd a Posso cr modl for dmographc sochascy from smlar assumpos

3 Supplmary Tabl. Probably mass fucos (pmfs) of sochasc cr modls wh dscr dvduals. Modl pmf Posso! gav bomal - vromal gav bomal - dmographc gav bomal - gamma Γ p Posso-bomal! Z gav-bomalbomal-vro. gav-bomalbomal-dmog. gav-bomalbomal-gamma Γ p Th pmf s h codoal probably Pr{ } b a s h bomal coffc ad Γ() s h gamma fuco. Paramrs: f ra of growh dsy dpd paramr (adul sarch ra) f ra of growh for a parcular codo of h vrom h umbr of fmals h probably ha a dvdual s fmal h shap paramr of h gamma dsrbuo for vromal sochascy h shap paramr of h gamma dsrbuo for dmographc hrogy. Paramrs ar dfd h Supplmary Mhods. gav-bomal-vromal cr modl. Ths s a modl for h combd ffcs of dmographc ad vromal sochascy. W allow h vrom o vary sochascally m spac or boh. Th modl dvlopm follows ha for h Posso cr modl. Th brh ra β ow vars sochascally m ad spac () rprsg vromal sochascy. Th umbr of brhs summd ovr all aduls a a parcular m ad plac s a sum of Possos so s also Posso: Posso ~ β. (S8) W rprs varao β m or spac as a gamma radom varabl wh ma β ad shap paramr : ~ amma β β. (S9) Sc h dsrbuo s codoal o h dsrbuo of β s also gamma bu wh ma β ad shap paramr amma ~ β β. (S) Th dsrbuo of oal offsprg s h a gamma mur of Possos (q. S8) whch s o form of h gav bomal dsrbuo (s.g. rf. 33): gom ~ β (S) whr β s h ma brh ra m or spac. As for h Posso cr modl (qs S2-S5) survval of offsprg s bomal: s S omal ~. (S2) Sc s gav bomal (q. S) h umbr of survvg offsprg from m a locao has a compoud bomal-gav bomal dsrbuo. Ths compoud dsrbuo rducs o a gav bomal dsrbuo: gom ~ (S3) whr aga β (-m). Th pmf s gv Supplmary Tabl. Th varac paramr of hs modl s o dsy dpd. Tha s h varac fal abudac vars oly wh h fal abudac ad dos o dpd o h al abudac. Ludwg 5 also drvd a gav bomal cr modl for dmographc ad vromal sochascy bu sc h dd o 3

4 corpora dsy dpd moraly h modl has a dffr paramrsao gav-bomal-dmographc cr modl. Ths s a modl for h combd ffcs of dmographc sochascy ad dmographc hrogy. Th modl dvlopm also follows ha for h Posso cr modl. L adul gv brh radomly a a cosa ra β ι spcfc o ha adul. Th h umbr of offsprg producd by adul a h bgg of h lfcycl s a Posso radom varabl: ~ Posso( β ) (S4) whr β s h ma umbr of brhs for adul. Th ma brh ra for a parcular adul rflcs h dcy of ha dvdual o produc mor or lss offsprg ha ohr aduls. or ampl a larg adul mgh cossly produc mor offsprg ha a small o. To capur varao brh ra bw dvduals w assum ha β follows a gamma dsrbuo wh ma β ad varac drmd by h shap paramr ad hrfor h dsrbuo of s gav bomal (gamma mur of Possos): ~ gom( β ). (S5) Survval s bomal (qs S2-S5) S ~ omal( s). (S6) ad so h dsrbuo of S s a compoud bomalgav bomal dsrbuo whch rducs o a gav bomal (as prvously): ( ) S ~ gom (S7) whr aga β (-m). Sc h sum of dpd gav bomals wh h sam paramr s also a gav bomal (s.g. rf. 33) h survvors summd ovr all aduls s: ( ) S ~ gom. (S8) Thus wh dmographc hrogy addo o dmographc sochascy brhs ad dahs has a gav bomal dsrbuo wh ma qual o h drmsc cr modl. Th pmf s gv Supplmary Tabl. Th aur of h varac h fal abudac for hs modl s parcularly rsg sc h varac paramr of h gav bomal ' s dsy-dpd; s a fuco of h al abudac. A low al abudac ' s small yldg largr varac o ma raos for whras as a hgh al abudac ' s larg ad h varac approachs h