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Optimal stock size and harvest rate in multistage life history models : Pacific salmon Oncorhynchus spp. Moussalli, Elie Ibrahim 1984

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OPTIMAL  STOCK  SIZE  MULTISTAGE PACIFIC  AND  HARVEST  L I F E HISTORY  RATE  IN  MODELS:  SALMON ONCORHYNCHUS S P P . By  ELIE  M.  Sc., American A  IBRAHIM  University  MOUSSALLI  Of  Beirut,  Lebanon,  1972  THESIS THE  S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF R E Q U I R E M E N T S FOR T H E D E G R E E OF MASTER OF S C I E N C E in THE F A C U L T Y OF GRADUATE S T U D I E S ( D e p a r t m e n t of R e s o u r c e Management S c i e n c e )  We  accept  THE  (c)  this thesis required  U N I V E R S I T Y OF June  Elie  as conforming standard  BRITISH 1984  to the  COLUMBIA  Ibrahim Moussalli,  1984  86  In p r e s e n t i n g  t h i s thesis i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may department o r by h i s o r her  be  granted by the head o f  representatives.  my  It i s  understood t h a t copying o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6  (3/81)  written  ABSTRACT A simple between and  habitat  harvest  implicit  life  h i s t o r y model t h a t a n a l y z e s capacity,  survival  rate for Pacific  in  nearly  salmon  the r e l a t i o n s h i p s  rates, optimal (  Oncorhynchus  a l l considerations  escapement, spp.), . i s  of salmon management.  T h i s model c o n s i s t s of d e n s i t y dependent e v e n t s a t spawning rearing, fry  and d e n s i t y independent prespawning m o r t a l i t y , e g g - t o -  s u r v i v a l and s m o l t - t o - a d u l t  model y i e l d s simple optimal with  improvement  or  and  life  the  optimal  ocean  escapement and h a r v e s t  rate.  The a n a l y s i s of t h i s  between  key  harvest  rate.  c y c l e such as  degradation  escapement and h a r v e s t in  survival.  relationships  escapement,  the salmonid  trends  and  flow  have simple  parameters  Interventions  regulation,  r a t e and  rate  will  thus  habitat  consequences on o p t i m a l  Similarly, detectable  survival  and  may  affect be  long  term  the o p t i m a l  incorporated  in  management d e c i s i o n s . If  there  is a  d e n s i t y dependent l i m i t  h i s t o r y , an i n c r e a s e i n the s u r v i v a l optimum is  increase and  rate w i l l  escapement and a higher optimum  no d e n s i t y dependent l i m i t in survival w i l l  a higher  by escapement  optimum  later  result  harvest  rate.  in  i n a lower  rate.  I f there  the . l i f e  i n a higher  i n the l i f e  result  harvest  optimum  history  an  escapement  I a l s o show t h a t management  r e q u i r e s i n f o r m a t i o n on h a b i t a t c a p a c i t y and s t o c k  p r o d u c t i v i t y ; whereas management by h a r v e s t the  later  rate  requires  only  latter. T h i s model  i s generalized to a multistage  form t o r e p r e s e n t  any h a r v e s t a b l e p o p u l a t i o n c o n s i s t i n g of s u c c e s s i v e Beverton and  Holt  type  density  dependent  stock-recruitment stages.  m u l t i s t a g e s t o c k - r e c r u i t m e n t model terms  of  arbitrary and  the  cumulative  stage.  those  parameters  The o p t i m a l  h a r v e s t rate are, a n a l y z e d  i n t e r m e d i a t e stage*  The  i s expressed up  This  recursively  in  t o and i n c l u d i n g  any  spawning stock s i z e  (escapement)  i n terms of the parameters of any  results  of  this  extension  of the salmonid model and g e n e r a l i z e them.  confirm  iv  TABLE OF CONTENTS  ABSTRACT  i i  L I S T OF TABLES  v  L I S T OF FIGURES  v i  ACKNOWLEDGEMENTS  v i i  INTRODUCTION CHAPTER ONE  1 L I F E HISTORY MODEL AND ANALYSIS  1.1 The g e n e r a l i z e d l i f e  9  h i s t o r y model  1.2 D e r i v a t i o n o f o p t i m a l e s c a p e m e n t a n d h a r v e s t  9 rate 13  1.3 A n a l y s i s o f t h e e f f e c t o f some k e y p a r a m e t e r s CHAPTER TWO  EXTENSION OF THE MODEL  ... 22 44  2.1 I n t r o d u c t i o n  44  2.2 The m u l t i s t a g e s t o c k a n d r e c r u i t m e n t m o d e l  45  2.3 A n a l y s i s o f i n t e r m e d i a t e p a r a m e t e r s  52  CHAPTER THREE LITERATURE CITED  DISCUSSION  64 69  V  L I S T OF TABLES  Table I S e n s i t i v i t y  of o p t i m a l escapement t o p a r a m e t e r s  of  the t h r e e models Table  II Sensitivity  of t h e t h r e e m o d e l s  39 of o p t i m a l h a r v e s t  rate to parameters 41  vi  L I S T OF FIGURES  Figure  1.1 A l t e r n a t i v e f u n c t i o n a l r e l a t i o n s h i p s b e t w e e n  d e p o s i t i o n and f r y p r o d u c t i o n  in Pacific  salmon  Figure  1.2 B e v e r t o n a n d H o l t t y p e r e c r u i t m e n t  Figure  1.3  Relationship  between  optimal  parameters of t h e u n c o n s t r a i n e d Figure  1.4  Relationship  escapement  optimal  escapement  1.5  Relationship  between  optimal  parameters of a c o n s t r a i n e d r e c t i l i n e a r Figure  1.6  Relationship  Figure  curvilinear  1.7 R e l a t i o n s h i p b e t w e e n o p t i m a l  escapement  Figure  1.8  Relationship  F i g u r e 2.1  B e v e r t o n and H o l t  and 28  r a t e and  function  harvest  31  rate  and  function  between o p t i m a l h a r v e s t  parameters of the r e c t i l i n e a r  and  function  parameters of the c o n s t r a i n e d c u r v i l i n e a r  34  r a t e and  function  type  stock  36 and  recruitment  curve Figure  46 2.2  Relationship  between o p t i m a l h a r v e s t  r a t e and  productivity F i g u r e 2.3 R e l a t i o n s h i p  55 between  optimal  stock  size  and  productivity Figure  24  f u n c t i o n . 26  between o p t i m a l h a r v e s t  parameters of unconstrained  15  and  f u n c t i o n ...  parameters of r e a r i n g c o n s t r a i n e d c u r v i l i n e a r Figure  6  f u n c t i o n ....  curvilinear  between  egg  2.4  capacity  Relationship  59 between  optimal  stock  size  and 62  ACKNOWLEDGEMENTS I s h o u l d l i k e t o t h a n k members o f my my  co-supervisors,  Dr.  Ray  committee  especially  H i l b o r n f o r p o i n t i n g t h e way o u t  when I was w a n d e r i n g i n b l i n d a l l e y s a n d D r .  Carl  Walters f o r  h i s i n s i g h t f u l comments a n d f o r s h o w i n g me t h e d e r i v a t i o n recursive  form  on  pages  50-52.  My t h a n k s a r e a l s o due t o D r .  J u s t i n Cook who w i t h a few s h o r t s t e p s c o n f i r m e d a n d pages  of  my  analysis,  and  comments he made on an e a r l i e r I a l s o w i s h t o t h a n k my critical  to  of the  Peter  Millington  simplified for helpful  draft. friend  Dr.  Thomas  Betlach  for  d i s c u s s i o n and f o r h i s h o s p i t a l i t y and c o m p a n i o n s h i p i n  t h e H i g h S i e r r a s when I n e e d e d Finally,  I  owe  t o g e t away f r o m F u n g u s  Land.  v e r y s p e c i a l t h a n k s t o K a t h l e e n Day.  She  e n c o u r a g e d a n d h e l p e d me i n many w a y s . T h i s work i s d e d i c a t e d t o my p a r e n t s who d i d n o t chance.  have  the  1 INTRODUCTION Conventional  wisdom o f s a l m o n management, b a s e d on R i c k e r ' s  (1954) s t o c k and r e c r u i t m e n t m o d e l , fishery  by  allowing  productivity.  sufficient  calls  escapement  regulating  future  Low e s c a p e m e n t s r e d u c e t h e l o n g t e r m y i e l d  from a  utilization  of the  h i g h escapements the s u s t a i n a b l e y i e l d mortality.  Using  to  resource,  but  thus  to  move  the  produces the best balance and  Ricker's  model,  a  Maximizing  between u t i l i z a t i o n  (1977), is  a  by  .  Keeney  and  Raiffa (1977),  more a p p r o p r i a t e t o o l  A n a l y s i s (MUA) a s  and Walker  et a l .  are  densities  (Paulik  Gulland to  However,  b a s e d on some t h e o r e t i c a l  form, w i t h compensatory m o r t a l i t i y a t h i g h  returns  (1983),  of t o t a l c a t c h .  