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Numerical experiments with least-squares catch-at-age analysis Lawson, Timothy Adair 1980

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NUMERICAL EXPERIMENTS WITH LEAST-SQUARES CATCH-AT-AGE ANALYSIS by  TIMOTHY ADAIR IAWSON E . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1977  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACUITY OF GRADUATE STUDIES (Department o f Zoology) and ( I n s t i t u t e o f Animal Resource Ecology)  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g tc the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA A p r i l , 1980  ©  Timothy  A d a i r Lawson 1S80  In p r e s e n t i n g t h i s  thesis in partial  an a d v a n c e d d e g r e e a t the  Library  I further for  shall  the U n i v e r s i t y  make i t  agree that  freely  of  extensive  s c h o l a r l y p u r p o s e s may be g r a n t e d  this  written  thesis for  It  of  the requirements  B r i t i s h Columbia,  available for  permission for  by h i s r e p r e s e n t a t i v e s . of  fulfilment  I agree  r e f e r e n c e and copying of  this  f i n a n c i a l gain shall  that  not  copying or  be a l l o w e d w i t h o u t  T i m o t h y A. Lawson  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  DE-6  BP  75-S1 1 E  April  29,  1980  thesis or  publication  permission.  D e p a r t m e n t nf Z o o l o g y  that  study.  by t h e Head o f my D e p a r t m e n t  i s understood  for  my  ABSTRACT  Three methods of a n a l y z i n g age composition of e x p l o i t e d p c p u l a t i c n s are compared assumptions  about  recruitment,  on  from the catches  the  basis  harvesting,  of  and  their  natural  m o r t a l i t y ; the manner i n which data e r r o r s are t r e a t e d ; and  how  information  in  contained  estimating population involves  in  the  catch  data  is  parameters. The a n a l y s i s of  utilized catch  curves  the s t r i c t e s t assumptions and uses the l e a s t amount of  i n f o r m a t i o n i n the data. Cohort a n a l y s i s , while having  the  relaxed  restricts  assumptions,  ignores  i n f o r m a t i o n to w i t h i n c o h o r t s . Doutleday  (1S76)  takes  the The  full  errors  and  least-sguares  account  of  approach  data  errors  i n f o r m a t i o n between c o h o r t s , but assumes that the age c h a r a c t e r i s t i c s c f a f i s h e r y are constant The  least-sguares  technigue  is  most  and  selection  over time.  modified  t o account f o r  changes i n the r e l a t i v e v a r i a n c e of data e r r o r s with age and prevent  unreasonable estimates  estimates  of catch-at-age  of  analyzing  about  values,  that  consistent  and  in  a  way  data,  parameter guarantees  results.  The method i s a p p l i e d to northwestern catch-at-age  catches-  error variances, e f f o r t  r e p r c d u c t i c n s t a t i s t i c s , and p r i o r i n f o r m a t i o n a l l simultaneously  to  of p o p u l a t i o n parameters. In i t s  most g e n e r a l form, the method i s capable at-age,  of  Atlantic  harp  seal  data. The r e s u l t s i n d i c a t e t h a t the s e a l p o p u l a t i o n  iii  is  more  abundant  than  previous analyses have shown. However,  p r o j e c t i o n s with a guota of 180,000 animals p r e d i c t a d e c l i n e i n the stock t c t w c - t b i r d s i t s present abundance over the next  ten  years,. Monte  Carlo  studies  were  performed  g e n e r a l r e l a t i o n s h i p between r e l i a b i l i t y  to  i n v e s t i g a t e the  of p o p u l a t i o n f o r e c a s t s  and the i n f o r m a t i o n content of the d a t a , as determined quantity,  data  errors,  and  the  contrast  in  e x p l o i t a t i o n r a t e s during the p e r i o d over which  by  abundance the  data  data and were  taken. The c o e f f i c i e n t of v a r i a t i o n f o r p r e d i c t i o n s of abundance ranged frcm 1% f o r high c o n t r a s t , high g u a n t i t y data s e t s t o 91% for  low  quantity,  low c o n t r a s t data. Estimates of the n a t u r a l  mortality  r a t e with the l e a s t - s g u a r e s technigue are p r e c i s e when  the data  set  moderate.  is  large,  contrast  is  high,  and  errors  are  iv  TABLE OF CONTENTS  ABSTRACT  i  i  TABLE CF CONTENTS .................................... i v L I S T OF TABLES  vi  L I S T OF FIGURES ........... AKNOWLEDGEMENIS  viii  . ..  ix  1, INTRODUCTION ........  1  2. CAICH-AT-AGE ANALYSIS ...........  5  2.1 C a t c h Curves, Cohort Sguares Approach  A n a l y s i s , And The L e a s t  6  2.2 F o r m u l a t i o n Cf The L e a s t - S q u a r e s Estimator  .14  2.2.1  .15  The M o d e l  2.2.2 The O b j e c t i v e F u n c t i o n 2.3 Seme M o d i f i c a t i o n s To T h e . l o g  18 Transform  M o d e l ... 21  2.3.1  W e i g h t i n g And A l t e r n a t i v e E s t i m a t i o n C r i t e r i a ....................... 21 2.3.2 U n r e a s o n a b l e C c n v e r g e n c e And C o n s t r a i n t s .. 23  2.3.3  M o d i f y i n g The M o d e l W i t h A S t o c k - R e c r u i t R e l a t i o n s h i p ..............................  2.3.4 The L e a s t - S q u a r e s Estimating  27  Approach F o r  N a t u r a l M o r t a l i t y ................ 28  2.4 I n t e r p r e t a t i o n Of The L e a s t - S g u a r e s  E s t i m a t e s ... 29  2.4.1 The S t a t e R e c o n s t r u c t i o n And F o r e c a s t i n g ...29 2.4.2 The C o v a r i a n c e M a t r i x And The R e l i a b i l i t y Of The E s t i m a t e s 32 2.4.3  The R e s i d u a l s And L a c k  Of F i t  2,.5 E x t e n s i o n s C f The L e a s t - S g u a r e s  Approach  .36 ........ 37  2.6  Summary  . 39  3m CASE STUDY: NORTHWESTERN ATLANTIC HABP SEALS 3.1  Data.  ,  .42 ,  43  3,.2 P c r m u l a t i c n Of The W e i g h t e d L e a s t - S q u a r e s E s t i m a t o r , N a t u r a l M o r t a l i t y Unknown 3,.3 E e s u l t s 3.4  W i t h The B a s i c  Model  ..  .44 ,  50  Initial E e s u l t s Using A Beverton-Holt S t o c k - E e c r u i t R e l a t i o n s h i p ......................  53  3.5 F i n a l E e s u l t s U s i n g A Beverton-Holt S t o c k - E e c r u i t E e l a t i o n s h i p ......................  61  3.6 P o p u l a t i o n 3.7  P r o j e c t i o n s With A Quota Of 180,000 .. 69  Discussion  77  4. NUMERICAL EXPERIMENTS WITH LEAST^SQUARES CATCH-AT-AGE ANALYSIS: A PINNIPED FISHERY .,  .80  4.1  F a c t o r s D e t e r m i n i n g The R e l i a b i l i t y Of The Estimates .......................................  4.2  The C a s e  4.3  Results  Study C o n t i n u e d And D i s c u s s i o n  84 .86  ..........................  5. NUMERICAL EXPERIMENTS WITH LEAST'SQUARES CATCH-AT-AGE ANALYSIS: A CLUPECID FISHERY  ,  88  105  5.1  F o u r C a s e s C f Low D a t a Q u a n t i t y .................106  5.2  Results  And D i s c u s s i o n  6. CONCLUSION 6.1  . . . . . . . . . . . . . . . . . . . . . . . . . . 115  ,  Summary  Literature Appendix  135 ............137  Cited A. REPARAMETEEIZATICN OF THE AGE STRUCTURE MODEL . .  ,,  139  BEVERTON-HOLT 142  vi  LIST OF TAELES  Table  1.  Data e r r o r c o e f f i c i e n t s c f v a r i a t i o n f o r northwestern A t l a n t i c harp s e a l c a t c h e s T a t - a g e . . 4 7  Table 2. Least-sguares catch-at-age a n a l y s i s parameter c o n s t r a i n t s f o r northwestern A t l a n t i c harp s e a l s ............................ 49 Table 3 .  R e s u l t s of l e a s t - s g u a r e s catch-at-age a n a l y s i s f o r northwestern A t l a n t i c harp s e a l s ........... 52  Table 4. Extreme values cf northwestern A t l a n t i c harp s e a l Eeverton-Holt s t c c k - r e c r u i t f u n c t i o n parameters a and p are determined from extreme values cf N o , A, and Sn ............... 56 Table 5. Parameter c o n s t r a i n t s of l e a s t - s q u a r e s catch-at-age a n a l y s i s with a Beverton-Holt s t c c k - r e c r u i t f u n c t i o n f o r northwestern A t l a n t i c harp s e a l s ............................ 58 Table 6. The age d i s t r i b u t i o n f o r northwestern A t l a n t i c harp s e a l s i n 1952, from i n i t i a l r e s u l t s of l e a s t - s g u a r e s catch-at-age a n a l y s i s with a Beverton-Hclt s t c c k - r e c r u i t f u n c t i o n , and the r e v i s e d d i s t r i b u t i o n ....................... 63 Table 7. Northwestern A t l a n t i c harp s e a l pup p r o d u c t i o n and annual pup e x p l o i t a t i o n r a t e s from the f i n a l l e a s t - s q u a r e s r e s u l t s are compared to r e s u l t s ficm s e q u e n t i a l p o p u l a t i o n a n a l y s i s ....68 Table 8 .  Northwestern A t l a n t i c harp s e a l age s e l e c t i o n f a c t o r s frcm the f i n a l l e a s t - s g u a r e s r e s u l t s ...71  Table  9, F o r e c a s t c f n o r t h w e s t e r n A t l a n t i c h a r p s e a l a b u n d a n c e f o r 1979 f r o m t h e f i n a l l e a s t s q u a r e s r e s u l t s ............................... 73  Table  10. Monte C a r l o r e s u l t s  f o r Case A  90  Table  11. Monte C a r l o r e s u l t s f o r C a s e B  .,,...93  Table  12. Monte C a r l o r e s u l t s f o r C a s e C  Table  13. F o r e c a s t e r r o r s f o r C a s e s A-C  Table  14. D a t a e r r c r c o e f f i c i e n t s o f v a r i a t i o n f o r C a s e s D-G  Table  15. L e a s t - s q u a r e s c a t c h - a t - a g e  ...  96  .................99  108  parameter  c o n s t r a i n t s f o r C a s e s D-G  114  Table  16. Monte C a r l o r e s u l t s  f o r C a s e D ...............117  Table  17. Monte C a r l o r e s u l t s  f o r C a s e E .,  Table  18. M c n t e C a r l o r e s u l t s  f o r Case F  ..123  Table  19. Monte C a r l o  r e s u l t s f o r Case G  126  T a b l e 20. F o r e c a s t e r r o r s f o r C a s e s D-G  ,..,.,120  ,.,.,,...,......129  L I S T 0F FIGURES  Figure  1. A p r o j e c t i o n o f t h e sum o f s g u a r e s s u r f a c e i n n - s p a c e t c two d i m e n s i o n s .................. 26  Figure  2. A B e v e r t o n - H o l t s t o c k - r e c r u i t d i s t r i b u t i o n  Figure  3. The age d i s t r i b u t i o n f o r n o r t h w e s t e r n A t l a n t i c h a r p s e a l s i n 1952, f r o m i n i t i a l r e s u l t s of least-sguares catch-at-age a n a l y s i s with a Beverton-Holt s t o c k - r e c r u i t function, and t h e r e v i s e d d i s t r i b u t i o n .................. 60  Figure  4. E s t i m a t e d t r e n d s i n n o r t h w e s t e r n A t l a n t i c h a r p s e a l a b u n d a n c e ........................... 65  ....34  s F i g u r e : 5. P o p u l a t i o n p r o j e c t i o n s f o r n o r t h w e s t e r n A t l a n t i c h a r p s e a l s u n d e r a q u o t a o f 180,000 ..76 Figure  6. The d a t a s e t " i n f o r m a t i o n - s p a c e " .............. 83  Figure  7. " T r u e "  histories  o f a b u n d a n c e f o r C a s e s D-G ..112  ix  ACKNOWLEDGEMENTS  Foremost, his  I w i s h t o t h a n k my s u p e r v i s o r , C . J . W a l t e r s , f o r  s u p p o r t and e n c o u r a g e m e n t d u r i n g t h e c o u r s e o f t h i s s t u d y . I  wculd a l s o their  like  helpful  providing  t c thank  N.J. W i l i m c v s k y  comments  on a n e a r l i e r  draft,  e s s e n t i a l i n f o r m a t i o n . Programming  P a t t e r s o n and B i l l assistance  and  P.A. L a r k i n f o r and P.F. L e t t f c r  advice  from  Mike  Webb was much a p p r e c i a t e d , a s was s e c r e t a r i a l  from Joan  Anderson  and Sandy M a s a i , . 1 am g r a t e f u l t o  Monica G u t i e r r e z f o r drawing t h e f i g u r e s .  A m a s t e r s d e g r e e t a k e s an a v e r a g e o f Institute: students  an are  experience severely  acknowledgements. thank and  In  sufficiently  tempted  to  three  years  at the  arduous  that  graduate  sentimentalize  t h i s r e s p e c t I would  in  their  particularly l i k e to  my o f f i c e - m a t e s : Max L e d b e t t e r , G r e g S t e e r ,  Larry  Smith,  C h r i s Wood; " t h e S c u l p i n s " : E r i c W o o d s w o r t h , J i m J o n e s , Ken  L e r t z m a n , and A r t h u r P o u c h e t ; and my h o u s e - m a t e s : Jchnson,  John  Lyon,  c r u c i a l moral support.  James  Currie-  K a r l N e u e n f e l d t , and M i k e Moore f o r t h e i r  1  1..INTRODUCTION  Chapter  The age c o m p o s i t i o n valuable  object  of  structure  of the  catch  abundance  of  mortality natural  rates to  study  by f i s h  contains  stock,  and  which  mortality,  population the  the  of harvests  h a s l o n g been c o n s i d e r e d population  a n a l y s t s . The age  information the  i t has  about  exploitation  been  the  and  subject.  age s t r u c t u r e o f t h e s t o c k ; t o g e t h e r  Recruitment,  affecting they  past  natural  and f i s h i n g a r e t h e p r o c e s s e s t h a t  numbers. Each p l a y s a r o l e i n  a  regulate  changes  have t h e i r  in  visible  outcome i n t h e age s t r u c t u r e o f t h e c a t c h . The  extent  to  which  depends, e s s e n t i a l l y , from  which  we  we  can  accurately  cn t h e i n f o r m a t i o n  assess  content  of  s t a r t . For a s e t of catches-at-age, quantity  a  stock  the  data  information  content  i s determined  p r i m a r i l y by t h e  of  degree  o f e r r o r with  w h i c h i t was m e a s u r e d , a n d t h e " c o n t r a s t " ,  the  range i n e x p l o i t a t i o n r a t e s and abundance d u r i n g  it  was  collected.  