Posso lm (g. 2 h ma ). Ths corass wh h modl for vromal sochascy (q. S3) whch h varac dos o dpd o h al abudac. gav-bomal-gamma cr modl. Ths s a modl for h combd ffcs of dmographc sochascy dmographc hrogy ad vromal sochascy. W comb h assumpos of h prvous hr modls spcfcally ha h umbr of offsprg producd by a adul s a Posso radom varabl varao brh ra bw dvduals wh a populao (dmographc hrogy) β follows a gamma dsrbuo wh ma β varao h populao ma a dffr ms ad locaos β follows a gamma dsrbuo wh ma β ad survval s bomal. Wh hs assumpos h dsrbuo of s a compoud gav bomal-gamma dsrbuo (gamma mur of gav bomals) wh ma qual o h drmsc cr modl ad wo varac paramrs ad. Th pmf s gv Supplmary Tabl. Posso-bomal cr modl. Ths s a modl for h combd ffcs of dmographc sochascy ad sochasc s drmao. Th drvao s smlar o h Posso cr modl. Th umbr of offsprg from fmal a h bgg of h lfcycl s a Posso radom varabl: ~ Posso β (S9) whr β s ow h ma umbr of brhs pr fmal rahr ha pr adul. Survval of offsprg s as bfor (qs S2-S5): S ~ omal( s) (S2) whr mporaly prdao volvs boh ss (q. S3). Addg up h survvg offsprg producd by all of h fmals ylds a Posso dsrbuo: S ~ Posso ( β ( m) ) (S2) whr s h umbr of fmals a h bgg of h lfcycl. Sc s a bomal radom varabl ~ omal (S22) whr s h probably ha a dvdual s fmal h dsrbuo of s a Posso-bomal dsrbuo (or roull mur of Possos) wh ma qual o h drmsc cr modl ad β(-m). Th pmf s gv Supplmary Tabl. 4

5 gav bomal-bomal-vromal cr modl. Ths s a modl for h combd ffcs of dmographc sochascy sochasc s drmao ad vromal sochascy. Th drvao s subsaally h sam as h Possobomal ad gav bomal-vromal cr modls. Addg up h survvg offsprg producd by all of h fmals ylds a gav bomal dsrbuo: S ~ gom ( β ( m) ). (S23) Sc s a bomal radom varabl h dsrbuo of s a gav-bomal-bomal dsrbuo (or roull mur of gav bomals) wh ma qual o h drmsc cr modl ad β(- m). Th pmf s gv Supplmary Tabl. gav bomal-bomal-dmographc cr modl. Ths s a modl for h combd ffcs of dmographc sochascy sochasc s drmao ad dmographc hrogy. Th drvao s subsaally h sam as h prvous modl. Th umbr of survvg offsprg from all fmals has a gav bomal dsrbuo: S ~ gom ( β ( m) ). (S24) Th dsrbuo of s hus a gav-bomalbomal dsrbuo wh ma qual o h drmsc cr modl ad β(-m). Th pmf s gv Supplmary Tabl. Th varac h fal abudac dpds o h al abudac whch corass wh h prvous modl for vromal sochascy (q. S23) whch h varac dos o dpd o h al abudac. gav bomal-bomal-gamma cr modl. Ths s a modl for h combd ffcs of all sochasc sourcs: dmographc sochascy sochasc s drmao vromal sochascy ad dmographc hrogy. Ths s h full modl. W comb h assumpos of h prvous hr modls whch cluds h assumpos for h gav bomal-bomal cr modl plus h assumpo ha h umbr of fmals s a bomal radom varabl. Wh hs assumpos h dsrbuo of s a compoud gav bomalbomal-gamma dsrbuo (gamma mur of gav bomal-bomals) wh ma qual o h drmsc cr modl ad β(-m). I addo o h wo varac paramrs ad h probably ha a dvdual s fmal also flucs h varac. Th pmf s gv Supplmary Tabl. Toal varac du o vromal sochascy or dmographc hrogy. To compar modls cludg vromal sochascy wh modls cludg dmographc hrogy w s h varac paramrs of hs modls (rspcvly ad ) so ha h oal varac was qual. Th oal varac s gv by grag h varac fuco ovr hus σ (S25) 2. d whch wh h oal varac for vromal sochascy ad dmographc hrogy ar quad ylds:. (S26) Supplmary