functional  about  confronts  f o r t h e complex r e a l i t i e s of salmon  exercises  1973;  of  ( 1 9 7 6 ) a n d a p p l i e d by K e e n e y  a simple maximization  management  Conclusions  resource  Indeed, b a l a n c i n g t h e needs  Multi-Attribute Utility  H i l b o r n and W a l t e r s  such  the  g r o u p s i n t h e f a c e o f u n c e r t a i n t y i s what  management t h a n all  of  which  the harvestable surplus i s u s u a l l y not the s o l e  m a n a g e r s most o f t e n formulated  surplus,  crowding.  o b j e c t i v e o f s a l m o n management. v a r i o u s user  common  f i s h e r y t o t h a t escapement l e v e l  r e d u c e d s u r v i v a l due t o  at  i s r e d u c e d by c r o w d i n g o r  o b j e c t i v e o f management i s t o m a x i m i z e t h e h a r v e s t a b l e and  the  ensure  s t o c k due t o i n s u f f i c i e n t  compensatory  for  1974;  Cushing  escapement  population  1973a,  a r e reached  b).  ex p o s t  f a c t o by a n a l y s i s o f h i s t o r i c a l  d a t a , a n d a r e a s s u m e d t o h o l d on  the average.  a  yolk  sac  Mortality during  exhaustion,  was  "critical  suggested  by  period" Hjort  following  (1914)  a s an  2  e x p l a n a t i o n f o r year European  c l a s s s t r e n g t h and f l u c t u a t i o n s of  fisheries.  hypothesis  are  Alternative  the  constant  hypotheses, which c l a i m history 1956;  stages  May  the  critical  that  mortality  at  r a t h e r than  intermediate  life  discontinuous  (Marr  However, f a c t o r s i n f l u e n c i n g t h e s e  stages  v e r y complex and t h e r e f o r e g e n e r a l l y n o t e x p l i c i t l y in  p o p u l a t i o n m o d e l s commonly u s e d i n f i s h e r i e s In  two  commonly  applied  r e c r u i t m e n t models put forward Holt  (1957),  history  the cumulative  by R i c k e r  spawning  namely  stock and  (1954) and B e v e r t o n in  stock  describes  and  the  corresponding  stock.  The  reaching  asymptotic  stock s i z e .  The R i c k e r c u r v e  dependence  in  recruitment  some  over  intermediate  implies  between  The  stages  Beverton reduction  very  information  i s available, their  may be i n c r e a s e d by i n c o r p o r a t i n g s u c h i n f o r m a t i o n .  some  W i t h o n l y two  t h e s e m o d e l s h a v e become a  history  stock  increasing  progressive  Where  or  Ricker  beyond  geometrically  a range o f s t o c k s i z e s .  management.  the  i s applicable for  rate as stock density increases.  fisheries life  other  shaped  recruitment  i m p l i e s an a r i t h m e t i c a l l y  parameters t o estimate, for  One o f  p o p u l a t i o n s w i t h low r e c r u i t m e n t a t h i g h  populations  Holt curve  life  recruitment,  dome  and H o l t curve  and  the  and  the  These models d e s c r i b e a r e l a t i o n s h i p  l e v e l s , whereas t h e Beverton  tool  the  stock p r o d u c t i v i t y ,  p r o d u c t i o n , from t h i s parent  density  incorporated  o f a p o p u l a t i o n a r e subsumed i n two p a r a m e t e r s .  carrying capacity.  curve  models,  are  management.  e f f e c t s of a l l stages  these parameters represents  a  period  s u r v i v a l and i n c r e a s i n g s u r v i v a l  i s cumulative  1974).  to  Northern  useful about  usefulness  3  The  most  common  application  of  Ricker's  r e c r u i t m e n t m o d e l , h o w e v e r , lumps t o g e t h e r the p r o d u c t i o n process marginal  question.  study  primary  without  i n the  Management  expensive  life  by  of  focusing history  stock  of complicated  tool  of  and  example,  of  the population  in  obviates the  steps  because  the  i t s objective i s  H o w e v e r , a s s u c h management i s b a s e d escapement,  stock  and  does n o t p r e s c r i b e escapement i n t h e f a c e  an o b s e r v e d  production  factors.  I f ,  temperature regime and d i s c h a r g e  of a salmon b e a r i n g r i v e r causes mortality  falling  details  i s escapement;  of c h a n g i n g h a b i t a t q u a l i t y o r o t h e r for  the  and  steps i n  of  recruitment  on a v e r a g e r e c r u i t s t o a g i v e n l e v e l o f analysis  on  intermediate  management  maximum s u s t a i n a b l e y i e l d .  recruitment  intermediate  a n d r e l i e s on t h e p r i n c i p l e  productivity  successive steps  stock  o r reduces smolt  some  prespawning  (Hartman e t a l .  1982), and  the  f l o w regime i s such t h a t a change i n prespawning  mortality  is  expected,  Stock and  recruitment will  not  history  survival  significant  rate  how d o e s t h e d e s i r e d e s c a p e m e n t c h a n g e ?  will  o n l y p r o v i d e a n a n s w e r b a s e d on p a s t  provide  a  events.  example.  1971  production  c o r r e l a t e d with increasing hatchery  smolt  releases.  followed  by  increasing  a  smolt  adults,  From  period  1961  1982),  to  million  i t  method t o i n c o r p o r a t e new c h a n g e s i n l i f e  O r e g o n c o h o ( O n c o r h y n c h u s k i s u t c h ) (ODFW another  data;  of  releases. crashed  fluctuating Returns  returns,  peaked  in  provide  of  This in  1976  level  was  s p i t e of  t o a r e c o r d l o w o f 1.1 m i l l i o n  f o l l o w i n g y e a r , and remained around t h e l a t t e r  adults  at  4.1  i n the  by 1981  as  4  releases  surpassed  contrary to  the  60 m i l l i o n  accepted  dependence i n the o c e a n i c 1978).  Clark  and  f a c t o r s , smolt nearshore  food  variation given  notion  and  observations  by  mordax  ; how  released  stock  Lasker  (1981)  could this level  s t o c k and  i n f o r m a t i o n be  o f c a t c h and  recruitment  above  life  limit  Several  1958, to  1962;  and  for  between  predicted such  anchovy to  i t does  a  from  as  the and  ( Enqraulis  determine  not  the  year? given  p o i n t t o w a r d s the need f o r t o o l s  that  (Ricker  of  1975)  determine  1954,  1958;  reported  harvesting Neave  1958,  an  upper  that  f r y i s approached as the  success  McNeil as  a  Holt's  d e p o s i t i o n and  h a b i t a t model of spawning s u c c e s s desirability  for  rates.  optimum l e v e l .  egg  for  for  d e n s i t y w h i c h i s b a s e d on B e v e r t o n and relationship  survival  escapement f o r t h a t b r o o d  records;  Thomas  spawning  for  allow  the average y i e l d  relationship  proxy  p r e d i c t s average returns t o a  workers  o f s p a w n e r s r e a c h e s an  uniform  used  h i s t o r y d a t a , when a v a i l a b l e , t o  policies. Wickett  examples  (Peterman  explanation  be  f o r the northern  of e x p e c t e d s u r v i v a l  The  could  density  explanatory  (a  the best  were  d e v e l o p e d by Bakun ( 1 9 7 3 , 1975)  s i z e b a s e d on h i s t o r i c a l  t h e use  use  smolts  two  upwelling  provide  no  cycle  of n e a r s h o r e o c e a n o g r a p h i c p a t t e r n s ,  appropriate Again,  is  that  Suppose t h a t ocean  cube of the wind-speed index aplied  there  show  lagged  availability)  of  that  (1983)  i n coho r e t u r n s .  cohort  These o b s e r v a t i o n s  phase of t h e coho l i f e  McCarl  releases  smolts.  (1964)  derived  f u n c t i o n of (1957)  finite  a  spawner  asymptotic  recruitment.  assumes a  density  McNeil's  habitat  i n w h i c h r a n d o m l y s p a w n i n g f e m a l e s may  of dig  5  up  r e d d s as  redd  their density  super i m p o s i t i o n  upper  limit  to  mortality,  of  the  is  thus  fry  by  maximum  (see F i g u r e  1.1).  h a b i t a t determines the  sustain  density. concept  m o r t a l i t i e s due  I use of  spawning  the  the  s e t s the  The  numerical  is  fisheries  number o f s u c c e s s f u l but  capacity  "equivalent"  whose  term " e f f e c t i v e females" t o convey the  same  dependence  of  fry production  resource  (Toews and the  use  conflict,  Moore 1982;  Tschaplinski  and  need f o r a d e c i s i o n - m a k i n g r a t i o n a l e have a l s o  t h e d e v e l o p m e n t and are stages  a p p l i c a t i o n of  regional  percent  1953)  as  L i s t e r and  other  or l o c a l  species  type  described.  