estimation  technique  results really efficiency  But  with  which  the  the  period  the r e s u l t s are not independent of the  a p p l i e d t o the data.  depends  data,  cn  both  the  The r e l i a b i l i t y  information  content  of the  and  the  analysis extracts the information.  S e v e r a l methods d e v e l o p e d t o a n a l y z e  catch-at-age  data  have  seen  numerous a p p l i c a t i o n s . However, g e n e r a l  guantitative  principles  relating  of population  parameters  the  reliability  cf estimates  2  (measured by  t h e i r b i a s and  variance)  t o a t t r i b u t e s of the  data  h a v e t e e n s l e w t c emerge. Dntil rates  and  recently,  abundance of f i s h  classes.  The  first,  fundamental  tc  under c o n s i d e r a b l e f o c u s e d on index  the  of  age  can  be  that  by  local  the  standardization  the  reguired,  "terminal"  ( o l d e s t age  the  last  a t t e n t i o n , termed the  history  by  by  of  method,  there  with c r i t i c i s m s as  changes  an  in  the  uneven m i x i n g o f  the  gear  Also, there  is  effort  saturation,  difficulties  involve  and the  fisheries.  usually  termed  a n a l y s i s , was I t uses the  seguential  developed p a r t l y  time-series  of  to  catches  i t s p a s t a b u n d a n c e . Though e f f o r t  must be  class)  come  i n d e p e n d e n t e s t i m a t e s of  e x p l o i t a t i o n r a t e and  the  rate  the of  mortality.  Over  1976).  1977)  of s u b - s t o c k s .  further  to reconstruct  data i s not  natural  stock,  a  that the e x p l o i t a t i o n  confounded  cf e f f o r t i n multi-gear type  as  is  T h i s a s s u m p t i o n has  to gear c o m p e t i t i o n , Still  major  has  catch-per-unit-effort  depletion  cf e f f o r t data.  of a y e a r - c l a s s  egually,  that  a n a l y s i s or c o h o r t use  e f f o r t data,  Eothschild  be  o f two  r e l a t i o n b e t w e e n e x p l o i t a t i o n r a t e and  due  second  population avoid  can  cooperation.  The  (e.g.  d i s t r i b u t i o n of the  nonlinear  search  attack  exploitation  catch-per-unit-effort  or,  observation  g r o u p s , and  evidence  that  and  tc fishing effort.  abundance  geographical  b e l o n g e d t o one  which uses c a t c h  abundance  i s proportional  used f o r e s t i m a t i n g  stocks  assumption  proportional rate  methods  This  few  a  least-squares  technigue  independent  years  cf  also  third  approach  a n a l y s e s the  effort  class  data,  but  has  received  (after  Doubleday  stock's does  catch~at-age not  reguire  3  estimates an  of t e r m i n a l e x p l o i t a t i o n r a t e s , nor, i n c e r t a i n  estimate  a general  cases,  of n a t u r a l m o r t a l i t y . In the l e a s t - s q u a r e s approach  model i s hypothesized  as to how observed  catches  have  a r i s e n as f u n c t i o n s c f h i s t c r i c a l r e c r u i t m e n t , e x p l o i t a t i o n , and natural  mortality  estimated  i n such a way t h a t as much i n f o r m a t i o n as p o s s i b l e i s  extracted  from a given data s e t .  Taking methods  parameters:  these  as i t s point c f departure  are  becoming inadequate  attempts  to  explore  then  that catch-per-unit-effort  of  fishing  fleets,  this  seme of the theory o f catch-at-rage  a n a l y s i s . In p a r t i c u l a r , my o b j e c t i v e s a r e , and suggest  are  with the t r e n d towards g r e a t e r  power, m o b i l i t y , and d i v e r s i f i c a t i o n paper  parameters  first,  to  examine  improvements to t h e l e a s t - s g u a r e s technique;  second,  t o use the l e a s t - s g u a r e s e s t i m a t o r as a v e h i c l e by which g e n e r a l quantitative and  r e l a t i o n s h i p s between the r e l i a b i l t y  of assessment  p r o p e r t i e s of the data can be formulated. In chapter  2 a comparison i s made between the l e a s t - s g u a r e s  approach and cohort a n a l y s i s together c a t c h curve, t c differences  point  among  cut  these  the  methods  with t h e i r f o r e r u n n e r , the  theoretical  similarities  and  of a n a l y z i n g age composition  data. Some m o d i f i c a t i o n s to the l e a s t - s q u a r e s approach are  then  chapter 3 an a p p l i c a t i o n i s given with  data  developed,  and  f o r northwestern hias  in  A t l a n t i c harp s e a l s . Chapter  4  considers  the  and variance of l e a s t - s g u a r e s estimates and p r e d i c t i o n s of  abundance, and how they are a f f e c t e d by data  quantity,  errors,  and  c o n t r a s t , with a s e r i e s of Monte C a r l o s t u d i e s using a model  of  the  northwest  Atlantic  harp  seal  population.  repeats t h e e r r o r a n a l y s i s , but with a model o f  a  Chapter 5  hypothetical  4  c l u p e c i d s t c c k . Chapter 6 presents  a summary and  conclusions.  5  C h a p t e r 2. CATCH-AT-AGE ANALYSIS  This  chapter  of catch-at-age  i s concerned with the t h e o r e t i c a l  analyses  i n g e n e r a l , and w i t h t h e  foundations  least-squares  approach i n particular,- A f t e r a comparison with other analyzing focussed  age c o m p o s i t i o n on  formulation 2.2),  the will  least-squares be  then modified  impair  (section 2 . 1 ) ,  data  presented  tc  deal  i t s practicality  i n a historical  the  history  abundance i s d i s c u s s e d  certain  as a g e n e r a l  The u s e o f l e a s t - s q u a r e s e s t i m a t e s reconstructing  of  c f population the  stock  comments  cn e v a l u a t i n g t h e r e l i a b i l i t y  forecast,  and a s s e s s i n g t h e r e l a t i v e  environmental  variation  t h e s i s i s concerned  with  in  detailed  light  (section that  (section 2.3). parameters  in  and i n f o r e c a s t i n g together  with  o f the estimates  importance  of  some  and t h e  stochastic  r e c r u i t m e n t . . The r e m a i n d e r o f t h e  the  t o a study  be  Its  shortcomings  technique  (section 2 . 4 ) ,  next  attention w i l l  estimator..  with  methods o f  application  developed  below  of a northwestern  population  and a f i c t i t i o u s c l u p e o i d  fishery.  of  the Atlantic  techniques pinniped  6  2.1  Catch  Curves,  Cohort  Consider  A n a l y s i s , and  three techniques  the Least Squares Approach..  to  evaluate  catch-at-age  a n a l y s i s o f c a t c h c u r v e s , c o h o r t a n a l y s i s , and a p p r o a c h . . How best  be  algebraically,  by  examining  comparing  the  (3)  fishing  e r r o r s are  recognized  assumed t o t e n e g l i g i b l e The the  use  early  cf  1900s  (summarized  in  (1)  mortality,  . F l u c t u a t i o n s i n a b u n d a n c e due and  or  (4)  ignored  Cohort  on  and  population  ( 1 9 4 9 ) . The 1968),  method has  Gulland  u s e d form described Dcutleday  virtual  since  been  ( 1 9 6 5 ) , Murphy  by Pope by  the  (1972).  Agger e t a l  ( 1 9 7 6 ) , and  The  developed  ( 1 9 6 5 ) , and  the  (1971)  Halters  right-hand  slope cf a l i n e  (MS  and  The  or  fitted  model i m p l i c i t  not  analysis.  the  by  basic  in  theme  also  termed  Bicker  (1948)  analysis by  are  models.  of  Jones  Fry  (1961,  g i v e n i t s most  widely  was  e x p l o r e d by Pope  first (1974),  1S76). survival  t o the  is  estimated  p o i n t s on t h e  descending  l i m b of the c a t c h c u r v e , a p l o t  a g a i n s t age.  natural  emmigration  l e a s t - s g u a r e s approach  In c a t c h curve a n a l y s i s , t o t a l from  the  analysis,  suggested  by  (2)  whether in  s e g u e n t i a l p o p u l a t i o n a n a l y s i s , was anticipated  which each i s  a long history, beginning  variations  1S75).  can  models  are ignored i n each of the  many  Eicker  upon  recruitment, and  another  underlying  t o i m m i g r a t i o n and  c a t c h c u r v e s has w^.th  the  assumptions  b a s e d . These a s s u m p t i o n s c o n c e r n  sampling  least-sguares  t h e methods s t a n d i n r e l a t i o n t o one  determined  mortality,  the  data:  (S)  of  log-catch-at-age  i n t h e method i s g i v e n  by:  7  where  C  is  the  ID i s t h e f i s h i n g  catch;  abundance i n number o f a n i m a l s ; age at  ( j = C,1>...,J  is  the  rate,  o f t h e number o f age j f i s h  where t h e number o f age j f i s h  age t , t h a t h a v e s u b s e q u e n t l y term  meanings.  "recruitment"  Here i t i s r e s e r v e d  age  cf f i r s t  the  number  capture of  vulnerability E =N  t,  0  (N  (N  t)  t)  to  i s t h e abundance and  cohort  a r e t h e number o f a  0  cohort  s u r v i v e d f r o m age t , t o age j .  has  come  to  have  a v a r i e t y of  f o r t h e number o f a n i m a l s  o r E) ; i t s h o u l d at  and an e x p l o i t a t i o n  the  at  n o t be c o n f u s e d  youngest  age  ) , n o r w i t h t h e number a t age 0  0  the  to  animals  when t = 0. Age t  of  the with  complete  (N ) , a l t h o u g h 0  i s t h e youngest a t which t h e p r o p o r t i o n  available to fishing  i s greater than  0.0  and age  i s t h e y o u n g e s t a t w h i c h t h e p r o p o r t i o n i s 1.0 . In  n a t u r a l l o g a r i t h m f o r m , E g . (2-1) becomes:  When l n C i s r e g r e s s e d ln  otherwise); N  referring  s u r v i v a l r a t e . I n o t h e r w o r d s , t h e c a t c h a t age j  product  The  of  unless noted  subscript  t h e y o u n g e s t age f u l l y e x p l o i t e d by t h e f i s h e r y , age t , ;  S i s the t o t a l  at  j i s a  mortality rate; N i s  S.  Equation  recruitment  (2^1)  of  and m o r t a l i t y . The a s s u m p t i o n s c a n be s u m m a r i z e d  as  (1)  mortality  i s constant  i s  recruitment  over  ever  the  i s  simplest  constant  possible  of  model  follows:  i s constant  a g a i n s t a g e , t h e s l o p e i s an e s t i m a t e  over  ages and y e a r s ;  years;  (2) n a t u r a l  (3) f i s h i n g m o r t a l i t y  a g e s and y e a r s ; and (4) e r r o r s a r e  recognized:  8  t h e y must have z e r o mean f o r t h e e s t i m a t e o f l n S t o he u n b i a s e d and  they  must be u n c o r r e l a t e d and h c m o s c e d a s t i c  f o r the estimate  t o be minimum v a r i a n c e . Total mortality  mortality  (S)  contains  both  natural  mortality  e s t i m a t e c f S, a n d an virtual  rate,  cohort  i s used  specific  fishing  fishery  i n which  negligible  made  about  t h e n m c a n be s e p a r a t e d f r o m t h e  estimate  of  N  t>  can  be  computed  class c f techniques, the catch history to  Sn  reconstruct  mortality rates.  numbers-at-age  (the  For  simplicity,  compared t o t h e r e s t o f t h e y e a r  i s the  Cohort  a n a l y s i s i s based  m, and C. The f i r s t Where  a  consider  1  a  season i s fishery,  on E g . ( 2 - 3 ) :  n a t u r a l s u r v i v a l r a t e over t h e p e r i o d between  h a r v e s t s and i i s a s u b s c r i p t r e f e r r i n g otherwise  (a Type  of  and a g e - y e a r -  natural mortality during the f i s h i n g  a f t e r B i c k e r 1S75).  age,.  been  mortality).  I n t h e second  unless  has  p o p u l a t i o n e s t i m a t e a t age t , , c o r r e c t e d f o r l o s s e s due  tc natural  Here  and n a t u r a l  c o m p o n e n t s ; t h e y c a n o n l y be s e p a r a t e d b y a s s u m i n g one  o r t h e o t h e r i s known. Once an a s s u m p t i o n the  fishing  noted.  to  i =  1,2,...,I  ( H e r e i n , we h a v e two s u b s c r i p t s f o r N,  s t a n d s f o r y e a r and t h e  subscripts  year:  second  stands  for  i n v o l v e a r i t h m e t i c o p e r a t i o n s , they a r e  s e p a r a t e d by a comma.) The u o d e l s i m p l y s a y s t h a t t h e number  in  a c o h o r t n e x t y e a r i s e q u a l t o t h e number p r e s e n t t h i s y e a r t h a t s u r v i v e e x p l o i t a t i o n and n a t u r a l m o r t a l i t y . R e a r r a n g i n g  Eg.(2-3)  9  gives  an  iterative  formula  f o r n u m b e r s - a t - a g e and  exploitation  rate:  from  which the  p a s t h i s t o r y o f t h e c o h o r t can be c o m p u t e d ,  an e s t i m a t e  o f Sn and  From  "terminal  this  of m f o r the e l d e s t exploitation  c a t c h d a t u m , an  estimate  obtained.  becomes  This  cf  the  N  age  in  r a t e " and  the  Eg. ( 2 - 4 ) ,  c f the  a b u n d a n c e and  last  age  i n the  s e r i e s a r e c a l c u l a t e d , and  age  of  first  capture. In the  for  exploitation cider  from  The  which  e x p l o i t a t i o n r a t e f o r the  iterative  so o n ,  is an  next-to-  back  to  the  scheme, n u m b e r s - a t - a g e  r a t e s f o r y o u n g e r a g e s depend on  ages.  abundance  ' '  yvl  estimate  and  series.  