A Statistical, Age-Structured, Life-History-Based Stock Assessment Model for Anadromous Alosa

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1 American Fisheries Sociey Symposium 35: , by he American Fisheries Sociey A Saisical, Age-Srucured, Life-Hisory-Based Sock Assessmen Model for Anadromous Alosa A. JAMIE F. GIBSON 1 Acadia Cenre for Esuarine Research, Pos Office Box 115, Acadia Universiy, Wolfville, Nova Scoia BOP 1XO, Canada; and Deparmen of Biology, Dalhousie Universiy, Halifax, Nova Scoia B3H 3J5, Canada RANSOM A. MYERS 2 Deparmen of Biology, Dalhousie Universiy, Halifax, Nova Scoia B3H 3J5, Canada Absrac. We presen a populaion dynamics model based on he life hisory of anadromous Alosa. The model is buil around a cach-a-age array o which we have added an addiional dimension o include previous spawning hisory. The previous spawning hisory deermines he number of imes ha a fish is exposed o riverine impacs ha conribue o overall moraliy raes. The model is designed o incorporae he kinds of daa ypically colleced for anadromous Alosa socks, such as cach-a-age daa, spawning escapemen couns a fish ladders, larval and juvenile abundance indices, and urbine moraliy esimaes. We show how he model can be adaped o individual socks and managemen quesions and how he model parameer esimaes can be obained using maximum likelihood. We demonsrae his approach wih examples for wo alewife Alosa pseudoharengus populaions in Nova Scoia. Inroducion The biology, fisheries, and daa for anadromous Alosa populaions differ from hose of marine fish. Consequenially, radiional fisheries models ha are designed for marine species are ofen no appropriae for Alosa and do no fully uilize he available daa. For example, riverine fisheries arge only maure fish whose availabiliy o he fishery depends no only on recruimen and moraliy raes bu also on variable mauriy schedules. Addiionally, anadromous fish are ofen affeced by oher anhropogenic aciviies such as hydroelecric generaion and barriers (Rulifson 1994, and survivorship differs depending upon wheher or no a fish is sexually maure and how many imes i has previous spawned. A fish s previous spawning hisory is available from is scales (Marcy This informaion can herefore be included in he model and used o separae marine sources of moraliy (a funcion of age, from riverine sources of moraliy (a funcion of previous spawning hisory. Auxiliary daa such as escapemen couns a fish 1 Corresponding auhor: presen address: Deparmen of Fisheries and Oceans, Diadromous Fish Division, Science Branch, Pos Office Box 1006, Darmouh, Nova Scoia B2Y 4A2, Canada. 2 ladders, larval abundance indices, and couns of seaward migraing juveniles are ofen available and can also be incorporaed ino an assessmen model. Fournier and Archibald (1982 and Deriso e al. (1985 developed he general heory for saisical cach-a-age models for sock assessmen ha allow auxiliary daa o be incorporaed ino he model. Wih he developmen of sofware ha allows complex, nonlinear models o be fied rapidly, hese echniques are being used more frequenly for sock assessmen. Here, we presen an age-srucured, life hisory model for anadromous Alosa and show how i can be linked o he kinds of daa ypically colleced for Alosa socks. This approach provides a flexible, dynamic framework for Alosa sock assessmen and can be used o address criical managemen issues. The Model The following model is based on he life cycles of he hree species of anadromous Alosa ha are indigenous o easern Norh America: American shad A. sapidissima, blueback herring A. aesivalis, and alewife A. pseudoharengus. Alhough differences exis among species and among populaions of he same species, he life cycles of anadromous Alosa have several shared characerisics. Aduls of hese species ascend rivers during he spring and spawn 275

2 276 GIBSON AND MYERS in lakes, pools, or sill waers wihin he waershed. Young of he year remain in freshwaer unil midsummer o lae fall when hey emigrae o he sea. Fish maure a 2 7 years of age, and mauriy schedules vary among populaions and among years. Mos populaions are ieroparous. Socks are ofen fished during he spawning migraion, alhough inercep fisheries exis in some regions. Legge and Carscadden (1978 describe he life hisory of American shad in greaer deail, while Loesch (1987 provides an overview of he life hisories of blueback herring and alewife. We model his life hisory as explained below. Of primary ineres is he number of fish of sex s (indexed m for male or f for female, age a, ha have spawned p imes previously and are reurning o he river in