specific  survival. vary  well Walker  Design c r i t e r i a  I n t e r n a l p u b l i c a t i o n , March with  as 1966;  with  local  conditions  the p a r t i c u l a r run  L a r k i n and  McDonald  using  " p r o b a b i l i t y o f use  criteria"  of  important v a r i a b l e s  for  1982). streams  (Skud  1958,  1968).  The  carrying  (Bovee 1978).  approach to h a b i t a t uses a h y d r o l o g i c a l model, concentrates few  on  successive  c a t e g o r y of h a b i t a t models a t t e m p t s t o a s c e r t a i n  capacity  based  biostandards  s u r v i v a l r a t e s between  ( S a l m o n i d Enhancement Program.  These s u r v i v a l r a t e s  1973;  the  Hartman  c a t e g o r y of s u c h models c a l c u l a t e s h a b i t a t c a p a c i t y  (Neave  on  such as between f o r e s t r y  a need f o r h a b i t a t b a s e d m o d e l s o f t h e  average  do  given  created  life  fry  at a  density  to redd s u p e r i m p o s i t i o n  and  which  reached  maximum c a r r y i n g  deposited  model  compensatory  capacity  1983),  One  this  population.  P r o b l e m s of and  to  increasing'  carrying  f e m a l e s , i . e . , t h o s e whose e g g s a r e not  According  the mechanism t h a t  emerging  and  asymptotically  increases.  (water depth, v e l o c i t y , temperature  This on  a and  6  Figure  1.1 A l t e r n a t i v e f u n c t i o n a l r e l a t i o n s h i p s b e t w e e n egg d e p o s i t i o n and f r y p r o d u c t i o n i n P a c i f i c s a l m o n ( a d a p t e d from M c N e i l l 1964).  EGGS  DEPOSITED  B.  substrate  composition),  usable area overall  and,  f o r each l i f e  (WUA) i s c a l c u l a t e d .  measure  of  stream  The sum o f t h e s e WUA's  carrying capacity.  dynamics approach of M c N e i l , the biostandards and  the  WUA  approach  of  life  h i s t o r y model t h a t  stages  l i n k e d by d e n s i t y i n d e p e n d e n t s u r v i v a l chapter  one  incorporates  anticipated  stages  in their  changes life  The p o p u l a t i o n  approach  density  are  in survival  available.  rates  are  escapements and h a r v e s t the  model  section  then  depends  on  salmonid  o f h a b i t a t c o n s t r a i n t , and i n  expressions derived.  f o r optimal  escapement  The q u e s t i o n o f how  optimal  r a t e s change w i t h v a r i o u s p a r a m e t e r s  graphically  the  presented  and  of  contrasted i n  model i s e x t e n d e d i n c h a p t e r  two s o t h a t i t  represents  h a r v e s t a b l e p o p u l a t i o n c h a r a c t e r i z e d by n s u c c e s s i v e d e n s i t y  dependent  stages.  T h i s m o d e l i s d e r i v e d i n s e c t i o n 2.2, a n d i n  s e c t i o n 2.3 t h e d e p e n d e n c e o f o p t i m a l rate  them  1.3.  The any  is  of  Two management  I n s e c t i o n 1.1 t h e g e n e r a l i z e d  1.2, c o r r e s p o n d i n g  harvest  impact  i n t h i s a n a l y s i s , escapement and h a r v e s t  model i s d e s c r i b e d f o r t h r e e c a s e s  and  such  r a t e s o f salmon a t v a r i o u s  r a t e , a n d I show t h a t t h e c h o i c e b e t w e e n  section  SEP  dependent  h i s t o r y parameters  c y c l e ; or i n f o r m a t i o n about the  considered  information  of  rates.  h a b i t a t m o d i f i c a t i o n , i n t o management d e c i s i o n s . tools  an  I c o n s i d e r one s u c h m o d e l t h a t a l l o w s t h e  - i n c o r p o r a t i o n of i n f o r m a t i o n about l i f e as  is  Bovee c a n a l l be g e n e r a l i z e d i n t o a  simple  In  stage, a weighted  on  parameters  of  stock  size  any i n t e r m e d i a t e s t a g e  g e n e r a l d i s c u s s i o n of r e s u l t s  i s presented  and  harvest  i s analyzed.  i n the chapter  A  three.  9  CHAPTER ONE  1.1 The g e n e r a l i z e d The  model  relationships rearing, This  L I F E HISTORY MODEL AND ANALYSIS  life  described  capacity  is  models.  prespawning to-adult  life  independent  implicit  on  history  survival  density  survival  rates  in  between.  a n d t h e SEP  factors  (s ), egg-to-smolt 1  dependent  s t a g e s , spawning and  i n M c N e i l ' s (1964)  Three  survival  model  i s based  a t two c r i t i c a l  and d e n s i t y  model  history  are  survival(s  (1982)  considered: ), and s m o l t -  2  s u r v i v a l (s ).  3 Equations such  a  (1.1) through  habitat-based  (1.6) a r e a s t e p w i s e d e s c r i p t i o n of  life  history  model.  Starting  from  escapement of a g i v e n brood year t h r o u g h a d u l t s  recruited  from  t h i s p a r t i c u l a r brood y e a r ,  (1.1) F  = 0.5E s  where F  s  =  number  of  female spawners,  1:1; and E = total  escapement;  assuming  a sex r a t i o of  10  (1.2) F  = F  g  s  s 1  where F  = females a r r i v i n g  t o t h e spawning  grounds;  9 s  F  = prespawning, density  1  =  e  where F  e  independent s u r v i v a l ; and,  a x  (1.3)  b + x  = e f f e c t i v e females;  x = density  o f female spawners  on t h e g r o u n d s  a = maximum number o f f e m a l e s t h a t c o u l d spawning  be a c c o m o d a t e d  in  area a v a i l a b l e A ; s  and b  = spawner h a l f s a t u r a t i o n c o n s t a n t r e f l e c t i n g  elasticity  of  territorial  saturation  capacity  behaviour, or  how  quickly  the  a i s reached.  V a r i a b l e s a a n d x c a n be more e x l p l i c i t l y  defined  as  A a =  and  s t  , where t = s p a w n e r t e r r i t o r y  size i n mVfemale;  11  F g X  =  .  A s  The  saturation  number o f  curve  biological  redd  curve.  T h i s c u r v e may  negative 1966). flow  location  (Dixon  McNeil's  in a uniform be and  (1.3)  formally  derived  Massey  or  Imperfect  represent  (1964)  model  of  a similar  territoriality  drying  1969)  up,  saturation  or  from  the  from  a  k=1.0  and  curve  fish will  thread  and  variation  i n the preceeding  production  as  Poisson "clumped"  (Southwood  in  hypotheses  gravel  ,be  best  produced.  is  marginal  random  quality. falling  spawners' d e n s i t y i n c r e a s e s .  the concept of f a l l i n g  choose the  to  of spawners would a l s o produce such a  c u r v e , a s w o u l d a l a r g e number o f v a r i a t i o n s on territoriality  a of  i s a g r a d a t i o n i n g r a v e l q u a l i t y , e i t h e r due  probability first,  can  h a b i t a t w i l l produce such a  b i n o m i a l d i s t r i b u t i o n w i t h parameter I f there  habitat  equation  hypotheses.  random  distribution  of  This  The  mating, common  marginal  fry  is similar  to  p r o d u c t i v i t y as used i n economic  production functions. The  effective  females are c a r r i e d to the next  step  of  egg  production,  (1.4)  12  where e = t o t a l number o f e g g s l a i d  by a b r o o d  f = f e c u n d i t y , as eggs/female,  year;  and smolt p r o d u c t i o n ;  and  es S = min  \  2  (surviving  smolts) (1.5)  A c (max s m o l t c a p a c i t y ) r where S = number s A  2 r  of s u r v i v i n g  = egg-to-smolt  survival;  = rearing area a v a i l a b l e  c = maximum number per u n i t and  smolts;  i n square  meters;  of smolts t h a t can rear  successfully  area;  finally,  (1.6) R = Ss 3  where R = a d u l t s r e c r u i t e d t o a brood s  3  Equation  year  = smolt-to-adult survival. (1.4)  generates  eggs  from  equivalent  equation  ( 1 . 5 ) , where s m o l t r e a r i n g c o n s t r a i n t s may  smaller  of  the  two  alternative  numbers  is  females; be met,  carried  in the  i n the  13  s u b s e q u e n t s t e p (1.6)  1.2  of a d u l t p r o d u c t i o n .  D e r i v a t i o n o f o p t i m a l e s c a p e m e n t and The  p r e v i o u s s e c t i o n summarized the l i f e  population  in  expression  for  escapement. section  harvest  six  steps.  adult  From  salmon  Symbols  have  these  production  the  rate  c y c l e of a  s t e p s we as  same m e a n i n g a s  a  now  salmon  d e r i v e an  function i n the  of  previous  .  