corresponding  corresponding in  HI,  the  given  the  estimates are:  (1)  recruitment i s year-specific;  (2) n a t u r a l m o r t a l i t y i s known  and  constant over  (3) f i s h i n g  ages and  a s s u m p t i o n s e m b o d i e d i n Eg. (2-4)  years;  year-specific;  and  (However,  Pope  1972,  variances  of the e s t i m a t e s g i v e n the s a m p l i n g  Compared  w i t h t h e use  assumptions it  (4)  (1) and  the  mortalities  sampling  A p p e n d i x C,  errors  offers  sguares  approach,  constrains fishing  are  age-  ignored.  formulas  for  the  error variances.)  cf catch curves, cohort  analysis  relaxes  ( 3 ) , but i s g u a l i t a t i v e l y  different  i n that  i g n o r e s the data e r r o r s i n the e s t i m a t i o n In c o n t r a s t , the  are  third  makes  class full  mortality  to  cf  use a  procedure.  technigues,  the  least-  of the e r r o r s t r u c t u r e , greater  extent  than  but does  10  cohcrt  analysis.  For  a Type 1 f i s h e r y t h e u n d e r l y i n g  model i s  usually written as:  i.e.,  the e x p l o i t a t i o n r a t e  specific  term,  y,  and  i s an  now age  a s s u m p t i o n s s u m m a r i z e d a r e : (1) (2) n a t u r a l other  the  product  selection  recruitment  of  factor, i s  but i s not n e c e s s a r i l y  yeara . . The  year-specific;  m o r t a l i t y i s c o n s t a n t o v e r ages a n d y e a r s  technigues),  a  (as i n t h e  known b e f o r e h a n d ;  f i s h i n g m o r t a l i t y i s e x p a n d e d i n t o an a g e - s p e c i f i c t e r m  constant  o v e r y e a r s and a y e a r - s p e c i f i c t e r m c o n s t a n t o v e r a g e s ; and the This  the  error  structure  last  property  i l l  the  general  E's, past  i s discussed  i n detail i n section  and  ages,.  parameters than there similarity)  However,  are data  ( t = 0, so t h a t 0  E  this  points  to  a  in some  many  approach c o l l a p s e  s i n g l e Sn, and t h e l a t t e r c o l l a p s e s  more  "collapse  ) i s constant over time.  c o h o r t a n a l y s i s and t h e l e a s t - s q u a r e s mortality  .) The  J0  t h e m's and Sn's t o g e t h e r i n t o S,  ( o r , more c o r r e c t l y ,  m  t o vary across a l l  results (C's):  E, = N  of  o f t h e p a r a m e t e r s i s n e c e s s a r y . The  curve technigue collapses assumes  2.2 .  o f E r e f e r s t o y e a r and t h e s u b s c r i p t s  m's, and S n ' s a r e , i n p r i n c i p l e , a l l o w e d  (assumed  procedure.  formulation:  Sn r e f e r t c b o t h age and y e a r .  years  (*l)  above m o d e l s c a n be e x p r e s s e d a s s p e c i a l c a s e s o f  Here t h e s u b s c r i p t and  forms t h e b a s i s o f t h e e s t i m a t i o n  (3)  1 1  catch and Both  natural  t h e m's t o  11  the  a's a n d y ' s . O t h e r b l o c k i n g schemes  tractable data,  solutions..In Walters  f o r instance,  Sn  is  y o u n g e r a g e s , and t h e a n n u a l years  lead  cohort  the  a n a l y s i s , at  models,  appears  allowed  t o vary  components c f e x p l o i t a t i o n  in  concern ncrtality  structure.  were  this  this  the  over  the  of  between t h e  of  the  the case, the least-sguares  apparent  statistical  Cohort for  properties  information  a g e s a r e made i s c o n t a i n e d within  the cohort;  described as s e q u e n t i a l : estimates o f c i d e r ages w i t h i n  upon  which t h e e s t i m a t e s  only  in  the  that i s , cohorts  data  cn t h o s e which  cr comparative:  are analyzed  f o r every  estimates  contained data  from  speak,  of  age  and  year;  the  of  the  same  f o r younger ages  independently. described  f o r any a g e a n d y e a r  as  depend  information  a b u n d a n c e f o r any age and y e a r  i n a l l of t h e c a t c h d a t a , though  method e n a b l e s to  other  estimates  the  f o r older  On t h e o t h e r h a n d , l e a s t - s g u a r e s a n a l y s i s i s b e s t integral,  rates  new d i m e n s i o n t o t h e e s t i m a t i o n .  a n a l y s i s i s best  the  are  l o s s i n p r e c i s i o n of estimating the  y o u n g e r a g e s depend on t h o s e  cohort;  error  approach  e x p l o i t a t i o n rates. I n f a c t , f a c t o r i z i n g the e x p l o i t a t i o n lends a completely  versus  underlying  trade-off  and t h e i n c l u s i o n  w o u l d be p r e f e r r e d i n a s m u c h a s i t s valued  f o r 24  years.  discussion  only  c o n s t r a i n t cn f i s h i n g If  b e t w e e n o l d e r and  merit of the l e a s t - s g u a r e s approach least  to  statistically  (MS 1976) a n a l y s i s o f h a r p s e a l  1  are collapsed t o 7 blocks of s i m i l a r Initially,  to  upon  a r e made i s  primarily  in  those  same a g e , y e a r , and c o h o r t . I n t h i s r e s p e c t t h e  e a c h datum i n a c o h o r t past  age  and  year  contribute t h i s information to the  to recall  its  memory,  so  e f f e c t s o f m o r t a l i t y and t o estimation  of  exploitation  12  rates of  and  abundance of other cohorts* For example, the 5 f i s h i n the l a s t  the number of age  depends  on  the estimate  i n the next to l a s t component  cf  estimate  year of the time  of the e x p l o i t a t i o n r a t e on age  year.  But  exploitation  in  information the  next  about to  the  last  4 fish annual  year i s not  contained  only i n the catch datum f o r age  just  the catch data f o r a l l ages t h a t year. I t i s contained  in  4 fish  series  i n the catch data f o r a l l ages that year and the youngest, the next, by "remember"  past  that year,  all  ages,  r e c r u i t m e n t , h a r v e s t i n g , and  as the r e s u l t of an "experimental  natural mortality catch  d e s i g n " i n v o l v i n g a complex of  effects,.  Though  all  data  e s t i m a t i o n of every appropriate  age,  c o n t a i n some i n f o r m a t i o n r e l e v a n t to the  unknown, the g r e a t e r p a r t i s i n data of year,  the e s t i m a t i o n of an age  it  each  e f f e c t s ; the a n a l y s i s depends on r e c o g n i z i n g c o n t r a s t s  among these  catch  except  v i r t u e of t h i s property of c o h o r t s to  e f f e c t s . In s h o r t , the l e a s t - s g u a r e s approach views  treatment  nor  data  or c o h o r t . The  i n f o r m a t i o n r e l e v a n t to  selection factor i s  strongest  f o r t h a t age,. For an annual e x p l o i t a t i o n  i s i n the c a t c h data  for  that  the  year.  Herein,  I  in  the  parameter, refer  to  catches t h a t are h i g h l y i n f o r m a t i v e of a p a r t i c u l a r parameter as that in  parameter's " s u p p o r t i n g " data. The  l a t e r chapters  variance  of  supporting  due  certain  t c the  strong  parameter  concept  w i l l be  relationship  estimates  and  between  the  the g u a n t i t y of  data.  Catches u l t i m a t e l y a r i s e as the r e s u l t of the between  usefull  interactions  three kinds cf e f f e c t s : r e c r u i t m e n t , h a r v e s t r a t e s , and  n a t u r a l m o r t a l i t y . &t t h i s p o i n t i t should te  emphasized  that,  13  of t i e three methods of a n a l y z i n g age here,  composition  data  c n l y the l e a s t - s q u a r e s approach i s capable,  cf p r o v i d i n g estimates concerning  cne  estimator  tc  of  analysis,  the  three the  is,  a  which  in principle,  of a l l t h r e e . I f p r i o r i n f o r m a t i o n  separate  enhanced.. T h i s  in  assuaes  factors,  effects sense,  natural  of the  the  ability  the  remaining  situation  mortality  The presents  high  between  fundamental  Somehow, we  problem  must diagncse  numbers  and  low  h a r v e s t r a t e s , or any between.. The  extracting  will  not  harvest  rates  were  the  possible  indeterminacy  from give  to  among  a technigue  observed  able  to  high in  rates,  efficient in separate  catches-at-age from  the  poor c o n t r a s t  r e s o l v e the problem. High q u a n t i t y , problem  low is  t h a t c f c a t c h per u n i t e f f o r t a n a l y s e s , where the i n e x p l a i n i n g c a t c h per  abundance and  catchability.  curves  of  uncertain  harvest  on the amount of i n f o r m a t i o n to be had  indeterminacy  The  result  the data, the a b i l i t y t o rise  data.  combinations  g u a n t i t y data s e t s with l a r g e e r r o r s and  tc  abundance  numbers and  e r r o r , high c o n t r a s t data s e t s ought to do w e l l . The identical  cohort  remaining  and  the  r a t e s , or low  n a t u r a l m o r t a l i t y . Given  information  be  is  the t e r m i n a l  d i a g n o s i s becomes even more complex and  depends d i r e c t l y  the  two  i n e x p l a i n i n g any c a t c h  c f the i n f i n i t e  the three e f f e c t s t h a t  data. Lew  harvest  whether catches  c o n s i d e r i n g the three-way abundance, and  of  abundance.  indeterminacy a  exists  for  and  e x p l o i t a t i o n r a t e s are known, i n order to estimate harvest r a t e s and  discussed  following  unit  effort  r e l a t i o n s h i p s emerge. The  i n v c l v e s the most c o n s t r a i n i n g  set  of  is  between  a n a l y s i s of catch assumptions  and  14  utilizes  the  analysis,  while having  the  catch  amount  equation,  information approach  least  to  the  most r e l a x e d  ignores  within  takes  of i n f o r m a t i o n i n the data.  full  assumptions  account  Finally,  of  the  the  data  2.2  dangerous assumption of c o n s t a n t  Formulation  of the Least-Squares  Consider  a  matrix  correspond  to years  method  model  a  and  of  and  uses  of  making  the  every  c o l u m n s t o ages,.  In  catches-at-age. that  The  we  can  the  together  g i v e t h e minimum sum  observed  and the  r e s p e c t t o the first,  the  function. ,  construct  those  observed  a  datum, b a s e d  a particular  matrix  values of the  of  function,  and  The is  sum t o be  predicted  the  specifies  unknowns w h i c h  of  sguares  minimized  unknown p a r a m e t e r s . I n what f o l l o w s , I  model, then  on  value  of sguared d i f f e r e n c e s between  predicted catches-at-age. objective  rows  least-squares  least-squares estimation c r i t e r i a  we t a k e a s o u r e s t i m a t e s  termed  i n which  t h a t s i m u l a t e s , or " p r e d i c t s " ,  f o r w h i c h t h e r e i s an  unknown,  selection.  data,  a s e t o f unknown p o p u l a t i o n p a r a m e t e r s . G i v e n for  least-sguares  Estimator.  catch-at-age  i s hypothesized  each catch-at-age  age  restricts  errors  i n f o r m a t i o n between c o h o r t s , though a t t h e c o s t possibly  concerning  t h e e r r o r s t r u c t u r e and  cohorts.  Cohort  the is with  consider,  problem of m i n i m i z i n g the o b j e c t i v e  15  2,2,1  The let  and  the f i r s t  the f i r s t  Eg. (2-5) in  model, y e a r i n t h e d a t a s e r i e s be d e s i g n a t e d y e a r  age,  can  age  be  0  (i.e.,  t=  0 ) , then the c a t c h e g u a t i o n .  a  w r i t t e n as a f u n c t i o n  of  number-at-age-(j-i+1)  y e a r 1, c r c f n u m b e r - a t - a g e - 0 , o r r e c r u i t m e n t , i n  depending  upon w h e t h e r i i s g r e a t e r o r l e s s t h a n j :  Equation  (2-7)  6),  age  term  (a) and an  c o n s t a n t o v e r a g e s and subscript  is  (m)  recruitment.  y e a r s ; and,  greater  Eg. (2-7)  so  hcmcscedastic. pointed objective  Pope that  (1974) the  Dcubleday  i s factored  into  for  those  (1976)  taking  variances and  Walters  logarithms would (MS  becomes:  the  instead  be  of more  1976)  o u t t h e l i n e a r i z i n g e f f e c t t h e l o g t r a n s f o r m has then  whose  subscript,  1 i s the parameter,  suggested  age  ( y ) ; Sn  catches  or equal to i t s year  error  f u n c t i o n . . E g . (2-7)  i - j ,  an  a n n u a l e x p l o i t a t i o n component  number p r e s e n t i n t h e c o h o r t i n y e a r of  year  t o t h e g e n e r a l f o r m u l a t i o n , Eg. ( 2 -  except: the e x p l o i t a t i o n rate  selection is  i s identical  1  both on  the  16  j  toy. +taa) + l n N  -  i0  Inyj - to a, - In N . l # i  jin^r, +  Z.\r\(\-y.^ a . J  * (i-oin 5n  i n  Z \n( l - ^ Q ^  I f n a t u r a l m o r t a l i t y i s assumed known and J+1 cf  ages, then there are 2I + 2J  , i «-j  there a r e I years  and  unknowns, i,.e., the f o l l o w i n g s e t  parameters: l n N„ , l n N  l n N, ,  1JL  ln N  , ln N  lo  7  ln N  l0  There  are  , ln y ,  ln  , l n a-j- •  only J age  is,  since  x  a, , l n a^  s e l e c t i o n terms because one  In a's must be f i x e d due  That  ,  I0  l n y , l n y^ {  cr  ,  to an indeterminacy  l n y's  the  results.  (i.e.,  the f i x e d Now  the  model.  added  to  s u b t r a c t e d from a l l the l n a's w i l l not  In  practice the  other  one  l n aj can  be  set  to  all  affect zero  terms then being s c a l e d r e l a t i v e to  term, consider  inaccuracies samples  and  aj = 1 ) ,  in  the l n y's and In a's add together to give the  l o g a r i t h m s cf the e x p l o i t a t i o n r a t e s , a constant the  of the l n y's  by  the  in  aging,  gear  type  data and or  errors,. e f f e c t s due  statistical  These  result  from  to the aggregation area.  