scusso sos o h modls. Svral sos ad varaos o our famly of modls ar possbl. Whl h modls dscrbd abov ad h ma clud dmographc sochascy all sags of h lfcycl (brhs dsy dpd ad dsy dpd moraly) w cludd vromal sochascy ad dmographc hrogy oly brhs. Moraly s also subjc o vromal sochascy ad dmographc hrogy. Howvr w show hr ha addg vromal sochascy ad dmographc hrogy moraly o our mchasc modls dos o alr our coclusos hr bcaus h ffc of moraly varac s glgbl compard o varac brhs or s ffcs ar fully accoud for by h modls dscrbd h ma. Icluso of sochascy dsy dpd moraly s sraghforward. or ampl a appropra form for sochasc varao h probably of dsy dpd survval (-m) s h ba dsrbuo. Th wh dmographc sochascy ad vromal sochascy ar h oly sourcs of moraly varao h rsulg modl s h Posso-ba-vromal cr modl. Sc h f ra of cras β(-m) s h mulpl of brhs ad dsy dpd survval h ffcs of vromal sochascy moraly o populao growh ad co ar dsgushabl from h ffcs of vromal sochascy brhs 5

6 (Supplmary gur 3). Th y ffc of vromal sochascy hr brhs or moraly s o h varac of ; h form of h dsrbuo s o mpora. urhrmor bcaus moraly s boudd bw ro ad o varac moraly s boudd rachg a mamum of m(-m). I coras varac brhs s ulmd. Thus w pc h gras poal for sochascy o b corbud by brhs. vrhlss wh moraly mas a mpora corbuo o sochascy s capurd phomologcally (as dmosrad Supplmary gur 3) by h or cr modls usd h ma whch modl varao as a gamma dsrbuo. 2 σ P m (v) m (dm) (v) (dm) dsrbuo ad h probably of dyg vars dpdly amog dvduals dmographc hrogy has o ffc o h varac of survval; h varac survval s always qual o h varac du o ordary dmographc sochascy for h sam ma probably of moraly. Th lac of a ffc of dmographc hrogy s dmosrad Supplmary gur 3 whr h varac m s s clos o h mamum possbl y h varac abudac h grao s dsgushabl from h Posso cr modl. Th ffc of vromal sochascy or dmographc hrogy moraly h dsy dpd cabalsm paramr s mor compl. Thr ar smlar rsrcos o h magud of h ffc of sochascy as h varac h dsy dpd probably of survval c s boudd by c(- c). Varac also chags h form of h ma modl hr by modfyg h cr paramrs or alrg h ma modl away from h cr form. Usg a gamma dsrbuo for w show by smulao ha sochascy rsuls a varac profl for ha pas wll o h rgh of h saoary po h cr produco fuco whch s dramac coras o modls wh sochascy m or (Supplmary gur 3). Ths faur s abs from our daa ad so w ar cofd ha hs s o mpora our laboraory sysm Supplmary gur 3 Varac h umbr of dvduals h grao as a fuco of h umbr of dvduals h curr grao for modls wh sochascy moraly. Modl paramrs: 5.5 m.9. sy dpd moraly (m) was cludd as a ba radom varabl for hr vromal sochascy (σ 2 m.2) m (v) or dmographc hrogy (σ 2 m.8) m (dm). To modl dsy dpd moraly varao was cludd as a gamma radom varabl ( 25; σ 2.) for hr vromal sochascy (v) or dmographc hrogy (dm). All modls clud dmographc sochascy as h Posso cr modl. Varacs wr smad by smulao. ac varacs for h Posso cr modl (P) ad gavbomal-vromal cr modl (; σ 2 2.5) ar show for rfrc (blac curvs). Th vrcal bar dcas h poso of h saoary po h cr produco fuco. Th ffc of dmographc hrogy dsy dpd moraly s svrly rsrcd so ha vr crass h varac abudac 7. Wh dvdual moraly s dscrbd by a roull obusss of h modl f. To am h robusss of h modl slco procss for our daa ad prmal dsg w grad arfcal daa assumg ha alrav modls o h bs fg g modl wr fac h ru modl ad h rfd h modls ad carrd ou modl slco usg AIC as do wh h ral daa. ach smulao prm was ad wh qual o ha usd h laboraory prm usg h sam lvl of rplcao ad wh modl paramrs as smad from h daa for h alrav modls (Tabl h ma ). W amd h ad d modls as alrav ru modls. Th smulaos show ha our prmal dsg was cll for dsgushg bw modls ad ha our frc ha h g modl was h bs fg modl s robus. Wh h ru modl h smulao was pur vromal sochascy () was corrcly dfd as h bs modl (ΔAIC > 2) 99.