Using equations expressed  (1.4),(1.5)  and  (1.6)  recruitment  may  be  as  F f s s e 2 3 R = min  •  (1.7) A c s r  Substituting and  for  3  effective  females  from e q u a t i o n s  ( 1 . 1 ) , (1.2)  (1.3), 0.5afs s s E 1 2 3 bA R -  min  s  +  0.5s  (1.8a) 1  E  <  ^  A c s r 3  =  R  2  ,  (1.8b)  14  g i v i n g two  expressions  Equation  for adult  production.  (1.8a) c a l c u l a t e s the rearing  adults  where t h e  only c o n s t r a i n t encountered i s spawning h a b i t a t .  periods (  of  0.  factor  r e a r i n g i n freshwater  gorbuscha  )  and  i n t h i s case i s  parameters  of  saturation.  the  area  kisutch  ) are  as  surplus  that  level is  escapement e n s u r e s t h a t  rather  is  at  maximized. habitat  f i s h e r y ; the  nerka  ) and  1.2  form  and  has  i s expressed i n equation  its  as  c o h o ( 0. it  escapement,  which  sustainable  Thus  defined,  production The  optimal  capacity,  as  term " o p t i m a l "  i t r e f e r s to the  harvesting  the  a  optimal  s e c t o r and  e x c l u d e d from t h i s d e f i n i t i o n .  f u n c t i o n R=f(E) i n F i g u r e f o r m and  the  limiting  r e c r u i t e d under  for  i t s saturation level.  f i s h component o f t h e  The  pink  In b o t h of t h e s e c a s e s  escapement  the  long  i n the c a s e of  and  determine  s p e c i a l i s e d in t h i s context,  economic a s p e c t s are  ( 0.  form. to  of  This  undergo  ) salmon.  area  and  r e l a t i o n s h i p that describes  sockeye  feasible  production  formulated,  spawning  examples of t h i s and  keta  constraints,  not  (1.8b) e x p r e s s e s a d u l t s  constraint;  desirable  defined  chum ( 0. the  do  h a b i t a t , as  functional  Equation  rearing  is  for salmonids that  no  of  where  valid  presents  number  produced  r e g i m e w o u l d be  habitat  maximum  The  habitatrelated derived  f a m i l i a r B e v e r t o n and  ( 1 . 8 a ) where R  i s the  is  Holt  number  1 of  adults  produced  c a l c u l a t e optimal  subject  e s c a p e m e n t we  to  no  rearing  constraint.  differentiate R  with 1  escapement  respect  To to  15  Figure  1.2 B e v e r t o n a n d H o l t t y p e r e c r u i t m e n t f u n c t i o n r e p r e s e n t i n g a f u n c t i o n a l r e l a t i o n s h i p between a d u l t s r e c r u i t e d and escapement.  E  *  i s the optimal 1  escapement t h a t maximizes a d u l t r e c u i t m e n t  (R  *  ) under 1  * no r e a r i n g a r e a  constraint (equation  1.112).  the o p t i m a l escapement under e f f e c t i v e  *  constraint  (equation  1.14, a n d E  escapement  f o r t h e r e c t i l i n e a r model  3  E  is 2 r e a r i n g area  i s the optimal (equation  1.16).  ESCAPEMENT  17  dR  0.5abfs s s A 12 3 s  1 =  dE  (bA + 0.5s E) s 1  > o  (1.9)  < 0  (1.10)  2  the second d e r i v a t i v e  2 d R  1 =  dE  confirms  2  (bA + 0.5s E) s 1  3  t h a t a d u l t s i n c r e a s e w i t h e s c a p e m e n t , a n d t h e y do s o a t  a decreasing model  _  2 0.5ab£s s s A 12 3 s  rate.  The h a r v e s t a b l e s u r p l u s i n t h i s  curvilinear  i s m a x i m i z e d ( R i c k e r 1973) by s e t t i n g e q u a t i o n  t o one a n d s o l v i n g f o r E,  dR  thus  0.5abfs s s A 12 3 s  1 =  dE  (1.9) e q u a l  (bA + 0.5s E) S 1  =  1.  (1.11)  2  * The  escapement  derived  from t h i s e x p r e s s i o n  (E  ) i s the 1  o p t i m a l escapement  f o r the case  where  rearing  area  does  not  18  present  a  c o n s t r a i n t , because E  *  maximizes s u r p l u s  production  1 (see F i g u r e  1.2)  •  0.5abfs s s A 12 3 s  1/2  0. 5s  I now d e r i v e an rearing be  area  returns  constraint  of  adults at R  2  s  1  expression  f o r optimal  escapement  c o n s t r a i n s t h e maximum number o f a d u l t s  noted i n Figure  adult  b A  .  .  when I t can  1.2 t h a t t h e f u n c t i o n R = f ( E ) shows i n c r e a s i n g  with  increasing  rearing  area  escapement  that  sets  subject  to  t h e maximum  T h i s means t h a t b e y o n d some  ceiling  the  returning escapement  * E  2  ,  adult  physical  returns R are constant.  space  survival (Chapman  and  and/or 1962,  food  availability,  premature 1966;  T h i s c e i l i n g may be s e t by  Mason  downstream and  that d r i v e s the r e a r i n g area  a saturable  functional  spawning a r e a . and  of  1965).  reduced smolts Another  c o n s t r a i n t i s t o assume  similar  to  that  A two-stage d e n s i t y dependent model would  t h e number o f s u r v i v i n g s m o l t s f r o m e q u a t i o n  r e p l a c e d by  in  emigration  Chapman  hypothesis  relationship  resulting  of  the  result  ( 1 . 5 ) c o u l d be  19  a' S =  (es ) 2  (1.5a)  b' + ( e s ) 2  where a' «= maximum number o f s m o l t s the r e a r i n g area a' = A  r  t h a t c o u l d be a c c o m o d a t e d  in  / c , and.  b' = s m o l t  half  s a t u r a t i o n parameter  b' = 1 s m o l t / m . 2  If  the f i r s t  ceiling  p r o p o s e d d e n s i t y dependent mechanism  holds,  then  rearranging  equation  of a  (1.8a),  numerical escapement  becomes  R b A s 0 . 5 a f s s s - 0.5s R 12 3 1  Given  the upper l i m i t  of  recruits,  set  .  at  (1.13)  R  «=  R  by  the  2 a v a i l a b l e r e a r i n g a r e a , I s u b s t i t u t e f o r R from equation a r r i v i n g a t a new  E  * 2  =  expression  for optimal  A cb A r s 0.5s  ( a f s -cA ) 1 2 r  .  (1.8b),  escapement  (1.14)  20  t  If  the  model  is  assumed  to operate  dependent s t a g e , as r e p r e s e n t e d e s c a p e m e n t has  a form s i m i l a r  i n c l u d e s t h e new repeated.  to equation  (1.14)  is  r e a r i n g a r e a c o n s t r a i n s salmon I have a l s o recruitment in Figure  considered  is a rectilinear  1.2),  represented  0.5 R  =  where  min  rearing  habitat  linearly thereafter this  the  upper  ( 1 . 1 2 ) , but  and  the  one  therefore w i l l  optimal  a  third  that not  escapement  production  model  f u n c t i o n of escapement  when  where  (dashed  line  = R  3  (1.15)  2  This  recruitment i s constant.  up  i s again  model to The  set only  however,  0.5  r  the r e a r i n g c a p a c i t y ; optimal  escapement  c  f s s 1 2  by  implies  case i s  A  be  by  l i m i t to production  recruitment  optimal  production.  constraints.  increasing  b',  (1.5a), then  E fs s s = R 12 3 3  A c s r  and  by e q u a t i o n  p a r a m e t e r s a' and  Equation  through a second d e n s i t y  (1.16)  in  21  Another rate.  tool  i n t h e management o f s a l m o n i d s  If optimal harvest rate i s defined  maximizes  the  (1.12),(1.14) expressed  long and  term  yield,  (1.16),  the  then  optimal  as  i s the  that  harvest  rate  which  f o r each of the  models  harvest  rate  may  be  as  HR  (1.17)  1 n n  where  n=1,2  o r 3, c o r r e s p o n d i n g  curvilinear, E  and  and  r e c t i l i n e a r models r e s p e c t i v e l y .  R i n equation  and  b A 2  =  1 -  Substituting  1/2  0.5fs s s 1 2 3  *  constrained  (1.17),  b t  HR  HR  t o the c u r v i l i n e a r ,  0.5s  s  s ( a f s -cA ) 13 2 r  (1.18)  (1.19)  22  HR  *  =  3  1  1 -  .  0.5 f s s s 12 3  (1.20)  1.3 A n a l y s i s o f t h e e f f e c t o f some k e y p a r a m e t e r s In  the  preceding  ( 1 . 1 6 ) were d e r i v e d three  section,  f o r optimal  regimes of c o n d i t i o n s .  harvest rates corresponding in  (1.12),(1.14)  escapement  t o these  (1.17),  regimes a r e a l s o  and  and  corresponding  B a s e d on e q u a t i o n  (1.18),(1.19)  equations  equations  (1.20).  How  to  optimal  specified do  optimal  e s c a p e m e n t s a n d h a r v e s t r a t e s c h a n g e when p r o d u c t i o n  parameters  change?  optimal  escapement  set  production  Alternatively,  and/or h a r v e s t  r a t e be,  what given  should a  the  changing  of  parameters? I  have  analyzed t h i s question f o r the r e l e v a n t parameters  i n each regime f o r o p t i m a l escapement  (equations  and  harvest  (1.16)),  and  for  (1.18),(1.19) and ( 1 . 