The  of  simplest  i>]  17  assumption are  i s t h a t i n t i e l o g t r a n s f o r m model t h e  normally  This i s the  distributed  same  catches-at-age age  as  w i t h z e r o mean and  assuming  are  lognormal,  have c o e f f i c i e n t s  that and  constant in  variance.  observing  the  i m p l i e s t h a t the c a t c h e s - a t -  cf v a r i a t i o n  y e a r s . T h a t i s , i f we  errors  errors i n ln C  (cv- ) c o n s t a n t over  ages  and  have:  then i t f e l l o w s t h a t :  = Var ( C i j )''  cv^  -,(e -0 V  i.e.,  c v i s c o n s t a n t . C-  C\y .  (The  normally mean  e  a product  I E l Cijl  Vl  (z-'o)  i s the t r u e catch-at-age,  n o t a t i o n e~N(/*,V)  distributed  of  X  and  then  by  v a r i a n c e V;  i s i t s variance.)  cf a s e r i e s cf  factors),  means t h e random v a r i a b l e  w i t h mean y u and  Var(e)  independent  the  Central  E[ e ]  variable;  lcgncrmal. order  He  that  with the  use  thus,  the  l n C^; - \  E £ ]  =  random  r e s u l t t h a t the  Limit  range of 0.1  t c 0 , 5 ) . While sampling  error  i t  in in  may  be  errors, i t i s likely  a C  of  the are  the  normal will  Eg. ( 2 - 9 )  i n Eg. ( 2 - 8 ) t h e  t o 11.8%, c o r r e s p o n d i n g  0.5SJ  lcgncrmal  sampling  to  is  (error  Theorem t h e sum  predicted catches  t h e range of  e  is  variables  i n s t e a d o f l n €[•  C;J , b u t  as  I f the e r r o r s i n C  l o g a r i t h m s cf the e r r o r s converge i n d i s t r i b u t i o n random  observed  be in  i s ignored,  a r e b i a s e d upward ( i n to v a l u e s of cv i n the reasonable  unreasonable  to  assume  t o suppose  18  a constant  c o e f f i c i e n t of v a r i a t i o n . I f the younger ages are not  w e l l sampled and i f aging the  coefficient  intermediate data w i l l methods for  of  i s also inaccurate f o r older variation  will  ages. In the case where cv  he is  smallest not  te i n a p p r o p r i a t e l y weighted and o r d i n a r y  i n the  constant,  the  least-sguares  e s t i m a t o r . . A remedy  w i l l not provide the most e f f i c i e n t  this situation  animals,  i s d i s c u s s e d i n s e c t i o n 2.3.1 .  2,2,.2 The :ob j e c t i v e f u n c t i o n . Ignoring data  error  f o r the moment t h e dangerous assumption  regarding  heteroscedasticity,  objective  the  least-sguares  f u n c t i o n i s given as:  predicted  We take as the estimates values  f o r the unknown  In  1*  Cn  U-il)  parameters that s e t  of  which minimize ? .  If  Eg. (2-8)  c o u l d be obtained regression. nonlinear minimized The function,  were  linear  in  the unknowns, the e s t i m a t e s  a n a l y t i c a l l y , as a simple  Unfortunately optimization  or  multiple  linear  the n o n l i n e a r terms make the use of a  algorithm  essential,  so  that  3?  is  numerically.. problem  of  minimizing  the  least-sguares  when the model i s n o n l i n e a r i n the  objective  parameters,  is a  s p e c i a l case of a g e n e r a l c l a s s c f problems i n which a n o n l i n e a r function  is  t c be optimized  parameters. The study subject  matter  of  with r e s p e c t t o one or more of i t s  c f s o l u t i o n s t c such problems make up what  is  variously  termed  the  mathematical  19  programming, n o n l i n e a r programming, c r n o n l i n e a r c o m p r e h e n s i v e d i s c u s s i o n cn n o n l i n e a r nonlinear  parameter  following  estimation,  comments, i s g i v e n by  is  minimized  starting  with  iteration  the  an  initial  values  and  s m a l l e r . The  at  each i t e r a t i o n  are  ®i»  initial ®z/  very  6  equal  The  ( 6  via  t h a t the zero  v e c t o r of f i r s t  - 0^  K+  is  3i  the  s o l v i n g the  at  becomes  each  smaller  the parameter  values  iteration  best  be  ci .  Oar  series  parameter -c9  |©  From  K  produces the  such as  the  values |  K  being  parameter e s t i m a t e s . must  The  satisfy  the  p a r t i a l d e r i v a t i v e s o f §E  be  6;  ~ H  cf  and  K  c f second  6, K  eguation  iteration  Newton's  i s defined  ^  K  5 H~'  k  is  by:  23>  gradient  derivatives  around  procedure:  unknowns,  optimization algorithm  iZ-\2.) v e c t o r of  with i s the  partial  Newton's method i s d e r i v e d by of  the  vector.  e ,  (the m a t r i x  the  t r u e minimum of 5>  -I  at  iterative  with a vector of  (or t h e N e w t c n - B a p h s c n ) method. The  evaluated  k  algorithm  as o u r  simplest practical  partial  an  of  at the  ) , the  0  These we t a k e  where  o f much o f  (1974).  some c o n v e r g e n c e c r i t e r i a ,  t c the  to  i s c a l l e d the o p t i m i z a t i o n a l g o r i t h m . Let  which g i v e s the  condition  source  updated such t h a t  which culminates  small.  vector  the  set cf r u l e s f o r improving  guess  that satisfy  Bard  guess  v e c t o r cf parameter estimates the  p r o g r a m m i n g as a p p l i e d  and  numerically  optimization..A  respect  (the v e c t o r of to  the  i n v e r s e of the  derivatives) t a k i n g the  including obtained  3  terms  unknowns)  Hessian  evaluated  first  matrix  at  B  K  .  Taylor s e r i e s expansion up  t o second o r d e r ,  from s e t t i n g the  gradient  of  and the  20  expansion  egual  to the zero v e c t o r . Eguation  (2-13)  illustrates  two p r o p e r t i e s ccmmon t c t h e c l a s s c f a l g o r i t h m s c a l l e d  gradient  methods: each i t e r a t i o n i n v o l v e s t h e c o m p u t a t i o n o f t h e g r a d i e n t of  1> and a m a t r i x  e g u a l o r , i n some  fashion,  related  to  the  Hessian. 3?  If  is  guadratic  and H i s p o s i t i v e d e f i n i t e ,  method g i v e s t h e minimum i n a  single  iteration..  This  Newton's follows  b e c a u s e E g . (2-12) d e f i n e s t h e v e c t o r f o r w h i c h t h e g r a d i e n t o f a guadratic hut  i s the zero  v e c t o r . I f IE i s n o t g u a d r a t i c ,  H i s always p o s i t i v e d e f i n i t e ,  though not  eguation  net  in  a single iteration.  be p o s i t i v e d e f i n i t e c r may  ether  t h e method  gradient  methods  still  Unfortunately, H either  be d i f f i c u l t t o  differ  as  converges,  calculate.  t o how t h e y  overcome  d r a w b a c k s . G a u s s and v a r i a b l e m e t r i c methods c i r c u m v e n t f o r second p a r t i a l and  may The these  t h e need  d e r i v a t i v e s , a s do d i r e c t i o n a l d i s c r i m i n a t i o n  M a r g u a r d t ' s method, which a l s o  deal  with  the  problem  of  p o s i t i v e d e f i n i t e n e s s . I n t h e h a r p s e a l example b e l o w and i n t h e numerical Powell  experiments of chapters  method  was  used:  this  4 and 5, t h e  Davidon-Fletcher-  i s a variable metric  s i m i l a r t c Newtcn's method, b u t w i t h s o p h i s t i c a t e d approximating  the i n v e r s e of the Hessian  without  algorithm,  schemes  performing  for the  matrix i n v e r s i o n . A m o d i f i c a t i o n o f E g . (2-8) was u s e d by Pope he  uinimized:  (1974) i n w h i c h  21  5  - X  I  observed  ' i l  1  In ''+'.jt»  )  _  predicted  by a v a r i a n t of the s t e e p e s t descent catch-ratios  was  2  In  method.  (2-13)  The  use  w i t h i n a c o h o r t , even though they reduced be  estimated.  Nevertheless,  algorithm he used was Eg. (2-11)  than  in  because  mere s e n s i t i v e Eg, (2-13),  transform technigue.  however, model  that  Solutions  there  to  are  a  serious its  Weighting  estimates  much  more  as  model  and  3.  coefficients  of  sampling  sum  weights  the  of  sguares data  if  e r r o r v a r i a t i o n are not more  than  about  three  o f magnitude, r e g a r d l e s s of the d i s t r i b u t i o n assumed f o r  the e r r o r s . T h i s makes i n t u i t i v e sense. I f ages  general  Model..  improperly  constant or the c a t c h data v a r i e s over orders  a  and a l t e r n a t i v e e s t i m a t i o n c r i t e r i a . .  f o r the l e g transform the  powerful  these problems are presented next  An o b j e c t i v e f u n c t i o n c o n s i s t i n g of a simple  either  in  problems with the l e g  usefulness  Some M o d i f i c a t i o n s to the Log Transform  2.3.1  optimization  initial  examples of t h e i r use are given i n chapter  2.3  the  he used the l a t t e r to provide a  impair to  catch-ratios  the number of unknowns  s t a r i n g p o i n t f c r the fcrmer. Even with algorithm,  log-  r e j e c t e d by Doubleday on the grounds t h a t they  i n t r o d u c e d a u t c c c r r e l a t e d e r r o r s t o the s u c c e s s i v e  to  of  were  less  accurate  than  for  the  o t h e r s , we  data  for  some  would want our  22  parameter e s t i m a t e s more  reliable  t o t e l e s s i n f l u e n c e d by t h e s e  data.  I f the catches  comparison with the r e s t , the  information  transform  m a g n i t u d e . Somehow, t h e d a t a If the  t h e sampling  model  were  a  Var (C) Our  estimates  would be  v a r i a n c e s , Var(C)  -I  (Eard  distribution  i s only  knowledge  because t h e l o g  t o w i t h i n one o r  must be p r o p e r l y  neglect  two  orders  weighted.  inverse  of  the  data  1974, C h a p t e r 4 ) . . I n t h e c a s e o f  and  approximately  the  a  nonlinear  optimal,  but  model t h e use o f still  reasonable.  about the v a r i a t i o n i n a p a r t i c u l a r  catch-at-age  datum i s r e s t r i c t e d , h o w e v e r , by t h e a b s e n c e o f r e p l i c a t i o n s the .  observation.  Nonetheless,  with  r e g u l a r i t i e s i n the e r r o r s , estimates computed  from  t h e same d a t a  (e.g., Gulland  1955).  information  estimating  the  i s data  some  assumptions  o f Var(C) c a n  u s e d t o compute t h e  Unfortunately,  information i s invariably such  of  e r r o r s were n o r m a l a n d u n c o r r e l a t e d , and i f  -I  non-normal  the  l i n e a r i n t h e unknowns, t h e w e i g h t s g i v i n g t h e  minimum v a r i a n c e error  would s e r i o u s l y  i n the large catches  reduces a l l the data  by  f o r some a g e s were huge i n  the estimator  contained  than  this  unavailable, error  some  variances  about  usually  be  catches-at-age  extremely  missing from published  of  data  valuable s e t s . When  alternative  means o f  i s reguired.  If  the  c o e f f i c i e n t s o f v a r i a t i o n depend on a g e , we h a v e :  (1-/*)  Of  course,  approximation for  the  C  are  themselves  f o r V a r (C) c a n s t i l l  C, so t h a t :  unknown,  be o b t a i n e d  but  a  useful  by s u b s t i t u t i n g C  23  This expression holds  without  d i s t r i b u t i o n c f the errors. estimates cf  If  little  The  weighted  '  information  be  least-squares  coefficients  exists,  formulated  to  various  explore  the  J j  objective  function  uses the  , to give:  [ obser ved  ^  =  the  of the a n a l y s i s to the weights.  b a s i c m o d e l . E g . (2-7)  1|>  regarding  of the a g e - s p e c i f i c  prior,  a l t e r n a t i v e assumptions should sensitivity  assumptions  A l l t h a t i s required are independent  ( c r educated guesses)  variation.  any  —  CVJ  C,j j  predicted  observed  Cj-  J  (l-\b)  or  i  ]  depending are  I  observed  on w h e t h e r  C  M  'J  —  predicted  C-,; J  1  -i  Varies  o r n e t t h e V a r (C) c a l c u l a t e d  ( 2  from the  -  l 7 )  data  available.  2, 3,.2 U n r e a s o n a b l e c o n v e r g e n c e and c o n s t r a i n t s . Here we a r e c o n c e r n e d w i t h t h e p r o b l e m o f i n s u r i n g t h a t t h e values are,  the  unknowns  which m i n i m i z e the o b j e c t i v e  function  i n f a c t , r e a s o n a b l e e s t i m a t e s o f t h o s e unknowns. Let  (n  of  the  = 2I+2J).  surface of  number The  of  parameters  in  the . e s t i m a t i o n  be  n  least-squares procedure i m p l i e s searching the  v a l u e s i n n - s p a c e . To g i v e a f e e l i n g f o r  how  the  24  sum  cf  squares  from n  tc  two  "abundance" 1)  and  year two  the  surface dimensions  parameters  representing  (recruitment  average cf the  certain  degree  rates).  least-sguares  of i n d e t e r m i n a c y  and  e x p l o i t a t i o n rates, depicted  by  their  unable  to  determine  n u m b e r s - a t - a g e and and  high  surfaces  can  estimates 0  whether  of the the  's, etc. Define  of  will  parameter  reasonable  greater  upper  t c T.  parameter  there  is a  model b e t w e e n a b u n d a n c e sum  of sguares  contours  This says that i f the  were t h e  minimum.  