% of smulao rus wh compard o h modl for pur dmographc hrogy (d). Th probably of wrogly accpg h d modl was rmly low ( smulao rus). Th md 6

7 modl (g) was rarly (2.%) dfd as h bs modl wh was h ru modl. Covrsly wh h ru modl was pur dmographc hrogy (d) was corrcly dfd as h bs modl 99.8% of smulao rus wh compard o h modl for pur vromal hrogy (). Th probably of wrogly accpg h modl was rmly low ( smulao rus). Th md modl (g) was rarly (.8%) dfd as h bs modl wh d was h ru modl. as ad masurm rror. Masurm rror our laboraory daa was glgbl (w sma lss ha o prc basd o rpa cous) bu ca b cosdrabl fld daa 34 ad has a larg fluc o smag co rs 35. To am h poal ffc of masurm rror o paramr smas w grad arfcal daa assumg ha h g modl was h ru modl o whch w addd crasg amous of masurm rror os ad h r-smad h modl paramrs. ach smulao prm was ad wh qual o ha usd h laboraory prm usg h sam lvl of rplcao ad wh "ru" modl paramrs as smad from h daa for h g modl (Tabl h ma ). W addd logormal rror o boh ad (roudd o h ars gr) smulag h commo fld suao whr h magud of h rror varac crass wh abudac. or parcular fld sysms a mchasc modl of h masurm procss would b dsrabl o proprly accou for h varous corbuos o h masurm rror (.g. rf 36). Th smulao rsuls show ha paramr smas wr basd by masurm rror (Supplmary gur 4). I parcular h varac paramrs whch ar crcal o smag co rs wr svrly basd by masurm rror. As masurm rror crasd h appar corbuo from vromal sochascy crasd rlav o ha from dmographc hrogy (small valus of or dca hghr varac) so ha a hgh lvls of masurm rror hr appar rlav mporac was rvrsd from h ru modl. Ths has mpora mplcaos for smag co rs bcaus udrsmag h rol of dmographc hrogy wll udrsma h co rs. Th smulao rsuls also show ha h mamum llhood smas of h bologcal ra paramrs ( ) wr ubasd bu ha h varac paramrs wr basd spcally (Supplmary gur 4; masurm rror qual o ro). W calculad bas corrcd smas of h varac paramrs by subracg h rlav dvao (aural logarhm scal) obsrvd h smulao sudy from h mamum llhood sma (Tabl ). ^ ^ ^ ^ Masurm rror (σ) Supplmary gur 4 Th ffc of logormal masurm rror o paramr smas from h gav-bomal-bomal-gamma cr modl. σ s h sadard dvao of h masurm rror o h aural logarhm scal. ach po s h ma of 85 rplca smulaos. Supplmary os udg. Ths sudy was fudd by S gra 565 o A.H. ad.a.m. Addoal rfrcs. 3. Karl S. & Taylor A.. A Iroduco o Sochasc Modlg. (Acadmc Prss Sa go 998). 32. Hlbor. & Magl M. Th cologcal cv: Cofrog Modls wh aa. (Prco Uvrsy Prss Prco w Jrsy 997). 33. Johso. L. Kmp A. W. & Ko S. Uvara scr srbuos. (Joh Wly ad Sos w Jrsy 25). 34. s. al. smag dsy dpdc procss os ad obsrvao rror. col. Moogr (26). 35. Holms.. smag rss dclg populaos wh poor daa. Proc. al. Acad. Sc. U. S. A (2). 36. Muhlfld C. C. Tapr M. L. Sapls.. & Shpard.. Obsrvr rror srucur bull rou rdd cous Moaa srams: Implcaos for frc o ru rdd umbrs. Tras. Am. sh. Soc (26). 7