2 0 ) ) . b,  A  to  prespawning,  s  and A . r  optimal  Parameters s , s 1 2 egg-to-smolt  half  saturation  and  3  respectively  smolt-to-adult  constant  b  is a  survival.  i sA . r  Optimal  territory  measure  a,  correspond  a s t h e maximum number o f  or p r o d u c t i v i t y of the stock.  and t h e r e a r i n g area s  and s  i s i n v e r s e l y r e l a t e d t o t h e female  dependence, A  (equations  females  c o u l d be accommodated i n t h e s p a w n i n g a r e a a v a i l a b l e  parameter The  rate  These p a r a m e t e r s a r e s , s , s , 1 2 3  Habitat c a p a c i t y a i s expressed that  (1.12),(J.14)  of  (this  size t ) . density  The s p a w n i n g a r e a i s  escapement  and  harvest  23  rate  expressions  were  these  parameters.  The  escapement  are  for equation ( 1 . 1 4 ) and is  in Figures to  at  a  w i t h r e s p e c t t o e a c h of  this  analysis  graphically in Figures  1.5a  1.4a  through  through 1.5d  for  1.3a  optimal  through  1.4g  for  equation It  note t h a t t h i s a n a l y s i s assumes t h a t o n l y  the  i s changing  while  v a l u e , and  i n g e n e r a l , as  for equation  1.3f  (1.16).  default  values are not,  of  in Figures  parameter i n question constant  results  shown  (1.12),  important  differentiated  the  others  are  t h a t the a b s o l u t e  relevant  as  the  kept  numerical  form  of  the  curves. The based  numerical  on  the  biostandards  following for  Salmon R i v e r . "typical"  used  physical  mainland r i v e r  r e a r i n g areas A  size  o f t = 10 m  carrying capacity i s a  -  these  results i s  characteristics ( 0.  i s a t r i b u t a r y of t h e  and  SEP  k i s u t c h ) i n the  Fraser  River  and  i n B r i t i s h Columbia, with  = A s  territory  to generate  a s t o c k of coho salmon  This  lower  s p a w n i n g and  example  =  83000  m, 2  and  a  a  equal female  r per  2  (A / t )  female. =  8300  Therefore  the h a b i t a t  females.  The  half  s saturation constant  b i s 0.1  is  f e m a l e f e c u n d i t y f i s 2500 e g g s / f e m a l e .  1  smolt/m  2  and  female/m , smolt  d e f a u l t v a l u e s of the the s u r v i v a l  rearing capacity c  2  rates are s  = 1  (  The  implying  1 that  there  is  no p r e s p a w n i n g m o r t a l i t y ) ,  s  = 0.012, and 2  0.15  (SEP  s  = 3  1982).  Under a c u r v i l i n e a r regime w i t h no rearing constraint, optimal escapement E i s an i n c r e a s i n g f u n c t i o n o f p r e s p a w n i n g 1  24  Figure  1.3 R e l a t i o n s h i p b e t w e e n o p t i m a l e s c a p e m e n t a n d parameters of t h e u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.12.  MAXIMUM EFFECTIVE FEMALES (»> (1000)  HALF SATURATION CONSTANT (b)  26  Figure  1.4 R e l a t i o n s h i p b e t w e e n o p t i m a l e s c a p e m e n t a n d parameters of r e a r i n g c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n corresponding to equation 1.14.  0  0  2  .6 PRESPAWNING SURVIVAL (»,) •  10 REARING AREA <A.) 10* m* '  1  .01  20  10 6 MAXIMUM EFFECTIVE FEMALE6 (•) 1000  SO  .02 EQQ-TO-8MOLT SURVIVAL <»„> 2  CO  J 5 SPAWNING AREA (A,) 10 m  a. O 8M0LT REARIG DEN8ITY (c>  .03  SO  28  Figure  1.5 R e l a t i o n s h i p b e t w e e n o p t i m a l e s c a p e m e n t a n d parameters of a c o n s t r a i n e d r e c t i l i n e a r f u n c t i o n corresponding to equation 1.16.  30  survival.  B u t a s shown i n F i g u r e  1.3a,  where o p t i m a l  i s p l o t t e d against prespawning s u r v i v a l  s , there  escapement  i s a threshold  1 below which i t i s o p t i m a l the p a r t i c u l a r  (using the present  stock to e x t i n c t i o n .  definition) to  For v a l u e s of s  fish  below  the  1 threshold,  the  recruitment  f u n c t i o n l i e s below the replacement  l i n e and t h e r e f o r e t h e s t o c k  is  not  viable.  Note  that  this  drastic result d e p e n d s on t h e a s s u m p t i o n t h a t t h e i n d e p e n d e n t v a r i a b l e (s i n t h i s case) i s c o n s t a n t h e n c e f o r t h . In a multi1 period context where s v a r i e s o v e r t i m e , t h i s r e s u l t may n o t 1 hold. The  optimal harvest  model (see e q u a t i o n threshold not  of  rate for t h i s unconstrained  (1.18) and  prespawning  persist.  Above  Figure  1.6a),  has  the  this  threshold,  graphs of F i g u r e s  1.3b  i s a l s o an i n c r e a s i n g f u n c t i o n  optimal  of  and s  1.3c and  2 critical  t h r e s h o l d of s u r v i v a l .  habitat  constraint,  related to carrying  the  shown i n F i g u r e s regime  optimal  spawning  capacity  of  i s , however,  the  density  harvest  rate  increases.  optimal s ;  could  escapement  again  with  a  3  escapement  i s positively  available,  spawning  l.3e.  and  hence  linearly to  the  h a b i t a t (parameter a ) , as  O p t i m a l escapement  under  this  n o n l i n e a r l y r e l a t e d to the h a l f s a t u r a t i o n  parameter b (Figure 1.3f). of  critical  U n d e r t h i s r e g i m e o f no r e a r i n g  area  1.3d a n d  a  s u r v i v a l below which the s t o c k  i n c r e a s e s a s y m p t o t i c a l l y as prespawning s u r v i v a l In  curvilinear  dependence  of  T h i s parameter effective  reflects  females.  the  degree  A stock with a  31  Figure  1.6 R e l a t i o n s h i p b e t w e e n o p t i m a l h a r v e s t r a t e and parameters of u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n corresponding to equation 1.18.  32  33  small a  b value indicates a higher  large  b  value,  p r o d u c t i v i t y than a stock  and t h u s t h e i n i t i a l  function  r i s e s more  capacity  i s r e a c h e d s o o n e r w i t h more p r o d u c t i v e  less  productive  0.05,  then d e c l i n e s  steeply  ones.  Under t h e  and  the  slope  Optimal  the  of the  production  habitat  saturation  stocks  escapement  with  than  increases  for b <  a t a s l o w e r r a t e f o r b > 0.05 ( F i g u r e  regime  of  rearing  area  constraint  with  1.3f).  (equation  * 1.14),  optimal  escapement  E  is  inversely  related  to  2 prespawning and egg-to-smolt hence and  to  spawning  t o t h e p a r a m e t e r a , a s shown i n F i g u r e s  1.4e r e s p e c t i v e l y .  related this  survival,  to  the  half  regime o p t i m a l  rearing  area  Optimal  escapement  i s an  and  1.4a, 1.4b, 1.4d, here  is  saturation constant b (Figure  escapement  area  increasing  A , a n d maximum r e a r i n g d e n s i t y  linearly  1.4g).  In  function  of  c, r e f l e c t i n g the  r fact  that  require 1.4 f  f o r a given  stock,  rectilinear  model  Optimal  smolt-to-adult increases,  escapement  survival  (Figure  increasing optimal  rate  the  1.4c a n d  constrained  1.5c).  but  parameters  (1.20) c o r r e s p o n d i n g t o  the  in three  of  As r e a r i n g p a r a m e t e r c  capacity  escapement  independent  of t h e h a b i t a t ,  (Figure  equations models  hence  1.5d).  1.6, 1.7 a n d 1.8 show t h e r e l a t i o n s h i p to  production  (Figures  in  1.5a a n d 1 . 5 b ) ,  so does t h e p r o d u c t i v e  Figures  in  i s i n v e r s e l y r e l a t e d t o prespawning and egg-  to-smolt s u r v i v a l (Figures  harvest  increases  ever l a r g e r r e a r i n g h a b i t a t c a p a c i t y ,  respectively).  linearly  marginal  of  (1.18),  optimal  (1.19) and  respectively.  e f f e c t o f e g g - t o - s m o l t s u r v i v a l ( s ) on t h e o p t i m a l 2  harvest  The rate  34  Figure  1.7 R e l a t i o n s h i p b e t w e e n o p t i m a l h a r v e s t parameters of the c o n s t r a i n e d c u r v i l i n e a r corresponding to equation 1.19.  r a t e and function  g  .06 -1 .16 HALF SATURATION CONSTANT (b)  •*  36  Figure  1.8 R e l a t i o n s h i p b e t w e e n o p t i m a l h a r v e s t r a t e and p a r a m e t e r s of the r e c t i l i n e a r f u n c t i o n c o r r e s p o n d i n g equation 1.20.  to  38  has a s i m i l a r 1.7b,  and  s a t u r a t i o n form f o r a l l t h r e e models ( F i g u r e  1.8b),  rectilinear  however t h e c e i l i n g  model.  