year, which we denoe as N,s,a,p. Assuming a nonselecive fishery, he number of eggs produced in year (Q is a funcion of he number of females in year (N,f,a,p, he exploiaion rae in ha year (u, and an age and previous spawning-specific fecundiy (f a,p and is given by, f, a, p 1 a, p ap, Q = N ( u f. Densiy-dependen naural moraliy wihin he spawning and nursery areas is hough o regulae Alosa populaion size (e.g., Crecco and Savoy This is o say ha he rae of larval naural moraliy (M larval varies among years as a densiydependen funcion of Q. We wrie his as M larval = g(q, where g is he funcion ha describes he naure of he densiy dependence. The effecs of environmenal variabiliy (ε can be incorporaed ino he model as deviaes around he densiy-dependen relaionship. Given he sex raio (v s, he number of offspring of each sex ha survive o migrae seaward in year (O,s is hen gq ( +ε O = Q e v. s, Two equaions are used o model he number of fish in he spawning run in each age, sex, and previous spawning hisory caegory. For fish ha have no previously spawned (p = 0, he number of fish of sex s and age a enering he river o spawn (downsream of he fishery in year is juv T N O e m e juv M a sa,,, 0 = as, a, s, a. Here, T juv is he insananeous rae of urbine moraliy for juvenile fish (zero for rivers ha are no s developed for hydroelecric generaion and M juv is he insananeous naural moraliy rae for immaure fish a sea. The mauriy schedule (m a,s,a is he probabiliy ha a fish of sex s ha is alive a age a will maure a age a. The mauriy schedule may vary among cohors, where he cohor year is given by a. For fish ha have spawned previously, he number of fish of sex s and age a ha spawned p imes previously enering he river (downsream of he fishery in year is N = N e sap,,, psa,, p, 0 adul adul FT k p+ Ms, a p k= p+ 1, where F is he insananeous rae of fishing moraliy, T adul is he insananeous rae of urbine moraliy for adul fish, and M adul sa, is he sex- and agespecific insananeous rae of naural moraliy for aduls (he relaionship beween F and he annual exploiaion rae u is F = log e [1 u ]. Female spawner biomass (SSB can be used as a proxy for he number of eggs if fecundiy is direcly proporional o weigh:,,,,,, ap, F SSB = N f a pw f a pe. Overcompensaion does no appear o be characerisic of Alosa populaions (Gibson and Myers 2003, his volume, and we herefore assume a Beveron Hol relaionship beween SSB and O (Hilborn and Walers This is a wo-parameer spawner recrui model where α is number of recruis produced annually per uni biomass of spawners (Myers e al and K is he half-sauraion consan. The full dynamic model becomes N sap,,, αssb K e juv a T v m e j uv M a p s a, s, a if = 0 ( 1 + SSB a/ =. adul adul Fk+ T p+ Ms a p, k p 1 = + N p, s, a p, 0e if p > 0 Two examples of how his model can be adaped o differen populaions are given below. The model is adaped o produce esimaes for variables corresponding o he daa ha exiss for he populaion. Model parameers are hen esimaed hrough minimizaion of he value of an objecive funcion ha relaes he model predicions o he observed age srucure, cach, and any auxiliary informaion (e.g., larval indices, escapemen couns ha may exis for he populaion. We programmed his model using AD Model Builder (Fournier AD Model Builder uses he

3 A STOCK ASSESSMENT MODEL FOR ANADROMOUS ALOSA 277 C++ auo-differeniaion library for rapid fiing of complex nonlinear models, has Bayesian and profile likelihood capabiliies, and is designed specifically for fiing hese ypes of models. Example 1: Margaree River, Nova Scoia, Alewife The Margaree River in Nova Scoia suppors an Alosa fishery ha is execued in-river using ip raps insalled along he bank (Chapu e al Alewives are he dominan componen of he cach. Exploiaion raes are conrolled using wihin-season closures ha were firs implemened in 1984 and have been subsequenly modified. Afer 1995 i was eviden ha he populaion size had declined, and furher closures were herefore inroduced in Here, our objecive is o use he model o deermine wheher his sraegy successfully reduced exploiaion raes o arge levels of abou 33% (Chapu e al The daa for he Margaree River alewife fishery consiss of he oal cach (C for he years , an esimae of he number of fish capured in each age and previous spawning hisory caegory (C,a,p for he years , and a larval abundance index (I for he years and The larval index is based on he number of yolk sac larvae capured using a plankon ne and is assumed o be measured before compensaion occurs in he populaion. Based on his assumpion, i is used as an index of spawner escapemen. We use