are  clearly y's,  of  sum  of  high  sguares  unreasonable  as  extreme  or  's  confine  lower  T h i s can  the  The  values,  c o n s t r a i n t s represent light  constraints  ancunt of independent or p r i o r  the  o r more i s not  estimator's  r e g i o n i n n-space  be done  constraints  a l l  for  y e t , i f t h a t one  wander o u t s i d e t h e  estimates,. and  or b e l i e v e a b l e , values  l i e w i t h i n T;  be  numbers-at-age  the  local  will  product  cn  quite the  v a l u e s , c r e a t i n g a r e c t a n g u l a r r e g i o n T'  approximation about  First,  catches  somehow  and  illustrates  a r e l u c k y , i n t h e c a s e o f two  must  so t h a t i t does n o t  imposing  diagram  the e s t i m a t i o n  reasonable,  t h e g l o b a l minimum, we  by  ( t h e age  i s low,  unknowns  p a r a m e t e r s . I f we  containing  year  T as t h e r e g i o n i n n - s p a c e c o n s i s t i n g o f  l o c a l m i n i m a o n l y one  search  i n the  model, i . e . , n e g a t i v e  points that represent set  numbers-at-age i n  technigue,.  rates. . Secondly,  h a v e more t h a n one  for  the  low e x p l o i t a t i o n r a t e s o r l o w  exploitation  Some v a l u e s  N;  of the data  of  The  i n the  average  parameters  generally inverse relationship.  information content  shows i t c o l l a p s e d  the  and  "exploitation"  components o f the h a r v e s t p r o p e r t i e s of the  1a  behaves, F i g u r e  prior  that  easily possible is  an  information  correspond  to  knowledge c o n c e r n i n g  a the  25  Figure  1a. .A p r o j e c t i o n o f t h e sum o f s g u a r e s surface in n - s p a c e t o two d i m e n s i o n s , . The p o i n t s A and B r e p r e s e n t l o c a l m i n i m a .  Figure  1b. The u n r e a s o n a b l e minimum (A) i s from t h e c o n s t r a i n e d r e g i o n T'.  excluded  " EXPLOITATION "  27  parameters, loose c o n s t r a i n t s to l i t t l e the  sensible  dimensicnal  extremes.  analogue  unreasonable  In  would  ninimum  is  the look  excluded  minimum, I n t h e p a t h o l o g i c a l c a s e T* than  cne  local  o r no  knowledge  optimistic like  case  Figure  from  1',  might  1b,  except  the  two-  where  the  l e a v i n g only  still  contain  minimum o r m i g h t n o t c o n t a i n any  one more  ( p l a c i n g some  parameter e s t i m a t e s  cn t h e i r c o n s t r a i n t  2.3.3  model w i t h a s t o c k - r e c r u i t r e l a t i o n s h i p . .  Modifying the  boundaries).  I n a p o p u l a t i o n f o r which r e c r u i t m e n t abundance, modelled  r e l a t i o n s h i p c a n be  incorporated  (N- ) 0  as harp s e a l s , t h e  reproductive  strongly process  by a d e t e r m i n i s t i c s t o c k - r e c r u i t r e l a t i o n s h i p .  form of the be  such  depends  in  expression  into  Eg, (2-7) will  be  the  with  the  If  stock-recruit  parameters.  unknown  be the can  by r e p l a c i n g r e c r u i t m e n t function.  This  i n t e r m s o f some component o f t h e s t o c k  seme unknown s t c c k - r e c r u i t recruitments,  the  can  postulated, this information  estimator  on  Instead  parameters  r e l a t i o n s h i p are estimated. Beformulating  of  the  of  the  and  annual  stock-recruit  Eg. (2-7) , we  get:  28  N.  ( I - y. i-i  a.  )  Sn  )-\  • • • 1  N:10  and  t  i s t h e age o f f i r s t  cn  past  0  (cr  present)  capture.  Because r e c r u i t m e n t  a d u l t abundance, the o b j e c t i v e f u n c t i o n  must be e v a l u a t e d r e c u r s i v e l y , cne y e a r the  sum  cannct year  cf  sguares.  Unless  at a time,  as unknown  2,3,4  0  ; t h u s , these  accumulating  t = 0, t h e s t o c k - r e c r u i t f u n c t i o n e  provide a value f o r recruitment i n up t o t  depends  initial  year  1,  nor  in  any  r e c r u i t m e n t s must be t r e a t e d  parameters,  The  least-sguares  approach  for  estimating  natural  mortality. If  natural  "reasonable" parameter  m o r t a l i t y i s unknown, i n s t e a d o f g u e s s i n g  value, the p o s s i b i l i t y  in  at a  e x i s t s t o estimate i t as  a  t h e o p t i m i z a t i o n . The b a s i c c o n d i t i o n f o r t h i s t o  be p o s s i b l e i s t h a t t h e r e must be enough c o n t r a s t i n t h e y's f o r the n a t u r a l "visible". 4, and 5. .  mortality  effects  Results using t h i s  on  total  survival  to  become  procedure are given i n chapters  3,  29  2.4  I n t e r p r e t a t i o n c f the least-Squares  2.4,1  The s t a t e r e c o n s t r u c t i o n and f o r e c a s t i n g , The  optimization  Eg. ( 2 - 7 ) , of  stock  exploitation  by  produces  estimates  Eq. (2-8), o r Eq. ( 2 - 1 8 ) .  the  the  Estimates.  year  can  he  From t h e s e , t h e  reconstructed  subtracting fishing  numbers-at-age  year  f o l l o w s frcm  the in  of a stock-recruit  recruitments  stock  i s  t > 0.  appropriate  When  0  have been e s t i m a t e d ,  statement  of recruitment  the  in form  t o the breeding  the  annual  i t i s possible to process  prediction,  about f u t u r e  recruitment.  breeding  stock  distribution  sizes  then  uses  a l l breeding  paragraphs.  the  the  The  r e l a t i n g the recruitments  from t h e s t a t e r e c o n s t r u c t i o n t o f i t  parameters.  Arguments a s t o t h e form that  following  over  o f the d i s t r i b u t i o n  stock,  1+  value  i s constant  the  have  o r t h e 1+  of the recruitment  o f making a s p e c i f i c  discussed  presupposes  recruitment  assume  the  f o r d c i n g s c , t h a t i s b a s e d on t h e a s s u m p t i o n t h a t  sizes,  methcd  of  Q  o f the s t o c h a s t i c nature  variation  the  of recruitment  p o p u l a t i o n , i f t = 0, i f  0  c a n make a p r o b a b i l i s t i c  relative  the  1+  I+1-t ,  the f o r e c a s t . Instead  A methcd  i s obtained  relationship  a prediction  component  predicted  themselves  take account  and  breeding  year  series  year.  parameters  the  population:  we  and a g e - y e a r - s p e c i f i c  i n the data  been e s t i n a t e d i n t h e c p t i m i z a t i i o n ,  in  history  m o r t a l i t y and n a t u r a l m o r t a l i t y from  i n the l a s t  the  population  past  rates calculated. A forecast of the population f o r  following the l a s t  When  f o r t h e unknowns i n  recruitment  of  such  a  distribution  depends cn a s e r i e s  usually  of m u l t i p l i c a t i v e  30  s u r v i v a l f a c t o r s , and note t h a t , by the C e n t r a l the  sum  cf  the  logarithms  cf  these  d i s t r i b u t e d . T h i s i m p l i e s recruitment 1975, Eeterman  1978),  distribution  is  Empirical  given  by  Limit  factors  is  i s lognormal  evidence Allen  Theorem,  (e.g. Walters  f o r the  (1973).  normally  lognormal  The  lognormal  d i s t r i b u t i o n has two parameters, one f o r the mean of the random v a r i a t e and one r e l a t e d to i t s r e l a t i v e We  wish  tc  construct  each breeding  stock  relating  breeding  any  size.  variance..  a s e r i e s of d i s t r i b u t i o n s , one f o r We  therefore  require  stock s i z e to i t s mean is  usually  stock-recruit  we  interpret  r e c r u i t f u n c t i o n without  Usually  a  function  recruitment..For  t h i s purpose we must r e i n t e r p r e t what relationship.  normal  meant  by  a  the stock-  regard to the s t o c h a s t i c p r o p e r t i e s  of  the r e l a t i o n s h i p i n nature; here i t w i l l be modified t o mean the average  recruitment  Our lognormal its  as  a f u n c t i o n of the breeding  stock  "stcck-recruit d i s t r i b u t i o n " written i n  size.  terms  of  p r o b a b i l i t y d e n s i t y f u n c t i o n i s given by:  (a-n) where  £(B | C)  recruitment average  i s the lcgnormal  probability density function f o r  (E) given any breeding  recruitment  stock s i z e  (P) ; h (P)  i s the  at P; and LT i s a v a r i a n c e parameter. From X  Eq. (2-1S) i t f c l l c w s t h a t : E£ and  Var( B|P ) = h (P) 2 (e^ - 1)  The c o e f f i c i e n t s i z e i s (e  B|P ] = h (P)  of v a r i a t i o n o f recruitment given breeding  - 1)  . . A f t e r the parameters  of  stock  the s t o c k - r e c r u i t  31  function  (h)  have  been  f i t using  a p p r o p r i a t e component c f t h e s t o c k , a s least-squares catch-at-age  cr where  T T T -  A  e s t i m a t e s , an e s t i m a t e  o f unknown  confidence  o  >  the  is' given  (2-20)  p a r a m e t e r s o f h.  x  predict  of  from  o f d a t a p o i n t s u s e d t o f i t h , and x i s  0  From h , <y , and an e s t i m a t e can  reconstructed  and  K«\  I - t i s t h e number  t h e number  recruitments  I [in K - \n h(P*Vl  x lo  the  recruitment interval:  in  of the breeding  year  1+1  as  h (P) ± 2 h ( P ) (e  display the probability distribution  stock  (P) ,  we  an a p p r o x i m a t e 9 5 %  -1)  ;  and,  of r e c r u i t m e n t  second,  graphically,  u s i n g Eg. ( 2 - 1 9 ) , Note  that  i f  the  parameters  least-squares catch-at-age possible  and,  recruitment. indicate  that  obtained  without  when  h are estimated  o p t i m i z a t i o n , no e s t i m a t e  therefore,  However,  of  no  the  estimate  numerical  of  the  studies  a stock-recruit  is  unimportant.  curve:  in  chapter  4  parameters  Consider  the  of  h  parameters  be  relationship; i . e . ,  p r o b l e m c n l y e x i s t s when s t o c h a s t i c v a r i a t i o n  interpretation.  1  variance  x  including  the  Q - is  (j i s l a r g e , b e t t e r r e s u l t s c a n p r o b a b l y  the  Sometimes  in  of  i n the  i n recruitment  w i l l n o t h a v e an of  the  obvious  Beverton-Holt  32  _P  h(PV  C2-2I)  ocP + p Fcr  age  directly  s t r u c t u r e models, the apparent.  obtained  by  meaningful derived:  recruitment formulas 1S75,  equilibrium rate  at  to  a  and  p>.  the  recruit  distribution  The  ccvariance  possible  a more  population  i s not can  no  age  structure  and  A  example  of  parameters  are  (N«>)  the  and  The  case  depend on  a  Sn  (Eicker and  Eeverton-Holt  u s i n g Eg. (2-19) and  be  biologically  unfished equilibrium  m a t r i x and  J, in stock-  Eq, (2-21) i s shown i n  reliability  a hypothetical situation  t c go b a c k i n t i m e and  in  the o r i g i n a l .  least-sguares w o u l d be  catch-at-age  different.  itself  a  tc  went b a c k  v a r i a b l e and  which  i t  data  variable  s e t were the  then  resulting  characterized  of by  variance  of the  we  would  have  e v e n t u a l l y we  sampling  the  fishery  another could  a  in  and  good  the  estimates unknown  probability the  unknown  e r r o r s . Each t i m e  realization  obtain  different  an a  be  measuring  used  with the v a r i a n c e of the e s t i m a t e of the  would  series slightly  analysis,  i n time, resampling  estimates,  estimates..  I n other words, the e s t i m a t e  randem  distribution, related  I f t h e new  of the  resample the f i s h e r y ,  t h e same v a r i a b l e s b u t c r e a t i n g a d a t a  is  p  2. .  Consider  frcm  and stock  such p o p u l a t i o n  b u t now An  in  unfished  half  III),  addition  2.4.2  and  are analagous t o the  appendix  Figure  ex  I n a p p e n d i x a two  the  a  F u r t h e r i n f o r m a t i o n about the  interpreting  way.  s i g n i f i c a n c e of  computing o f our idea  we new  random of  its  33  Figure  2. A Eeverton-Hclt stock-recruit distribution. The distribution parameters are OL= 6.67 x 10"*, (* = 0.33, and or = 0.29 ( c o r r e s p o n d i n g t o N«> = 1000, A = 1.5, and a coefficient of v a r i a t i o n f o r r e c r u i t m e n t of 0.30, f o r a model w i t h o u t age s t r u c t u r e ) .  34  35  variance.  Of course, the s i t u a t i o n i s only h y p o t h e t i c a l , yet  need t c knew the variances cf our estimates i n order their  reliability..  variance and l i t t l e Bard  A  the  parameter  c o v a r i a n c e matrix and are themselves  assess  with a s m a l l  bias.  (1974,p.177,Eq. (7-5-13))  approximating  regarded  trustworthy estimate i s one  to  we  gives  covariance  a  formula  matrix  for  from the data  the g r a d i e n t c f $ . The r e s u l t s , he warns,  random v a r i a b l e s and,  unfortunately,  should  be  as no more than a rough e s t i m a t e , c o r r e c t to w i t h i n an  crder of magnitude. A  second  possibility  is  to  estimate  c o v a r i a n c e matrix using a Monte C a r l e approach,. to  simulate  the  resampling  computer, c r e a t i n g s e t s fishery,  mimicking  of  process  fake  observation  number generator. Each observed  data errors  C has  as  the The  described from with its  a  parameter idea here i s above  model a  mean  on of  a the  pseudo-random the  "true"  A  catch  (C) , and a r e l a t i v e v a r i a n c e , s p e c i f i e d beforehand,  of cv..  Each s e t of fake data i s the r e a l i z a t i o n of a d i f f e r e n t seguence of  random  numbers; each data s e t g i v e s r i s e t o a unique  set of  parameter e s t i m a t e s . A f t e r a number of r e p l i c a t i o n s of the generation  and  data  analysis  v a r i a n c e can be c a l c u l a t e d population  process,  data  estimates of b i a s and  f o r each estimated parameter and  the  f o r e c a s t . E x p l o r a t i o n s with the Monte C a r l o approach  w i l l be the s u b j e c t c f chapters 4 and  5.  