The  optimal  harvest  r e l a t e d to the h a l f  s a t u r a t i o n constant  curvilinear  ( F i g u r e s 1.6f  model  on b i n t h e r e c t i l i n e a r m o d e l .  i s reached  and  sooner i n the  rate  is  inversely  (b) i n b o t h c a s e s  1.7g), b u t d o e s n o t  Spawning a r e a  (A  1.6b,  ) and  of  the  depend  effective  s females  (a) h a v e  unconstrained  no  curvilinear  only a small e f f e c t 1.7e).  influence  Changes  on  model  the  harvest  (Figures  i n the c o n s t r a i n e d case  i n the r e a r i n g area  rate  1.6d  and  the  1.6f),  and  1.7d  and  (Figures  (A ) h a v e an  in  impact  on  the  r optimal harvest The  s e n s i t i v i t y o f o p t i m a l e s c a p e m e n t and  rate  to  Table  I and  the  various  and  in  1 percent  Table  sensitive  to  is  shows  as used  importance that  egg-to-smolt  harvest  defined  escapement or h a r v e s t default  value  In t h i s sense s e n s i t i v i t y  of e l a s t i c i t y  I  Sensitivity  i n c r e a s e i n the  measures the r e l a t i v e  points.  optimal  model.  t h e t h r e e m o d e l s a r e shqwn i n  optimal  considered.  to the concept  of  II respectively.  change  to a  parameter  parameters  Table  percent  response  r a t e only i n the c o n s t r a i n e d c u r v i l i n e a r  in  survival  escapement and  of  the  terminology,  of parameters at  (s )  rate in  i s equivalent  economic  optimal  as  different is  highly  smolt-to-adult  2  survival  (s ) 3  under  however,  under  the  the  unconstrained  constrained  curvilinear  p a r a m e t e r s have n e g l i g i b l e or n u l l e f f e c t s . that  other  parameters  have  only  curvilinear  small  model;  model these  Table effects  I also  same shows  in a l l three  e I S e n s i t i v i t y of o p t i m a l escapement t o p a r a m e t e r s of t h e t h r e e models r e p r e s e n t i n g change i n o p t i m a l escapement due t o an i n c r e a s e i n p a r a m e t e r v a l u e s f o r c u r v i l i n e a r , c o n s t r a i n e d c u r v i l i n e a r , and r e c t i l i n e a r models. Numbers i n p a r e n t h e s e s a r e p e r c e n t c h a n g e s i n o p t i m a l escapement f o r a 1 p e r c e n t i n c r e a s e i n d e f a u l t parameter values.  40  Parameter and value  1 1.0 s  Unconstrained curvilinear model  Constrained curvilinear model  Rectilinear model  increases  decreases  decreases  (0.5)  (1.0)  (1.0)  increases  decreases  decreases  (14.6)  (1.5)  (1.0)  increases  no c h a n g e  no c h a n g e  (14.6)  (0)  (0)  increases  increases  (1.0)  (1.0)  no c h a n g e  increases  (0) increases  (0)  (1.5)  (1.0)  increases  decreases  no c h a n g e  (0.5)  (1.5)  (0)  decreases  increases  no c h a n g e  (5.0)  (1.0)  (0)  no c h a n g e  increases  increases  (0)  (1.5)  (1.0)  2 0.012 s 3 0.15 A  1  s 83000 A  m  2  no c h a n g e  1  r 83000  m  2  t 10  m  b 0.1  F  2  /m  2  9  c 1 smolt/m  2  Spawning and r e a r i n g a r e a s a r e p h y s i c a l parameters S a l m o n R i v e r ( B r i t i s h C o l u m b i a ) , n o t SEP b i o s t a n d a r d s . 1  of  the  41  T a b l e I I S e n s i t i v i t y of o p t i m a l h a r v e s t r a t e t o p a r a m e t e r s of the t h r e e models r e p r e s e n t i n g change i n o p t i m a l h a r v e s t r a t e due t o an i n c r e a s e i n p a r m e t e r v a l u e s u n d e r c u r v i l i n e a r , c o n s t r a i n e d c u r v i l i n e a r , and r e c t i l i n e a r models. Numbers i n p a r e n t h e s e s a r e p e r c e n t c h a n g e s i n optimal harvest rate for a 1 percent increase i n d e f a u l t parameter v a l u e s .  42  Parameter and value  unconstrained curvilinear model  constrained curvilinear model  Rectilinear model  increases  increases  increases  (1.0)  (2.0)  (0.3)  increases  increases  increases  (1.0)  (3.0)  (3.0)  increases  increases  increases  0.15  (1.0)  (2.0)  (0.3)  A  no c h a n g e  increases  no c h a n g e  (0)  (1.0)  (0)  no c h a n g e  no c h a n g e  no c h a n g e  (0)  (0)  (0)  decreases  decreases  no c h a n g e  (1.0)  (3.0)  (0)  decreases  decreases  no change  (1.0)  (2.0)  (0)  no c h a n g e  decreases  no c h a n g e  (0)  (1.0)  (0)  s 1 1.0 s 2 0.012 s 3  1  s 83000 A  m  2  1  r 83000  m  2  t 10  m  2  b 0.1  F  /m  2  9 c 1 smolt/m  2  Spawning and rearing a r e a s a r e p h y s i c a l parameters of the S a l m o n R i v e r ( B r i t i s h C o l u m b i a ) , n o t SEP b i o s t a n d a r d s . 1  43  models. most  In contrast  T a b l e I I shows t h a t o p t i m a l  harvest rate i s  s e n s i t i v e t o changes i n the parameters of  the  c u r v i l i n e a r model. These If there goes  '.  r e s u l t s show t h e i m p o r t a n c e o f r e a r i n g  i s no r e a r i n g c o n s t r a i n t  up  with  constrained  higher  then  prespawning  s u r v i v a l , but i f r e a r i n g  the  constraints.  optimal  survival  or  i s l i m i t e d the optimal  escapement egg-to-smolt  escapement  goes  down. Finally, optimal rates not  lead t o higher  limiting,  i f  better  differences  between  harvest rate are i l l u m i n a t i n g .  is limiting. So  the  optimal  we were u n c e r t a i n off  furthermore,  using do  a not  rearing  area  e s c a p e m e n t s when r e a r i n g  harvest rate  and  i s higher  f o r both  is area  cases.  a b o u t r e a r i n g c o n s t r a i n t s , we m i g h t be  harvest depend  w h e r e a s optimum e s c a p e m e n t s d o . substantially  escapement  Higher presmolt s u r v i v a l  e s c a p e m e n t s when  but t o lower o p t i m a l The o p t i m a l  optimal  less information  rate upon  policy. spawning  Harvest rate  Harvest  rates,  or rearing policies  t h a n optimum e s c a p e m e n t  areas, require  policies.  44  CHAPTER TWO  EXTENSION OF THE MODEL  2.1 I n t r o d u c t i o n In  chapter  density  one  dependent  survival  rates,  are  life  stages and  c y c l e was a n a l y z e d . multistage  a  separated  representing  density  life  relationships,  constitutes  the  input  stages  the  as  e t aJL  ( 1964)  curves  about  characteristics.  stock one  this at  and stage  idea  and  intermediate  based  on  a  variety  premises (1964)  e x p l a i n e d t h e phenomenon o f c y c l i c d o m i n a n c e i n t h e Adams  River  due  salmon  ( 0.  nerka  Ward  of  and L a r k i n  sockeye  history  effects  stages  regarding  from  Using  a  p r o d u c e d a number o f t h e o r e t i c a l  s t o c k and r e c r u i t m e n t life  by  successive  next.  e m p l o y i n g compensatory and depensatory Larkin  history  accomplished  such t h a t t h e output  to  life  t h e model i s extended t o  This i s  history  independent  a g e n e r a l i z e d salmonid  f o r m s u c h t h a t more i n t e r m e d i a t e l i f e  intermediate  stages,  m o d e l c o n s i s t i n g o f two by  In t h i s chapter  formally inc6rporated.  recruitment  history  ) i n terms of depensatory  t o predation, during the l a c u s t r i n e stage.  A  mortality  logical  next  s t e p i s t o d e r i v e an a n a l y t i c a l model t h a t i n c o r p o r a t e s a l l l i f e history  stages.  This  model  will  then  be  used t o d e r i v e a  functional  form f o r t h e o p t i m a l s t o c k s i z e and h a r v e s t  terms  the  of  optimal harvest  parameters  o f any s t a g e  r a t e and stock  i n the l i f e  s i z e are again defined  which maximize the l o n g term h a r v e s t a b l e s u r p l u s . of t h i s chapter general  form  rate  cycle. as  in The  those  The r e m a i n d e r  i s d i v i d e d i n t o two s e c t i o n s : i n s e c t i o n 2.2 t h e of  a  multistage  stock-recruitment  function i s  45  derived, while rate,  i n s e c t i o n 2.3  s t o c k s i z e and  f u n c t i o n a l f o r m and different  the r e l a t i o n s h i p  and  parameters  formulation  a  Instead  of  represent  2.2  The  capacity  harvest  intermediate parameters i s presented. are  than  presented  in  the  parameter  "escapement",  I  in  previous  p r o d u c t i v i t y parameter p r e p l a c e s the h a l f b;  between  a  The  slightly  chapter.  