only he second half (P > 0 of he full dynamics model. The model (Table 1 is se up as a combined sex model ha esimaes he number of virgin fish in each age-class ha ener he river in each year (N -p,a-p,0, he exploiaion rae in each year, and a cachabiliy coefficien for larval alewife (q for a oal of 108 esimaed parameers. This approach is similar o radiional saisical cach-a-age models (Quinn and Deriso 1999 excep he previous spawning hisory is incorporaed ino he model by adding anoher dimension o he cach-a-age array. This increases he number of observaions of oal moraliy raes in each year. We assumed a consan value of 0.6 for M adul for all year- and age-classes. While his value is lower han ha assumed by some oher auhors (e.g., Crecco and Gibson [1990] used M adul = 1.0, we have esimaed nonfishing moraliy on rivers where naural moraliy is confounded wih urbine moraliy and obained values close o 0.6. We assumed he larval index in year was linearly relaed o spawning escapemen in year hrough a cachabiliy coefficien (also assumed consan across years. We used a lognormal error srucure obs pred for he cach (C and C are he observed and prediced caches in year and he larval index obs pred (I and I are he observed and prediced larval indices in year. We used a mulinomial error srucure for he number of fish in each year, age, and obs previous spawning caegory ( π is he number ap,, of fish of age a ha have spawned p imes previously wihin a sample colleced in year, and pred p ap,, Table 1. The equaions used for he Margaree River alewife model. Type of equaion Dynamics Equaion N = N e ap,, pa, p, 0 adul Fk+ Mp k p = + 1 C = N u ap,, ap,, C = ( N, a, pu a p I = q [ N, a, p( 1 u] a p Log-likelihoods (non-consan porions Objecive funcion obs pred 2 l cach = (logec log ec = π ap e ap obs pred l composiion log p,,,, a p obs pred 2 l larval = (loge I log e I OBV = ( λ 1 l composiion + λ l cach + λ l larval 2 3

4 278 GIBSON AND MYERS is he prediced proporion of fish in each age and previous spawning caegory in ha year. We fi he model by minimizing he value of an objecive funcion ha is he sum of he negaive log-likelihoods for he cach, larval index, and number of fish in each year age previous spawning caegory. The relaive conribuion of each likelihood o he objecive funcion was conrolled by a se of weighing values (λ i seleced o keep any one par of he objecive funcion from dominaing he fi. Because of difficulies inerpreing juvenile (Jessop 1994 and larval abundance indices, we weighed he larval index so ha is conribuion was abou one half ha of he oher componens. The core of he model is shown in Figure 1. The cach is pariioned by cohor and age a mauriy, and he abundance of fish in each caegory is projeced hrough ime. As shown in Figure 1, cohor sizes and he proporions mauring a each age vary among year-classes. Exploiaion raes and he number of virgin fish in each age caegory are esimaed in he model, and prediced caches and spawning escapemens (Figure 2 are calculaed from model oupu. The fi o he larval index is no as good as he fi o he cach or composiion daa because of he smaller weighing facor used for hese daa. The increased wihin-season closures have reduced exploiaion raes from an average of 0.79 ( ime period o 0.39 ( Spawning escapemen has increased as a resul of hese closures. Example 2: The Gaspereau River, Nova Scoia, Alewife Fishery The Gaspereau River, also in Nova Scoia, suppors an alewife populaion ha shows he characerisics of a heavily impaced sock (Gibson and Myers The sock is fished commercially as fish ascend he river o spawn, and he waershed has been exensively modified for hydroelecric generaion. Waer managemen policies are being developed o reduce impacs on his populaion, including improvemens o fish passage faciliies. Here, our objecive was o use he life hisory model o deermine he biomass of fish ha need o reach he spawning areas o produce maximum susainable yield (SSB MSY in he fishery. Daa for he Gaspereau River alewife sock and fishery are limied. The daa consis of he caches for he years (we use in his model because of uncerainy in he process ha resuled in large caches in he mid-1970s; spawning escapemen couns (E a a fish ladder jus upriver from he fishery for he years , 1995, and ; and he sex, age, and spawning hisory composiion for all years when couns were conduced excep 1995 (Gibson and Myers We se up he model (Table 2 o esimae he log of he mean asympoic recruimen (R 0 and a recruimen deviae (ε for each year around he spawner recrui relaionship. Mean asympoic recruimen can be inerpreed as he median carrying