36  2.4.3  The r e s i d u a l s a n d l a c k o f f i t . For  the  differences with  l o g transform  between  observed  model,  the  residuals  l n C and p r e d i c t e d  residuals  o b s e r v e d C and p r e d i c t e d the  residuals  are  exactly  are  the  l n C computed  differences  C.) I f t h e model were  the  leastbetween  exactly  correct  w o u l d be r e a l i z a t i o n s c f t h e e r r o r s i n o b s e r v i n g  C (or C ) . I f t h e r e  be  the  t h e s o l u t i o n t o t h e o p t i m i z a t i o n . . (For t h e w e i g h t e d  sguares  ln  model  zero.  were no s u c h e r r o r s , t h e  residuals  would  The i m p l i c a t i o n i s t h a t t o d e t e r m i n e how t h e  d i s t r i b u t i o n o f t h e r e s i d u a l s d e p a r t s from t h a t o f t h e e r r o r s i s to  d e t e r m i n e how w e l l t h e model s t r u c t u r e f i t s For  the l o g transform  model we p o s t u l a t e d  t e r m was n o r m a l l y d i s t r i b u t e d w i t h that  the  residuals  errors  the (S ) z  the  freguency  (e.g.,  with  model f i t s  be  by V a r ( C ) w i l l  of  normality,  a  made on n o r m a l p r o b a b i l i t y case  h a v e a r e s i d u a l mean s q u a r e  are  "good",  o f t h e r e s i d u a l mean s q u a r e g r e a t e r  than  1.0, *.  a basic  to  cchcrt  to  e r r o r s i n t h e model. T h i s  advantage  cf  least-sguares  the  s  ,  relationship  techniques  as  a n a l y s i s : knowledge o f t h e magnitude o f t h e knowledge o f  goodness  of  f i t , and  versa. P l o t s of the r e s i d u a l s against  any  a s c e r t a i n whether t h e  assumption be  V, a n d  1976). I n t h e weighted l e a s t - s g u a r e s  e r r o r s i n the data i m p l i e s vice  To  error  e x a c t l y . I n f a c t , i f t h e Var (C)  attributed  illustrates compared  the can  the  a v a l u e c l o s e t o 1.0 i f t h e V a r ( C ) a r e a c c u r a t e a n d i f  proportion must  plot  Dcubleday  residuals divided  with  that  z e r o mean a n d v a r i a n c e  independent.  are consistent  cumulative paper  were  the data.  non-independence. I f patterns  a g e , y e a r , and c o h o r t i n the r e s i d u a l s e x i s t ,  reveal or i f  37  t h e y a r e c f an u n a c c e p t a b l e ignore  outliers,  deficient.  Lack  apparently  of  selection  model  f i t can  erroneous  reformulating  data  natural  reduced  with  ones  mortality.  i s g i v e n by D r a p e r  fact,  incorporating Annual  considered  be  the a  more  effort  data  of  (f ) can (  be  be  by  to  somehow replacing  c o n s i s t e n t , o r by  assumptions  about  age-  classic  treatment  of  capable  of  (1966)..  Approach.  least-sguares  variety  to  either  The  and S m i t h  2.5 E x t e n s i o n s o f t h e L e a s t - S g u a r e s  In  i s  t h e model w i t h d i f f e r e n t  cr  residuals  the  magnitude, or i t i s unreasonable  data  approach other  included  i s than  by  catches-at-age.  substituting  the  expression:  Yi * intc  %  {  $i  (  1-12)  Eg. (2-7) , Eg. (2-8) , o r E g . (2-18) . H e r e t h e g ' s , t h e a n n u a l  catchability Assumptions  c o e f f i c i e n t s , are estimated  instead  of  the  y's.  a b o u t r e g u l a r i t i e s i n t h e g's l e a d t o a r e d u c t i o n i n  t h e number o f p a r a m e t e r s . a n a l y s i s i f independent Fertility  data  E f f o r t d a t a c a n o n l y be used  estimates of c a t c h a b i l i t y  can  i n cohort  are a v a i l a b l e .  a l s o be i n c l u d e d i n t h e l e a s t - s g u a r e s  e s t i m a t o r . For example, i f a g e - s p e c i f i c annual  fecundities  and  sex r a t i o s a r e monitored, t h e g e n e r a l e x p r e s s i o n f c r r e c r u i t m e n t in  E g , (2-18) becomes:  38  N  = h(a p,... P _ J  i0  i  where  bj  I n the  simplest case, N  in  i s the  (  recruitment  can  still  function The  i  fecundity-at-age i s egual  io  t e used t o f i t the  from the  c o r r e c t e d f o r t h e sex t c P,-^  Allen  u s i n g a mcdel  catches-at-age,  where  r,  and  be  the  for  given  the  composition  total  annual  catch,  whale  instead  (\-^-fj_ ) Sn (  +  rN,-.,  ,z*ifel  of  (2-2.4-)  r a t e , i s estimated  along  g. .  capable  cf  the  was  developed  approach serves  of o t h e r f i s h e r i e s extracting  matrix of  the  specific  fecundities  and  sex  for  statistics.  information  catches,  strictly  as a p a r a d i g m  f r o m enormous  matrix  v a r i a n c e s , the s e r i e s of e f f o r t d a t a , p l u s the  age-year  is  , i-i  amounts o f d a t a : t h e errcr  density  by:  catches-at-age,  is  Eg. (2-23)  (1977) d e s c r i b e s a method o f a n a l y z i n g  a n a l y s i s of a v a r i e t y It  Eg. ( 2 - 1 8 ) ,  a p p l i e d e v e n i f t h e age  Though t h e l e a s t - s q u a r e s t e c h n i g u e to analyze  variablity  of  r e p r e s e n t i n g the r e c r u i t m e n t  w i t h K, , Sn,  . I f strong  stock-recruit probability  N, H  ratio.  state reconstruction.  approach can  N  (z-23)  +  d o e s n o t a l l o w t h e use  unavailable. data  )  ratios.  of  catch  matrices It  not  of  only  39  p r o v i d e s a framework f o r the simultaneous age  structure  cf  the  catch,  (2) the  consistent  past,  each,  fishery s t a t i s t i c s  In  the  resulting these  i.e.,  represented  the estimated  ty  the  the (3)  assessment  is  t h r e e s p e c i e s of independently.  information  a much s m a l l e r s e t of  parameters  (1) and  have u s u a l l y been i n v e s t i g a t e d  The l e a s t - s g u a r e s technigue a l l o w s a l l collectively  of  effort statistics,  r e p r o d u c t i o n data, but guarantees with  analysis  to  be  statistics,  (and t h e i r c o v a r i a n c e matrix)  of  a l s o be used to estimate c a t c h a b i l i t y  and  our model eguations. The  method  can  n a t u r a l m o r t a l i t y frcm tag tagged  recovery  f i s h e d t h a t are r e l e a s e d a t one  data.  If  the  number  of  p o i n t i n time i s N , then 0  a model f o r subseguent r e c o v e r i e s (C) i s given  by:  (Z-25) where  the time u n i t s can be on the order of months, r a t h e r than  years. I t i s p o s s i b l e t c natural  2.6  mortality  use  Eg.(2-25)  to  estimate  separate  r a t e s f o r a s e r i e s of time i n t e r v a l s ,  estimate  catchability  coefficients  areas and  gear types through  for  which the f i s h  distinct  and/or  statistical  migrate.  Summary.  Aside of  frcm  g u e s t i c n s cf b i a s and e f f i c i e n c y , a comparison  c a t c h curves, cohort a n a l y s i s , and  can be made cn the b a s i s model  and  the  manner  of  the  the l e a s t - s g u a r e s approach  assumptions  underlying  each  i n which the i n f o r m a t i o n i n the data i s  40  utilized. utilize  Catch  curves  involve  the  strictest  restricts  information  w i t h i n c o h o r t s . The l e a s t - s g u a r e s a p p r o a c h e x t r a c t s t h e most  i n f o r m a t i o n , b u t assumes t h e age the f i s h e r y The at-age  are constant  ever  selection  characteristics  of  time.  f u n d a m e n t a l p r o b l e m t o be r e s o l v e d i n e x p l a i n i n g c a t c h data  exploitation, In  i s the three-way indeterminacy  least-sguares  (1976),  catch equation numerically.  and  tc  Walters  i s u s e d and  method (MS  the  the  sum  of  error  and  parameter  values  s p e c i a l cases function,  weights  when  stochastic  natural mortality  as  informative.  results  The  an  minimized  convergence modifications  inverse  of  recruitment  unknown,  are when  the  the  the  data  possible  bounds,. M o d i f i c a t i o n s f o r  effects  of  is  constraining  w i t h u p p e r and l o w e r  include modelling  (1974),  m o d e l . These i n c l u d e u s i n g t h e  with the  as  squares  t h a t some  weighted l e a s t - s g u a r e s c r i t e r i a variances  Pope  and u n r e a s o n a b l e  and i t i s s u g g e s t e d l o g transform  of  1976), the l o g a r i t h m of t h e  Problems o f weighting  h a v e been d i s c u s s e d made  between abundance,  and n a t u r a l m o r t a l i t y .  the  Boubleday  be  and  t h e l e a s t i n f o r m a t i o n . C o h o r t a n a l y s i s h a s t h e most l a x  a s s u m p t i o n s , b u t i g n o r e s d a t a e r r o r s and to  assumptions  with a s t o c k - r e c r u i t small, the  and  data  treating  i s  highly  a n a l y s i s allow a h i s t o r y of  s t o c k a b u n d a n c e t o be r e c o n s t r u c t e d , f r o m w h i c h a p r e d i c t i o n c a n be  made. When a n n u a l  recruitments are estimated  the o p t i m i z a t i o n , i t i s p o s s i b l e t o f i t a function technigue  relating i s suggested  recruitment  to  for fitting  w h i c h assumes t h e r e l a t i v e  probability  breeding a  as parameters i n  stock  lognormal  variance c f recruitment  density size. . a  distribution i s constant.  41  The  parameter  u n r e l i a b l e and Examination  of  can  covariance  matrix obtained  be estimated using  a  Mcnte  from the data i s Carlo  the r e s i d u a l s r e v e a l s l a c k of f i t and  to b e t t e r s t r u c t u r a l assumptions.  approach. may  point  42  Chapter  3. CASE STDDY: NOBTHWESTEBN ATLANTIC HABP SEALS.  The  seal data presented  a n a l y s i s o f harp  serves  two  study:  first,  chapter  purposes  2,  i t  and,  in  relation  illustrates  seccnd,  The  harp  approach  mammals, t h u s t h e i r  fcr  o b j e c t i v e s of  methodology  the r e s u l t s w i l l  fluctuations  developed  s p e c i e s . This suggests the harp  for  including  a  stock-recruit  reasons.  least-squares They a r e  marine  observed  over  II,  of  26  ages  t h a t , i n the p e r i o d over  which  then  1970s  the  set  is  ( L e t t and  and  27  years.  the data  were  s e t and  strong  rates)  imply  likely test  contrast that  h i g h . The case  for  the i n t r o d u c t i o n of guotas  Benjaminsen 1977).  harp  the  in  both  These two abundance  case  large,  intense harvesting pressure f o l l o w i n g  increased after  most  i n the e s t i m a t i o n  factors and  with It is  collected  (1952-1S78), t h e abundance of s e a l s d e c l i n e d d r a m a t i c a l l y result  such  s e a l s a r e a good t e s t  data  be  4..  relationship  scheme. What i s more i m p o r t a n t , t h e catches-at-age  in  as a r e c h a r a c t e r i s t i c o f  fish  the  p r o v i d e a model t o  of chapter  the f o l l o w i n g  chapter  p a t t e r n o f r e c r u i t m e n t s h o u l d n o t show  stochastic  suspected  this  s e a l s were c h o s e n t o i l l u s t r a t e t h e  primarily  violent  to o v e r a l l  the  used i n t h e n u m e r i c a l e x p e r i m e n t s  in  as  World  i n the (large  a War  early data  exploitation  i n f o r m a t i o n content of the data s e t i s  seals should,  estimating  therefore,  be  a  suitable  n a t u r a l mortality. L a s t l y , the  data  13  h a v e p r e v i o u s l y been population  analyzed  with  a  a n a l y s i s ( L e t t e t a l 1978,  variant  of  sequential  Mohn e t a l 1978)  and  so  a  b a s i s f o r c o m p a r i s o n o f t h e r e s u l t s i.s p r o v i d e d . Here, the approach  data  were a n a l y z e d  without  Estimates  of the  analysis  was  including  data  first a  under  ( s e c t i o n s 3.2  assumptions  an  guestion  to  i s p o s e d : How  Then, u s i n g  the  alternative  but  (sections  this 3.4  time and  assess  t h e e f f e c t on  quota  of  180,000  p r o j e c t i o n s over ten  with age),  including  3.5), the  a  so  the  about  the  3,3),. As e a c h weighting  set  "best" r e s i d u a l s  the a n a l y s i s i s performed  stock-recruit relationship  s e a l p o p u l a t i o n o f the  years  This  from t h e  the  weights?  These r e s u l t s are s u b s e q u e n t l y  animals,  of  scheme,  a n a l y s i s t o the  scheme t h a t g i v e s t h e  (that g i v i n g reduced trends again,  assumptions  and  s e n s i t i v e i s the  weighting  relationship.  were u n a v a i l a b l e ,  various  magnitude c f the e r r o r s leads  least-sguares  stcck-recruit  error variances  repeated  using the  i s done by  used  current  to  annual  making s t o c h a s t i c  p r e d i c t i o n f o r 1979  (section  3.6).  3.1  Data.  The  harp  hunters^  landsmen,  n a t i v e s . The hiking take the in  seal  or  fishery large  vessels,  landsmen are those snowmobiling  s e a l s by c l u b b i n g  n e t s . The  consists  of  four  small  hunters  who  categories  v e s s e l s , and work  from  of  Arctic shore,  over the i c e , or using s m a l l b o a t s , them, s h o c t i n g  l a r g e v e s s e l s are  them, o r d r o w n i n g  s h i p s g r e a t e r t h a n 65  feet  to  them with  44  crews  of  in  rafting  St,  the  25-30 men  i c e c f f the  The  small  by  hunters,  work from t h e  their  way  Arctic  and  derive  (1S78), 1976.  The  individual  Formulation  The function model for  the  mcdel and when  the  errors  model  are c r a f t  like  Finally,  natives.  