saturation  constant  c replaces i t s counterpart use  the  term  A  "stock  a .  size"  to  numbers o f s p a w n i n g a d u l t s .  m u l t i s t a g e s t o c k and  Consider i n F i g u r e 2.1  Beverton and  and  recruitment  model  H o l t ' s stock-recruitment curve  w r i t t e n i n the  shown  form  c  where R = number o f  recruits,  S = number o f  spawners,  p = p r o d u c t i v i t y parameter,  and  c = c a p a c i t y parameter. In a h y p o t h e t i c a l juveniles  and  holds  both  for  adults, stages,  population assume but  consisting  that that  of  two  stages,  the above f u n c t i o n a l form the  parameters  are  stage  46  F i g u r e 2.1 B e v e r t o n and H o l t t y p e s t o c k and corresponding to equation 2.1.  recruitment  curve  RECRUITS (R)  CO  TJ  >.  Z  m CO CO  o  48  specific.  Then  I +  1  where  R  1  = juveniles  S = spawning  JL S c,  produced  adults,  represents the f i r s t  s t a g e ; and  i +  h_  c  R  1  2  where R  2  = Adults  produced  R  = j u v e n i l e s from s t a g e 1 the second stage.  Substituting  for R  1, a n d e q u a t i o n  (2.3) r e p r e s e n t s  we g e t 1  ^(1^)5 T h i s may  be r e p e a t e d  f o r more s t a g e s  and a g e n e r a l  form  for  n  49  s t a g e s may be r e p r e s e n t e d by  Equation  ( 2 . 5 ) may be r e w r i t t e n  as  P„ 5 £ 5  12 J.)  +  where  P.- W is  a  composite  f.  of the p r o d u c t i v i t y parameters of n s u c c e s s i v e  s t a g e s , and  From e q u a t i o n  (2.8)  we  can  solve  for  the  overall  capacity  50  parameter C , r e s u l t i n g i n n  c „  Thus,  the  --  -  cumulative  composite p r o d u c t i v i t y  8  (2-9)  '  —  capacity  parameter  parameter  P  and  i s a function the  stage  of the  specific  n capacity  c . i  Next,  I  solve  for P  and n  overall productivity some  stage  n,  may  and c a p a c i t y be d e f i n e d  C  recursively  such that the  n parameter  up t o a n d  i n terms of t h e (n-1)th  including stage.  T h u s e q u a t i o n ( 2 . 8 ) may be e x p r e s s e d a s  (z.iz)  51  However,  U  55  therefore equation  '  —  *  ( 2 . 1 4 ) by P  p .Jand  *i-i  ( 2 . 1 2 ) may  " ^  D i v i d i n g equation  r  +  be  v  r e w r i t t e n as  -  n-1  |  gives  ,  (  t h e o v e r a l l c a p a c i t y p a r a m e t e r may  be e x p r e s s e d  2  , ) 5  recursively  as  C  n  —  This  derivation  (Z./4)  —  =  allows  —n-'  us  to  write  the  stock-recruitment  r e l a t i o n s h i p r e c u r s i v e l y s u c h t h a t r e c r u i t s t o any  stage  n  may  52  be  expressed  1)th stage.  i n t e r m s o f t h e c o m p o s i t e p a r a m e t e r s up t o t h e ( n Thus t h e r e c u r s i v e f o r m o f e q u a t i o n  ( 2 . 6 ) becomes  when s u b s t i t u t i o n s a r e made.  2.3 A n a l y s i s o f i n t e r m e d i a t e H a v i n g an e x p r e s s i o n relationshp,  i t  is  parameters  for  now  a  feasible  multistage  stock-recruitment  t o t o examine t h e e f f e c t of  c h a n g e s i n i n t e r m e d i a t e p a r a m e t e r s on management d e c i s i o n s . in  chapter  one, I f i r s t  s i z e and h a r v e s t recruitment respect  d e r i v e an e x p r e s s i o n  f o r optimum  r a t e from t h e g e n e r a l m u l t i s t a g e  function.  t o S, s e t  the  form  Then I d i f f e r e n t i a t e e q u a t i o n derivative  equal  to  one  to  of  As stock the  (2.6) w i t h maximize  * surplus Thus  over  replacement,  and s o l v e f o r t h e o p t i m a l  stock S .  53  and  Using equation  (  -  <-  *  (2.8), S  may  be  rewritten  2i9)  as  Yz  1%. If t h i s l a s t represented  result  i s combined w i t h the g e n e r a l m u l t i s t a g e  by e q u a t i o n  (2.6), the r e s u l t i n g o p t i m a l  form  recruitment  is  f?  ^  T h i s e x p r e s s i o n may equation  (2.19),  be  giving  /  5  s i m p l i f i e d by  substituting  for  S  \  from  54  To  calculate  the  optimal  harvest  model I e x p r e s s the o p t i m a l s t o c k  rate H  i n terms of H  f r o m t h e above as  (2-23)  or  2.2+)  Substituting  for R  from e q u a t i o n  (2.22),  gives  r ]/2  (2.25.  w h i c h c o u l d be made an e x p l i c i t  function  o f any component p  , as J  in  U  -  i  "is I  fi  -y  »•  T h i s r e l a t i o n s h i p i s shown g r a p h i c a l l y The  optimal  stock  2  size i n equation  i n F i g u r e 2.2 ( 2 . 2 0 ) may  be  . considered  55  F i g u r e 2.2 R e l a t i o n s h i p b e t w e e n o p t i m a l h a r v e s t r a t e a n d p r o d u c t i v i t y parameter p f o r any i n t e r m e d i a t e s t a g e j J i n an n s t a g e s t o c k a n d r e c r u i t m e n t m o d e l c o r r e s p o n d i n g to-equation  (2.26).  at which harvest  f  p  i s the p r o d u c t i v i t y j r a t e becomes p o s i t i v e .  threshold  <5C  57  t o be f u n c t i o n  of a s e r i e s of parameters p  and c J  j,  w h e r e 2 < j < n.  explicit  function  (2.20) may  Thus e q u a t i o n  of any i n t e r m e d i a t e  *  ^  5  -  o f any  stage  J be r e w r i t t e n a s an  p r o d u c t i v i t y parameter p : j  (2.21)  1  where  and  (7.11c) are  constants  associated  r e l a t i o n s h i p between S  with  and p  is  the  j t h stage.  therefore  The  functional  dependent  on  the  58  sign  of  the d e r i v a t i v e of equation  (2.20) w i t h r e s p e c t t o p .  This d e r i v a t i v e i s  and  c a n be e i t h e r p o s i t i v e o r n e g a t i v e d e p e n d i n g on t h e a b s o l u t e  v a l u e s o f t h e component t e r m s . g r a p h i c a l l y a s shown  may  be  parameter c  where  be  expressed  stock  size  expressed  by  equation  r e w r i t t e n i n t e r m s o f any i n t e r m e d i a t e c a p a c i t y  such t h a t j  ( 2 . 2 7 ) may  i n F i g u r e 2.3 .  Likewise, the optimal (2.20)  Equation  59  F i g u r e 2.3 R e l a t i o n s h i p b e t w e e n o p t i m a l s t o c k s i z e and p r o d u c t i v i t y parameter p f o r any i n t e r m e d i a t e s t a g e j j i n an n s t a g e s t o c k and r e c r u i t m e n t m o d e l c o r r e s p o n d i n g to equation it  (2.27).  p  t  i s optimal to harvest  inflection point p  i s the t h r e s h o l d below j the stock to e x t i n c t i o n .  i s where t h e d e r i v a t i v e j  (2.28) e q u a l s  z e r o and  S  *  r e a c h e s a maximum.  in  which The  equation  61  Equation Figure  (29) has t h e f a m i l i a r 2.4  with  an i n i t i a l  saturation  type  form  shown  in  s l o p e o f 1 / l a n d an a s y m p t o t e o f 1 3  1 / 1 . 1 2 These r e s u l t s c o n f i r m one  and  generalize  intermediate  life  them  history  the conclusions to  any  stages.  reached  harvestable  in  chapter  population  with  62  F i g u r e 2.4 R e l a t i o n s h i p b e t w e e n o p t i m a l s t o c k s i z e a n d c a p a c i t y p a r a m e t e r c f o r any i n t e r m e d i a t e s t a g e j i n an j n s t a g e s t o c k and r e c r u i t m e n t model c o r r e s p o n d i n g t o equation (2.29).  63  CAPACITY (c. )  64  CHAPTER THREE S t o c k and management  recruitment  of  predicted  or  allow  the  the  extent  recruitment Factors  (Lasker 1983)  sum  1980)  marine  Tyler of  oceanographic  believe  recruitment  the  1983;  models  be  stock  do  changes i n l i f e  mechanisms  acting  history life  disaggregate  components. s u r v i v a l (Lasker  Kreutz et a l .  1982)  measurements  conditions of  like  of s u r v i v a l .  salmonids  is  albeit  to not  during  i t i s possible to  into constituent  variables,  point wind  1979,  to  To  velocity Huyer  the extent  coupled  to  w i t h a margin of t h e one  the  that  specific  uncertainty,  p r e s e n t e d h e r e may  be  tool.  