capaciy of he nursery areas rescaled by survival o he age a recruimen, seleced here as age 3. As such, all facors affecing survival from he egg o age 3 are incorporaed ino he spawner recrui relaionship. We used a logarihmic form of he Beveron Hol model, parameerized in erms of R 0, using he subsiuion K = R 0 /α. The logarihmic form consrains recruimen o be posiive during he esimaion process and resuls in a muliplicaive error srucure for recruimen. The SSB MSY can be esimaed from he spawner recrui relaionship. For semelparous species, given a spawner recrui funcion R = f(s, he spawning escapemen a he maximum susainable yield (MSY occurs where f (S = 1 (Quinn and Deriso For an ieroparous species, if fishing occurs jus before spawning, naural moraliy during he fishing season is negligible, he fishery is nonselecive, and fish are fully grown when enering he fishery, he siuaion is analogous. The SSB MSY occurs a he poin where he firs derivaive of he spawner recrui relaionship equals he inverse of he rae a which recruis produce replacemen spawners in he absence of fishing moraliy, denoed as SPR F = 0 (Gibson and Myers 2003: When fishing occurs on fish ha are no fully grown, his relaionship underesimaes he rue SSB MSY (Deriso For he Beveron Hol spawner recrui model, Thus, 1 f ( S =. SPR F= 0 α f ( S = ( 1 + / K. 2 SSB MSY = K SSB MSY SPR α F= 0 K. Because he daa for his populaion are very limied, a number of resricive assumpions are made. Foremos, we rea he exploiaion rae as

5 A STOCK ASSESSMENT MODEL FOR ANADROMOUS ALOSA 279 Figure 1. Par of he observed ( and prediced (lines cach-a-age arrays for he Margaree River, Nova Scoia, alewife populaion. Caches are pariioned by cohor year (righ column and age a mauriy (op labels. The year (boom labels is he year of capure.

6 280 GIBSON AND MYERS Figure 2. Observed ( and prediced (solid lines oal caches and larval indices and prediced exploiaion raes and spawning escapemens for he Margaree River, Nova Scoia, alewife populaion. The dashed lines show 95% confidence inervals for he esimaed values based on normal approximaions. Table 2. The equaions used for he Gaspereau River alewife model. Type of equaion Dynamics Equaion loge R = log e( + log e( loge + SSB R / + α SSB 3 1 ε α 0 juv M ( a 3 N R v m e sa,,, 0 = a+ 3 s sa, C = N u sap,,, sap,,, C = ( N, s, a, pu s a Esap,,, = Nsap,,,( 1 u N = E e + 1, s, a+ 1, p+ 1, s, a, p p adul M Reference poin Log-likelihoods (non-consan porions ( SSB = E sap,,, w sap,,, s a p SSB MSY = K SPR α F= 0 K obs pred 2 l cach = (logec log ec = π sap e sap obs pred l composiion log p,,,,,, s a p obs pred 2 l escapemen = (logee log ee Objecive funcion OBV =- (l1lcomposiion + l2lcach + l3lescapemen

7 A STOCK ASSESSMENT MODEL FOR ANADROMOUS ALOSA 281 known. For years where he cach and spawning escapemen are known, he exploiaion rae can be calculaed direcly. We use he mean exploiaion rae for years ha lack escapemen couns. We also rea he mauriy schedule as fixed across cohors. While his assumpion is no realisic, he daa are insufficien o esimae a mauriy schedule for each cohor (we only have age daa for 7 of he 21 years. We fixed α = 60.7 (a low spawner abundance, 1 kg of spawners can produce 60.7 age-3 recruis annually. This value was aken from a mea-analysis of five oher populaions (Gibson and Myers As in he previous example, we fi he model o he daa by minimizing an objecive funcion value ha is he weighed sum of he nonconsan porions of he negaive log-likelihoods of he caches, escapemen couns, and sex age previous spawning composiions of he spawning run. While he esimaed recruimen deviaes for each year have wide sandard errors (Table 3, he mean asympoic recruimen and he spawning biomass a MSY are well deermined by he model. These reference poins are consisen wih hose produced by a mea-analysis of alewife populaions (Gibson and Myers As weighed, he model racks he coun very closely (Figure 3 and fis he cach reasonably well excep during he ime period. We believe he large residuals in his ime period may be due o annual variabiliy in he mauriy schedules and he possibiliy ha he mean exploiaion rae may no be indicaive of he exploiaion raes in hese years. However, we simply do no have he daa o invesigae hese hypoheses; only he cach is known during his ime period. The esimaed SSB MSY implies ha MSY occurs wih an equilibrium spawner abundance of abou 400,000 fish. Curren spawner abundance is ypically 10 40% of his level. The mean asympoic recruimen for his populaion is abou 1.7 million fish. The esimaed recruimen is lower han his value in 14 of he 19 