and  a  history  discussed  small  boats  t o make  Weighted  Canadian  problem  that the  is  data  and  and  Least-Squares  to  represent used  (1977), L e t t  method  in Lett  feet  vessel  catch  the  35-65  large  fisheries;  and  moulting  there are the  Benjaminsen  of  the  serious  total  shore  Gulf  p u p p i n g and  (1978) f o r ages 0 t o 25  given  Eq. (2-*16) and  year, the  by  Eg.(2-7)  w e i g h t s from  recruitment r e f e r r e d to  age-specific  years  used  to  the here  et  al  1952  to  weight  Benjaminsen  (1977).  Estimator,  Natural  used  Eq. ( 2 - 1 5 ) .  i s estimated below as  the  coefficients  c o n s t r a i n t s , the  the o p t i m i z a t i o n a l g o r i t h m cv's  was  o b j e c t i v e f u n c t i o n , the  ( c v ) , the  The  Newfoundland o r the  who,  f o r the  f a r from  Unknown,  wherein  each  ship,  longliners,  individual  Mohn e t a l  of  the  seals in their  men  in Lett  samples are  of  ice floes.  of the  sampling  from  s h i p , sometimes u s i n g  freguencies  presented  and  Mortality  the  west G r e e n l a n d  effects  those  coast  groups cf  amcng t h e  age  combined are  small  out  v e s s e l s , or  operated  and  strike  Lawrence, to h a r v e s t  patches.  3.2  who  were i n i t i a l l y  are  as  with  the  objective  Eg. (2-7)  is  a separate  parameter  " b a s i c " model, G i v e n formulation of  variation  initial  parameter  the  is  the  complete  of the  data  estimates,  specified. .  derived  i n d i s c u s s i o n with  P.F.Lett  45  as  very c r u d e a p p r o x i m a t i o n s ,  with  the  data.  twice  t c c o r r e c t f o r trends  t h o u g h b a s e d cn L e t t ^ s  I n the course of the  weighting  of  three  the  same p a t t e r n : t h o s e f o r pup  largest  schemes  i n the  proportion  j u v e n i l e s are  of the  smaller  a n a l y s i s t h e y were  1).  (Table  The  catches  they  are  very  concerning tighter was  the  wide,  constraints  e a s i e s t t o age;  in  implying  the  A  o r no  case  converge  usually  to  a  no  unigue  starting  p o i n t s , so  solution  obtained  is  guarantee  is  independent  of  an  catch-at-age  the  T',  i f the  from  s o l u t i o n obtained  are  identical,  unigue.  In  we  algorithm  data  initial i t  is  study  estimate. to  constraints  starting  near  confident  a v a r i a n t of the  the  that  of the  Davidon-  a l l  parameters  (in  one  instance  more t h a n a hundred) a t a c o n s i s t e n t l y a c c e p t a b l e More  the  produced s o l u t i o n s s i m i l a r to at l e a s t  figures for  convergence.  will  possible  given the  have r e a s o n t o be  this  but i t  otherwise.  a n d - l o w - e x p l o i t a t i c n - r a t e s e x t r e m e and  Fletcher-Powell algorithm  of  analyses,  for  that  defining  numbering  made  demonstrate  1b:  six significant  be  tc  crucial  s t r u c t u r e shewn i n F i g u r e  is  information  different  the  solution  prior  whole,  arbitrarily  exploit  opposite  age  from  analyzing  its  the  achieved  that  Luckily, in  high-abundances-  On  b a s i s of previous  solution  i t  older  accuracy of  could  t h o u g h t more o b j e c t i v e r e s u l t s c c u l d be There i s  the  those f o r  those f o r  T a b l e 2.  little  values..  cn  exhibit  age.  presented  parameter  total  are s m a l l e s t , being  t h a n t h o s e f o r a d u l t s ; and  c o n s t r a i n t s are  a  In each, the cv's  s a m p l e and  decrease with  modified  r e s i d u a l s , producing  a g e s a r e l a r g e s t , as b o t h numbers o f a n i m a l s and determination  familiarity  precisely,  the  rate  low-abundance-high-  46  Table 1. Data errcr northwestern age.  coefficients cf variation for A t l a n t i c harp s e a l catches-at-  Age 0  1-2  3-7  8-20  21-25  Weighting  #1  .025  .05  .05  .15  .35  Weighting  #2  .10  .20  .15  .20  .35  Weighting  #3  .20  .25  .20  .25  .40  48  Table 2. L e a s t - s q u a r e s catch-at-age a n a l y s i s parameter constraints f o r northwestern A t l a n t i c harp seals.  A l l numbers-at-age i n year 1: [ 1, i o  6  ]  A l l annual components o f e x p l o i t a t i o n : [.001, 1.0 ] A l l age s e l e c t i o n f a c t o r s : [.001, 1.0 ] A l l annual recruitments: [ 1, 5 x 1 0 ] 6  S u r v i v a l r a t e through n a t u r a l [ 0.7, 1.0 ]  mortality  50  exploitation i n i t i a l constraints  cn  p o i n t s were t a k e n  the  N,: 's  and  c o n s t r a i n t s cn t h e a ' s , t h e were t a k e n  3.3  The  the  mean s g u a r e and  reduced  tried  was  are  three  to  in  still  be  weightings  the  too  residuals.  the e f f e c t further,  high  for  represent  Table  I t i s c l e a r t h a t the technigue weights,  gives  a  fishing mortality,  is  and the  range  unigue and  combination  abundance;  these  harp s e a l  data.  estimates  of  each  among  abundance p a r a m e t e r s . the  more  unexpected.  o f v a l u e s . Some are  shown  seal  data._  lower  sensitive  There  implies  a  will  Each  radically  always  the i n f o r m a t i o n  the  in  natural mortality,  natural mortality, exploitation The  of  i s not robust t o l a r g e  at l e a s t f o r the harp  i t i s net  covariance  data,  wide  third  although  e x p l a n a t i o n o f t h e c a t c h h i s t o r y . Whereas t h i s  unfortunate  strcng  relatively  A  of i n c r e a s i n g  rough  from the r e s u l t s of each o p t i m i z a t i o n  different  a  ages;  Weighting  statistics  weighting  residual  f o r younger ages than  spanning  changes i n the  points  w i t h much  variation  3 .  upper  #2,  trends  variation  considered  Together the  of the  opposite  t e s t e d . Weighting  i n order t c assess  the c o e f f i c i e n t s of values  The  lower  a t r e n d of i n c r e a s i n g r e s i d u a l s a t younger  effectively was  Sn.  995?  the  w e i g h t i n g scheme l e d t o a q u i t e h i g h  a second w e i g h t i n g  weighting  ' s , and  10  of  extremes.  h i g h e r c o e f f i c i e n t s of v a r i a t i o n #1,  101X  B a s i c Model.  initial  therefore  y ' s , and  t c te the reverse  R e s u l t s With  the N  as  estimator  result be  rate,  content  will  be  a  to  of the  51  Tafcle 3 .  Results of least-squares catch-at-age analysis for northwestern Atlantic harp s e a l s . The l e a s t - s g u a r e s r e s u l t s a r e c o m p a r e d tc results frcm seguential population analysis. ,  52  .2 S  mean Sn  ^ i  0  mean N  i0  With the  basic model, weighting  #1  2 3 . 97  .94  .47  384  With the  basic model, weighting  #2  3 . 98  .87  .17  1080  With the  basic model, weighting  #3  2 . 23  .92  .12  1386  With a Beverton-Holt s t o c k - r e c r u i t f u n c t i o n , w e i g h t i n g #2  4 . 51  .90  .33  509  .90  .47  423  Sequential  population  analysis  53  weights. and  I f the i n f o r m a t i o n content  high  guantity  covariance enough  over  tc  give  reliable  be  The  enhanced,  ability  to by  d a t a e r r o r v a r i a n c e s as w e i g h t s available  but  information  4 Initial  used)  (assumptions)  modifications the l a t t e r  not  to  the  i s given  Results  effects  results,  first,  might  including estimate  be  an  the  reduced  estimate  natural  (information that  of  mortality  using accurate e s t i m a t e s of  and, into  quality  a wide range o f s t o c k s i z e s ,  among t h e t h r e e k i n d s o f  natural mortality. would  data  i s h i g h , i . e . , high  is  the  generally  s e c o n d , by i n c o r p o r a t i n g e x t r a  the e s t i m a t i o n  model and  in  the  form  of  the c o n s t r a i n t s .  An e x a m p l e  of  Beverton-Holt  Stock-Recruit  next.  Using  a  Relationship.  Mechanisms  for  density-dependent  a b u n d a n c e h a v e been p r o p o s e d t o o p e r a t e the  mortality  rate  c f pups, the  pregnancy r a t e  ( L e t t e t a l 1978).  possibilities  for  All  three  would  to  at  mean age This  be  least  6  simplificaticn  is  a r b i t r a r i l y , I c h o s e t c use s t o c k was 1,06  d e f i n e d a s t h e sum  ( f o l l o w i n g L e t t and  ways:  a in  explicitly,  recruitment be  required,  through the  number  of  Eg.(2-18). but  this  p a r a m e t e r s and  i t is  obtained  each.  therefore,  a Bevertcn-Holt curve. of a l l adult animals,  B e n j a m i n s e n , who  seal  o f m a t u r i t y , and  suggests  included  doubtful that r e l i a b l e estimates could Seme  i n three  t h e s t o c k - r e c r u i t f u n c t i o n (h)  mechanisms c o u l d  lead  r e g u l a t i o n of harp  The  for  somewhat breeding  multiplied  assumed t h a t 6%  of  by the  54  breeding for  p o p u l a t i o n was  changes  neglects  in  the  changes  c o n s t a n t a t age  5.  be  a.  and  in  the  We  have:  Z  Lett et a l  remains  to  optimization  on  frcm  the  The  are  the  p  were  terms  large  true  sinusoidal. the  of  the  should  relationship, As b e f o r e , i t weights,  the  parameter e s t i m a t e s .  determined  analagous  cf  shown  Hgo ,  by  deriving  eguilibrium  unacceptable in  3.2  due  an  unfished  Eg. (A-8)  A ,  and  Sn  (Table  5,. W e i g h t i n g and  #2  was  initial  but  cx and 4). used,  p» The  with  estimates  . using  the  s t o c k - r e c r u i t curve  t o an a n c m a l y i n  the f i r s t  due  and  extreme v a l u e s of  algorithm  y e a r . The  age  the  t c low  to  25  guality  are and  estimates  freguencies f o r  model a r e shown i n F i g u r e 3  e s t i m a t e s f o r a g e s 22  variances  t c Eg. (A-7)  i n Table  results  the B e v e r t o n - H o l t The  the  initial  calculating  in section  numbers-at-age for  Sn,  then  initial  considered  to  and  in  values  as  assumed  Beverton-Holt curve  constraints,  Davidon-Fletcher-Eowell  determined  was  but  (Noo ) , t h e r e c r u i t m e n t r a t e a t h a l f t h e u n f i s h e d  Eg. ( 3 - 1 ) ;  constraints  mortality,  m a t u r i t y , which  is slightly  and  (A) , and  extreme  pup  (3-1)  the  them  population size  using  suggest  a  for  eguilibrium  of  and  accounts  NJ;  algorithm,  Constraints expression  age  approximation  specify  model i m p l i c i t l y  rate  p a r e t h e unknowns. The  c o n s i d e r e d a s an  which  2 5 ) . The  pregnancy  Pi » 1.06 where  ever  were of 1952  ..  unreasonable,  with  g u a n t i t y of s u p p o r t i n g  55  Table 4. Extreme values of northwestern A t l a n t i c harp s e a l Beverton-flolt stock-recruit parameters CL and (b are determined frcm extreme values o f N. , * , and Sn,  Sn  X  N  a  CO  3  .80  .10  10  6  -3,.833 x IO"  .80  .10  10  7  -3..833 x I O  .925  .50  10  6  .925  .50  IO  7  5  18,.279  - 6  18..279  -5 1..570 x 10  -3..752  1..570 x 10"  -3..752  3  6  57  Table 5.  Parameter c o n s t r a i n t s of l e a s t - s q u a r e s c a t c h at-age a n a l y s i s with a Beverton-Holt stockrecruit f u n c t i o n , f o r northwestern A t l a n t i c harp s e a l s .  A l l  numbers-at-age [ 1,  A l l  1:  io ] 6  annual components o f [.001,  A l l  i n year  1  exploitation:  ]  age s e l e c t i o n f a c t o r s : [.001,  Stock-recruit [ 0, Stock-recruit [ Survival  0,  rate  1  ]  parameter 1.570  x  10~5  parameter 18.279  1.0  ]  3:  ]  through  [ 0.7,  a:  natural ]  mortality:  59  Figure  3. The age distribution for northwestern Atlantic harp seals in 1952, from (A) i n i t i a l r e s u l t s cf least-squares catch-atage a n a l y s i s with a Beverton-Hclt stockrecruit f u n c t i o n and (E) the revised distributicn.  AGE  DISTRIBUTION  (PERCENTAGE  09  OF  THE  IN I +  1952 POPULATION)  61  data.  Cne  s o l u t i o n to t h i s problem  results  by  postulating  relative  to adjacent  distribution  6,  using t h i s  pre-estimated  Eg, ( 3 - 1 ) ,  except:  >j  where The  pr-  total  N|j .  is  the  the  size.  on  The  Final  values  for  3).  The  distribution  ,  the  size, N  pre-estimated l+  was  same w e i g h t s and frcm  Eesults  initial  a g e s 22  to  resulting  e s t i m a t i o n was  in a  model  age  25 age  repeated  identical  I ~ j * T  , i s the  1+  the  to  (3-2) p o p u l a t i o n a t age j .  unknown i n s t e a d  of  each  d i s t r i b u t i o n , the c o n s t r a i n t s  c o n s t r a i n e d t o ( 10  6  ,10  L e t t e t a l ' s s i m u l a t i o n r e s u l t s o f maximum  points derived  3.5  modify  p r o p o r t i o n of the t o t a l  were as b e f o r e , e x c e p t N roughly  Figure  p^j  m  population  With  reasonable  to  ages, then n o r m a l i z i n g  (Table  N  is  algorithm  were  used,  ], b a s e d  7  population  with  initial  the c o n s t r a i n t s .  using  the  Bevertcn-Holt  Stock-Recruit  Relationship.  Some s t a t i s t i c s Table 3  and  the  from the state  seguential population Figure  4 .  