W h e r e a s we  affect  h i s t o r y stages cannot  expected  t h e n a h a b i t a t based model such as  survival  when  o r n e a r s h o r e t e m p e r a t u r e r e g i m e s ( K r u s e and  estimate  the  t o t a l of t h e s e mechanisms i s  and  elucidated,  physical  survival  a useful  for  particularly  g o v e r n i n g m a r i n e g r o w t h and  use  to  be  of that  function  K r u s e and  potential  life  tools  H o w e v e r , t h e s e p a r a m e t e r s a r e assumed  incorporation  To  useful  p a r a m e t e r s , namely p r o d u c t i v i t y of  h i s t o r y e v e n t s may  1981;  The  the average; thus stock  events.  the  intermediate  habitat capacity.  h o l d on  are  populations,  controlled.  c o m p r e s s e d i n two and  models  exploited  m e c h a n i s m s a c t i n g on  DISCUSSION  are  r a t e s , we  such r a t e s that  temperature  are  of  and  critical  and  smolt-to-adult  to understand the  C l a r k and  upwelling,  structure,  to modify  beginning  (ODFW 1982;  early  temporal pattern  less likely  as  i f we  McCarl  ocean  factors If  we  o c e a n s u r v i v a l d e p e n d s on  the  interpreted observe that  1983).  that  by such a  near-shore structure  65  i s c h a n g i n g , we c o u l d  i n c r e a s e or decrease r e q u i s i t e  escapement  goals i n response. The in  implications  prespawning  of expected changes,  mortality,  on  management  readily  o b v i o u s ; i n d e e d t h e y may  example,  a fisherman w i t h a short time  higher  catches  its toll.  before  appear  A r e g u l a t o r y agency,  compensate  for  increase  decisions  are  not  in conflict.  For  t o be horizon  would  an e x p e c t e d p r e s p a w n i n g  advocate  mortality  on t h e o t h e r h a n d , w i t h  t e r m mandate f o r t h e r e s o u r c e w o u l d to  s u c h a s an  takes  a  long  recommend h i g h e r e s c a p e m e n t s  the higher expected prespawning  mortality.  U s i n g a s p e c i f i c d e c i s i o n r u l e , namely t o maximize  the l o n g term  harvestable surplus, the r e s u l t s i n  resolve  apparent explicit  conflict  and  incorporation  parameters,  such  as  chapter  one  using  three  illustrate, of  expected  survival  changes  rates,  this  models,  in  life  the  history  i n management d e c i s i o n s .  T h i s d e c i s i o n r u l e h o w e v e r , i m p l i c i t l y a s s u m e s t h a t b o t h manager and  f i s h e r m a n a r e o p e r a t i n g u n d e r an  namely  zero.  Incorporating  a  a p p l y i n g t h e d e c i s i o n r u l e may Hilborn  the a n a l y s i s no  rate  while and  t h e most s u r p r i s i n g and c o u n t e r i n t u i t i v e r e s u l t  of  i s the importance of r e a r i n g c o n s t r a i n t s .  rearing  rearing  constraint  is  If there  t h e n t h e o p t i m a l escapement goes  s u r v i v a l or egg-to-smolt  survival,  i n F i g u r e 2.3.  of  any  intermediate  M o s t human i n t e r v e n t i o n w i l l  This  stage  affect  up but  l i m i t e d t h e o p t i m a l e s c a p e m e n t g o e s down.  c o n c l u s i o n holds f o r parameters shown  modify  dicount  rate,  i t s outcome ( M o u s s a l l i  w i t h higher prespawning if  positive  discount  1983).  Perhaps  is  identical  as  these  66  survival  rates.  prespawning survival  s u r v i v a l due  due  enrichment, affect  Hydrological  to  canalization,  and Brownley and Walker  1981).  may  t o temperature changes,  siltation  egg-to-smolt  manipulation  or  modification  bank  survival  protection  or  change  egg-to-smolt  of f l o w . etc.  (Cederholm et a l .  should a l l  1981;  Scrivener  Though t h e s e e f f e c t s a r e documented  1981; H a r t m a n e t a l .  1982),  to  my  Stream  (Lister  knowledge  the  c o n s i d e r a t i o n g i v e n t o such h y d r o l o g i c a l m a n i p u l a t i o n s , does not extend s u f f i c i e n t l y In  a  life  t o management  history  implication.  m o d e l s u c h a s t h e one p r e s e n t e d h e r e ,  d i s a g g r e g a t i n g the p r o d u c t i o n f u n c t i o n  i n t o component s t e p s  further  enhancement  implications  Alternative  proposals  improvements, few, may stage  lake  the  production  enhancement,  fertilization,  .  life  to  that  s p a c e due production  history  To do t h i s , a  if  a  spreadsheet  ( M o u s s a l l i and H i l b o r n is  of  salmonid  a  well  version  One  as  impact a t as  of  stream  the  model  on  a  misallocated.  simulation  that  rearing  increase  e v a l u a t e d and c o m p a r e d .  computational  framework  Thus u s e d , t h i s  model  for evaluating alternative  proposals.  Inasmuch as l o c a l  knowledge  available,  the  such  outcome  of  fry  V a r i o u s enhancement p r o j e c t s a r e  s i m u l a t e d o v e r t w e n t y y e a r s ' t i m e ; d i s c o u n t e d b e n e f i t s and are  been  microcomputer  outcome o f s u c h a  projects  overall  has  h a b i t a t b o t t l e n e c k e x i s t s such as smolt then  bed  specific  i n terms of  this  configuration  1983).  to t e r r i t o r i a l i t y , are  as  such  activities.  o r bank p r o t e c t i o n t o name a  be e v a l u a t e d i n t e r m s o f t h e i r  in  adapted  for  has  of  stream  simulations  costs  provides  a  enhancement reaches  is  i s increasingly  67  reliable. Disaggregating presented  a  allows  production  function-  and  habitat, Hartman  such as those  1982;  Columbia, two  Holtby  the west c o a s t  Scrivener  and  modifications  opening  transfer  of  including  the  of  Andersen  canopy  biological  benthic  and  s a l m o n i d s can  as the d r i v i n g free  of  i n t o the  densities  are  term  (35-50  increases  drift  a l l increase.  by  the  habitat.  resource  of a n o t h e r As Ursin  between  term  (1-15  r a t e of  energy  Primary  production,  These  predators  impacts  t h e s t r e a m and  disappear  the  increased  the second growth f o r e s t . usually  negative  and  not  be a d e q u a t e t o  I t i s p o s s i b l e t o use long term e f f e c t s  be e x p l i c i t l y  incorporated  long  gradient transport  destroying  the present the  The  model  exploitation  i n management  plans  one.  species  1982;  history,  the  study British  T h e y a r e most p r o n o u n c e d i n low  i n a manner t h a t a l l o w s t h e one  short  f i n e s which then accumulate i n the g r a v e l  spawning  (Holtby  Island,  f o o d o r g a n i s m s and  s t r e a m s where m e c h a n i c a l e n e r g y may introduced  In the  community.  reduced  years).  anadromous  (1982) d i f f e r e n t i a t e d  f o r c e o f s u n l i g h t on  nutrients  to  In a ten-year  Vancouver  second group of h a b i t a t changes a r e  of  1982).  groups of l o g g i n g r e l a t e d e f f e c t s :  years)  manner  g e n e r a t e d by d e f o r e s t a t i o n  H a r t m a n and  o f a w a t e r s h e d on  the  the the e v a l u a t i o n of p o t e n t i a l l y p r e d i c t a b l e  e f f e c t s of l o n g term anthropogenic fish  in  i n t e r a c t i o n s become more t r a c t a b l e ( J o n e s  Sheldon et a l .  disaggregating  f e a s i b l e and  thus  may  a help  1982)  at  intermediate  production formulate  function  stages  1982; in  becomes  multispecies  life more  fisheries  68  management. This  study  intermediate  life  future empirical explicitly  points  t o the importance of f a c t o r s a c t i n g a t  h i s t o r y stages studies  incorporated  define  i n harvestable such  populations.  factors,  i n management d e c i s i o n s .  they  may  As be  69  LITERATURE CITED  Bakun, A. 1973. C o a s t a l u p w e l l i n g i n d i c e s , w e s t c o a s t o f N o r t h A m e r i c a , 1946-1971. U.S. D e p t . Commerce, NOAA Tech. R e p t . NMFS SSRF 6 7 1 . 103pp. Bakun, A. 1975. D a i l y a n d w e e k l y u p w e l l i n g i n d i c e s , w e s t c o a s t of N o r t h A m e r i c a , 1 9 6 7 - 1 9 7 3 . U.S. D e p t . Commerce, NOAA T e c h . R e p t . NMFS SSFR 6 9 3 . 114pp. Beverton, R.J.H. a n d S . J . H o l t . 1957. On t h e d y n a m i c s o f exploited fish populations. U.K. M i n . A g r . , F i s h . I n v e s t i g . , S e r . 2, 19:533 p . Bovee, K.D. 1978. P r o b a b i l i t y of use c r i t e r i a f o r t h e f a m i l y s a l m o n i d a e . 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