years ha were included in he model. Discussion Table 3. Parameer esimaes for he Gaspereau River alewife populaion obained from he saisical life hisory model. While he sandard errors for he recruimen deviaes are large relaive o heir esimaes, he mean asympoic recruimen (R 0 ; number of fish and spawning biomass (SSB MSY ; kg a maximum susainable yield are comparaively well esimaed. Year and Recruimen Sandard variable deviae error R 0 1,647, ,330 SSB MSY 91,939 33,607 In his paper, we have presened a general, lifehisory-based model for he populaion dynamics of anadromous Alosa and shown how he model can be used for sock assessmen by adaping i o he daa colleced for individual populaions. This approach has several advanages, he foremos being ha he life hisory of ineres is specifically modeled. For Alosa, daa such as spawning escapemen couns a fish ladders, larval and juvenile abundance indices, couns of emigraing juveniles, previous spawning hisory, indices of he number of posspawning fish (Olney and Hoenig 2001, and informaion abou oher sources of moraliy can be incorporaed ino he assessmen process. One of he mos useful pieces of informaion ofen colleced for Alosa ha is no ypically available for marine species is he number of imes ha a fish has previously spawned (available from a fish s scales. Riverine impacs such as fishing or urbine passage do no affec immaure fish a sea. When he number of imes a fish has previously spawned is known, his variable can be used o deermine he number of imes ha he fish has been exposed o riverine impacs by adding an exra dimension o he cach-a-age array. Addiionally, when daa are pariioned by sex and age a mauriy as well as age, he number of observaions of a cohor each year increases (from one o eigh for a populaion ha maures over 4 years. Assuming an adequae sample size, his increase

8 282 GIBSON AND MYERS Figure 3. Observed ( and prediced (lines oal caches and spawner escapemen couns and he exploiaion raes (assumed known and prediced number of age-3 recruis for he Gaspereau River, Nova Scoia, alewife populaion. The dashed lines show he number of spawners a maximum susainable yield (upper righ panel and he median asympoic recruimen (lower righ panel for his populaion. The gray shaded areas are 90% confidence inervals for hese reference poins. improves he researcher s abiliy o esimae moraliy raes or oher parameers ha are held consan across hese caegories. While no specific o Alosa populaions, saisical mehods of fiing sock assessmen models have a major advanage over virual populaion analyses: models can be fi o inermien daa ses. For many smaller fisheries, such as he Gaspereau River example presened here, assessmen daa are no colleced each year. Virual populaion analysis-based mehods use a backward summaion for which esimaes of he numbers a age are required for each year (Hilborn and Walers While fiing hrough long periods when age daa were no colleced requires ha addiional consrains be placed on he model, saisical mehods can be used o esimae abundance when all ha is known is he size of he cach. Our life hisory model can be furher generalized where appropriae for some populaions. As presened, we have reaed Alosa fisheries as nonselecive. When selecive fishing gear such as gill nes are used, seleciviy models (e.g., Millar and Hols 1997 can be incorporaed ino our framework, allowing he error in he seleciviy parameer esimaion o be carried forward hrough he assessmen. Similarly, when cachabiliy varies wih fish abundance (e.g., Harley e al. 2001, hese relaionships can also be included in he model. Mea-analyic approaches o sock assessmen and fisheries biology are becoming more common (e.g., Myers e al. 1999; Gibson and Myers In order for hese approaches o be successful, some sandardizaion among daa ses (e.g., a sandard definiion of recruimen for spawner recrui meaanalysis is required. When a general life hisory model is used for he assessmens, a basis for hese kinds of sandardizaions is provided. A presen, we have no generalized our compuer code so ha i can be applied o a generic sock. The modeling framework is inended o be flexible and o be adaped o he daa and biology of specific populaions. Thus far, we have fi he model o four alewife populaions, each wih differen kinds of daa. The compuer code, and assisance wih adaping i o oher populaions, is available from he auhors. Acknowledgmens The auhors hank Gerald Chapu for generously sharing he daa for he Margaree River sock. Shelon Harley, Kuris Trzcinski, and wo anonymous reviewers provided commens on an earlier version of his manuscrip.

9 A STOCK ASSESSMENT MODEL FOR ANADROMOUS ALOSA 283 References Chapu, G., P. LeBlanc, and R. Crawford Assessmen of he Margaree River gaspereau fishery, 1997 o Deparmen of Fisheries and Oceans, Canadian Sock Assessmen Secrearia Research Documen 2001/046. Crecco, V., and T. Savoy Effecs of climaic and densiy-dependen facors on inra-annual moraliy of larval American shad. Pages in R. D. Hoy, edior. 10h annual larval fish conference. American Fisheries Sociey, Symposium 2, Behesda, Maryland. Crecco, V. A., and M. Gibson Sock assessmen of river herring from seleced Alanic coas rivers. Alanic Saes Marine Fisheries Commission, Special Repor 19, Washingon, D.C. Deriso, R. B Harvesing sraegies and parameer esimaion for an age-srucured model. Canadian Journal of Fisheries and Aquaic Sciences 37: Deriso, R. B., T. J. Quinn II., and P. R. Neal Cach-age analysis wih auxiliary informaion. Canadian Journal of Fisheries and Aquaic Sciences 42: Fournier, D An inroducion o AD Model Builder for use in nonlinear modelling and saisics. Oer Research Ld., Nanaimo, Briish Columbia. Fournier, D. A., and C. P. Archibald A general heory for analyzing cach a age daa. Canadian Journal of Fisheries and Aquaic Sciences 39: Gibson, A. J. F., and R. A. Myers Gaspereau River alewife sock saus repor. Deparmen of Fisheries and Oceans, Canadian Sock Assessmen Secrearia Research Documen 2001/061. Gibson, A. J. F., and R. A. Myers A mea-analysis of he habia carrying capaciy and he maximum reproducive rae of anadromous alewife in easern Norh America. Pages in K. E. Limburg and J. R. Waldman, ediors. Biodiversiy, saus, and conservaion of he world s shads. American Fisheries Sociey, Symposium 35, Behesda, Maryland. Harley, S. J., R. A. Myers, and A. Dunn Is cachper-uni-effor proporional o abundance? Canadian Journal of Fisheries and Aquaic Sciences 58: Hilborn, R., and C. J. Walers Quaniaive fisheries sock assessmen: choice, dynamics and uncerainy. Chapman and Hall, New York. Jessop, B Relaions beween sock and environmenal variables, and an index of abundance, for juvenile alewives and blueback herring. Norh American Journal of Fisheries Managemen 14: Legge, W. C., and J. E. Carscadden Laiudinal variaion in he reproducive characerisics of American shad (Alosa sapidissima: evidence for populaion specific life hisory sraegies in fish. Journal of he Fisheries Research Board of Canada 35: Loesch, J. G Overview of he life hisory aspecs of anadromous alewife and blueback herring in freshwaer habias. Pages in M. J. Dadswell, R. J. Klauda, C. M. Moffi, R. L. Saunders, R. A. Rulifson, and J. E. Cooper, ediors. Common sraegies of anadromous and caadromous fishes. American Fisheries Sociey, Symposium 1, Behesda, Maryland. Marcy, B. C., Jr Age deerminaions from scales of Alosa pseudoharengus (Wilson and Alosa aesivalis (Michill in Connecicu waers. Transacions of he American Fisheries Sociey 98: Millar, R. B., and R. Hols Esimaion of gillne and hook seleciviy using log-linear models. ICES Journal of Marine Science 54: Myers, R. A., K. G. Bowen, and N. J. Barrowman The maximum reproducive rae of fish a low populaion sizes. Canadian Journal of Fisheries and Aquaic Sciences 56: Olney, J. E., and J. M. Hoenig Managing a fishery under moraorium: assessmen opporuniies for Virginia s socks on American shad. Fisheries 26(2:6 11. Quinn, T. J., and R. B. Deriso Quaniaive fish dynamics. Oxford Universiy Press, New York. Rulifson, R. A Saus of anadromous Alosa along he eas coas of Norh America. Pages in J. E. Cooper, R. T. Eades, R. J. Klauda, and J. G. Loesch, ediors. Anadromous Alosa symposium. American Fisheries Sociey, Tidewaer Chaper, Behesda, Maryland.

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