mortality method  In  is  population  Sn = 0.9, has  the  presented  compared  in  to  the  Mohn e t a l  in  f o r s u r v i v a l through n a t u r a l but  the  least-sguares  been more a b u n d a n t w i t h  e x p l o i t a t i o n r a t e s . L e t t e t a l assumed a v a l u e on  are  a n a l y s i s o f L e t t e t a l and  same i n b o t h ,  the  results  reconstruction  s h o r t , the e s t i m a t e  i s the  shews  final  a v a i l a b l e evidence from 7 sources  of Sn = 0.9  (summarized  lower based  therein),  62  Table 6, The age distribution for northwestern Atlantic harp seals i n 1952, from i n i t i a l results of least-sguares catch-at-age analysis with a Beverton-Holt s t o c k - r e c r u i t f u n c t i o n , and the r e v i s e d d i s t r i b u t i o n .  Age  1 2 3 4 5 6 7 8 . 9  Initial  results  (%)  Revised  5.78 6.19 5.05 5.03 5.22 3.97 4.33 1.61 3.33  6.70 7.17 5.85 5.83 6.05 4.60 5.02 1.97 3.86  10 11 12 13 14 15 16 17 18 19  3.49 3.25 3.01 2.94 3.72 3.59 1.88 3.02 2.69 2.83  4.04 3.77 3.49 3.41 4.31 4.16 2.18 3.50 3.12 3.28  20 21 22 23 24 25  2.87 3.01 0.30 4.56 8.82 9.53  3.33 3.49 3.19 2.90 2.60 2.32  [%)  64  F i g u r e 4. E s t i m a t e d trends i n northwestern A t l a n t i c harp seal abundance, 1952-1979. Final results cf least-squares catch-at-age analysis with a Eeverton-Holt stock-recruit function are compared to results from sequential population analysis.  66  each  d e p e n d i n g , i n seme manner, on  the  a n a l y s i s of s u r v i v o r s h i p  indices. Pup  p r o d u c t i o n and  year  in  occur  i n the  errors fit  Table  are  years  guite  age  well,  with  sequential steadily  1958,  might  frcm  over  1952  to  have t h e  mid-1970s, b u t  leveling  o u t and  2.53  million  over  300%.  in  view o f  with  the The  estimated the  size  While  the  either  4.3  The  small  the  1+  unexploited  changes  other  the in  a.  of  years f i t  different  and  from  which  then  |+  increasing  ) i s 8.86  compared  the  the  decrease  increase.  The the  thereafter..  The  m i l l i o n , compared  may  a difference  to cf  appear i n c r e d u l o u s  result  is  consistent  population  estimates estimate  are cf  population  (N )  density-dependence in  in level.  the  1952  was  of  4.2  twice  the  suggests  that  The  Noo which i s v e r y  recruitment r a t e at h a l f  using  least-sguares  This  unreliable.  was  w  L e t t e t a l ' s value  inconsistency  eguilibrium  The  1+  with  model under f u l l  greatest  with  effect  of  35,000 p u p s .  unexploited  millicn,  of the  lack  population decreasing u n t i l not  (N  The  data.  equilibrium  c r both  probably  although  least-sguares figure  catch-at-age  analysis i s that estimated  i n 1952  production  the  estimates,  previous analyses, s t i l l ,  as  to  by  prediction  1977.  for seguential population analysis,  Lett-Benjaminsen  millicn.  than  mid-1970s  results  population  time,  analysis the  i n pup  and  1+ p o p u l a t i o n i s s l i g h t l y  least-sguares  1+  1971,  attributable  most e r r o r s l e s s  population  compared  least-sguares catch  1961,  be  selection  trend i n  r a t e s are  greatest differences  1956,  years  in  The  The  exploitation  f o r which t h e  large:  f o r these  changes  7 .  pup  error  sensitive the  is to  unfished  67  Table 7. Northwestern Atlantic harp seal pup production and annual pup e x p l o i t a t i o n r a t e s frcm the f i n a l least-squares results are compared to results from sequential population analysis. Values for pup p r o d u c t i o n are i n thousands.  final least-squares results  sequential population analysis  Year  N  i0  Vo  1952 1953 1954 1955 1956 1957 1958 1959  692 680 668 656 642 625 609 593  .27 .21 .23 .28 .21 .31 .42 .37  559 549 558 560 566 567 555 518  .35 .36 .33 .46 .61 .30 .27 .47  1960 1961 1962 1963 1964 1965 1966 1967 1968 1969  578 563 552 533 514 497 491 475 454 437  .44  .39 .60 .53 .34 .54 .51 .35 .44  498 468 433 403 381 362 364 358 362 368  .33 .37 .49 .71 .71 .52 .62 .78 .44 .64  1970 1971 1972 1973 1974 1975 1976 1977 1978  430 413 400 391 380 370 368 366 363  .45 .20 .21 .30 .32 .28 .23 .12 .25  361 347 329 329 317 308 312 327 349  .61 .61 .36 .31 .37 .46 .42 .40 .34  .10-  N  i0  •'i  0  69  equilibrium  population  were 1 . 0 8 2 x 10~fe The  that the animals  c o u l d s i m p l y be an a r t i f a c t ever  The  2.283  358,000  state  prediction  aillicn  seal hunters), or  o f assuming n a t u r a l m o r t a l i t y i s  f c r 1979 i s g i v e n i n T a b l e 9 . The  and 3 5 8 , 0 0 0 ,  f o r sequential model  techniques,  compared  population  to  1.397  million  and  a n a l y s i s combined w i t h t h e  (Mchn e t a l 1 9 7 8 ) .  p r e d i c t i o n c f pup p r o d u c t i o n ether  learn t o avoid  i t i snot  method p r e d i c t s a 1+ p o p u l a t i o n a n d pup p r o d u c t i o n  Lett-Eenjaminsen  frcm  (e.g.,  as t h e  a l l ages.  least-sguares cf  a a n d p>  and 2 . 5 4 6 , r e s p e c t i v e l y .  g e t c i d e r . T h i s c c u l d be r e a s o n a b l e  constant  as 0.24 .  with age, implying decreased v u l n e r a b i l i t y  inconceivable it  was e s t i m a t e d  a g e s e l e c t i o n f a c t o r s a r e shown i n T a b l e 8. They show a  definite trend animals  (A)  The  least-squares  i s c o n s i s t e n t with recent  e.g., tagging  estimates  s t u d i e s and t h e DeLury  m e t h o d , s u m m a r i z e d i n L e t t e t a l a n d Mohn e t a l . The  final  catches.  model e x p l a i n e d  However,  parameters r e l a t i v e  3.6 Population  The  i t should  be  noted  t c t h e amount o f d a t a  that  t h e number  cf  i slarge.  P r o j e c t i o n s With a Quota o f 1 8 0 , 0 0 0 .  c u r r e n t t o t a l a l l o w a b l e c a t c h o f harp s e a l s i s 1 8 0 , 0 0 0 . .  Projections c f population guota  85% c f t h e v a r i a t i o n i n w e i g h t e d  s i z e over t h e next t e n years  were made by Mohn e t a l ( 1 9 7 8 ,  Ben j a m i h s e n m o d e l . Under t h r e e dependence, i n each case they  T a b l e 3)  using  with  this  the Lett-  d i f f e r e n t assumptions of d e n s i t y found t h e stock t o i n c r e a s e  slowly  70  Table 8. Northwestern A t l a n t i c harp s e a l age s e l e c t i o n f a c t o r s from the f i n a l l e a s t - s g u a r e s r e s u l t s .  71  Age  0 1 2 3 4 5 6 7 8 9  1.0000 .0427 .0521 .0445 .0431 .0436 .0386 .0368 .0366 .0309  10 11 12 13 14 15 16 17 18 19  .0321 .0266 .0279 .0243 .0246 .0265 .0231 .0193 .0184 .0149  20 21 22 23 24 25  .0139 .0076 .0019 .0053 .0053 .0017  72  Table 9. F o r e c a s t of northwestern A t l a n t i c harp s e a l abundance f o r 1979 from the f i n a l leastsguares r e s u l t s .  73  Age  Abundance  0 1 2 3 4 5 6 7 8 9  358258 246069 259367 203080 170089 147326 138262 142983 132969 85896  10 11 12 13 14 15 16 17 18 19  78033 83390 59253 50710 64342 42126 32875 46693 62517 36142  20 21 22 23 24 25  37540 31307 34532 36583 30907 30298  1+  2283289  74  but  steadily.  population  was  mortality a  For  was  the  selectcn the  results  catches  estimates  lew  C.05  a  .  It  i s net  gave  for over  a both  question  1+  is:  ages  estimated  using  Eq. (3-2) ,  plus  or  runs.  distributed.  had  The  &  and  almost  i n the  represents  or  a  one-third  do  age with  of  standard  variability of  variation  f o r pup 0.1,  identical  while  different  high  0.1  two  lognormal  Two  coefficient  values  smaller  using  minus  catches. 0.2,  to  the  and high  deviations,  Mchn e t a l ' s p r o j e c t i o n s , t h o s e  This  natural  and  5.  decrease  Why  through  .  mean  catches,  projection  continual  years,  p  were u s e d .  a s s u i t p t i o n , t u t with  cases.  and  initial  the  stochastic  assumption  a  modelled  c f 0. 2 f o r t h e  shewn i n F i g u r e  ten  o(  100  f o r the  In c o n t r a s t t o show  was  normally  value  0.4  survival  the  b r o k e n down o v e r  5 show t h e  variability  variability  variabilty so  had  here  random f l u c t u a t i o n s i n r e c r u i t m e n t ,  assumed  assumptions about  given  normalizing  of  cf  mimicked  recruitment,  ty  in Figure  were  assumption  q u o t a was  Recruitment  deviations  distribution  The  the  calculated  least-squares  standard  of  and  factors.  The  projections  p r e d i c t i o n f o r 1979,  0.9,  distributon  the  the  stock decline  of  two  cf about  the  of  present  methods  give  given 3.6%  700,000  here  per  year  animals  population. such  The  different  forecasts ? Two potential greater  explanations embodied than  that  come  in  the  embodied  density-dependent  functions in  fertility  sampled  rates  to  mind.  First,  Lett-Eenjaminsen in the  s i n c e the  the  the  reproductive  model  Beverton-Holt  L-B early  model  were  1950s and  could  be  model.  The  f i t  with  estimates  of  75  f i g u r e 5. P o p u l a t i o n projections Atlantic harp seals 180,000..  for under  northwestern a guota of  77  abundance from a s e q u e n t i a l p o p u l a t i o n sizes  were t o o  lew,  p c t e n t i a l i n the proportion  of  abundance, productivity.  as i s s u g g e s t e d h e r e ,  model the  i f  would  stcck  second  p o t e n t i a l s are s i m i l a r , but r a t e at the to  the  than be the  stock  reproductive  However,  since  the  decreases with i n c r e a s i n g  L-B  model  explanation the  underestimates  i s that the decrease  in  reproductive recruitment  by l e a s t - " s q u a r e s ,  relative  from s e q u e n t i a l p o p u l a t i o n a n a l y s i s , i s g r e a t e r  corresponding  case,  the  that  then the  biased.  population estimated  populatcn the  the  high  be  recruited  anything  The  a n a l y s i s . I f these  with the  decrease i n e x p l o i t a t i o n r a t e . This d i f f e r e n c e being  l a r g e enough  to  must  produce  predicted decline.  3, 7 D i s c u s s i o n .  The First,  harp  the  seal  analysis i s deficient i n several respects.  c c e f f i c i e n t s of v a r i a t i o n  f o r the  data  assigned  somewhat  Better  estimates  can  calculated  from the  A  sampling  experiment  wculd  arbitrarily.. historical also  be  age  worthwhile.  c o e f f i c i e n t s o f v a r i a t i o n t o be how  they  change w i t h  Fecundity for  harp  and  seals  year  i n the  This  large will  f i x e d a c c u r a t e l y and  were  enable will  be  the  reveal  age.  s e x r a t i o d a t a h a v e been s i n c e t h e e a r l y 1950s, but  t h i s a n a l y s i s . I f estimates proportions-at-age  samples.  errors  are net  annually  included in  of both a g e - s p e c i f i c sex r a t i o s  of females whelping can  catch-at-age  collected  data  s e r i e s , then,  be  and  s u p p l i e d f o r each  using  Eg. ( 2 - 2 3 ) ,  the  78  least-sguares historical bring  state  reconstruction  reproduction  statistics.  t h e r e s u l t s more i n l i n e  population  analysis,  will  which  be  consistent  I expect t h a t doing  with  Lett  already  et  with  so  will  a l ' s sequential  i n c o r p o r a t e s some o f t h i s  infcrmaticn. Some o f t h e pup c a t c h p r e d i c t i o n large,  probably  as  result  of  years  selection  with  large  pup  assumption  be  modified  a t 1.0,  the  data  such  that  but a r e estimated  T h i s w o u l d h a v e t h e same that  unreasonably  in  c a t c h p r e d i c t i o n e r r o r s , t h e pup  f a c t o r s a r e not f i x e d  optimization.  are  v a r i a t i o n i n t h e age s e l e c t i o n  f a c t o r s f o r p u p s . The a n a l y s i s c a n those  errors  affect  errors  as  the  f o r those  i n the extreme  catches  are  negligible. Another source exists  o f s t r u c t u r a l weakness  in  the  i f n a t u r a l m o r t a l i t y d e c r e a s e s w i t h a g e , t h o u g h how  wculd a f f e c t t h e e s t i m a t e s  i s not c l e a r .  A  related  is  t h a t t h e age s e l e c t i o n f a c t o r s a r e n o t i n f a c t  age  f o r older The  estimate  estimator of  a  has  natural  powerful  tool  whether t h i s t u r n s not  by  are  reliable,  covariances in  this  possibility v a r i a b l e with  succeeded  mortality.  in  producing  Apparently,  a  reasonable  i n i t s a b i l i t y to  approach c o u l d  f o r analyzing catch-at-age  o u t t o be t h e c a s e w i l l  prove  data.  best  be  The p r o b l e m i s t o d e t e r m i n e  whether  to  However,  determined  p r o v i d i n g mere c a s e s t u d i e s i n w h i c h r e a s o n a b l e  are obtained.  this  animals.  e x t r a c t i n f c r m a t i c n , the l e a s t - s g u a r e s be  formulation  the  results results  and t h i s i m p l i e s a k n o w l e d g e o f t h e v a r i a n c e s a n d  of the estimates.  respect, net only  Simulation studies w i l l  providing estimates  be  useful  o f the parameter  79  variances,  but  This  i s the  point  the  variance  of  the  state  f o c u s c f c h a p t e r s 4 and  prediction 5. .  as  well.  80  Chapter  4. NUMEBIC AL EXPERIMENTS WITH LEAST-SQUABES  CATCH-AT-AGE  ANALYSIS: A PINNIPED FISHEEY  2  Chapter  was c o n c e r n e d  m e t h o d o l o g y , and c h a p t e r this  chapter  relationship attributes  and  with developing  3 with a p p l y i n g i t t o  t h e n e x t , an a t t e m p t  between of data  the  type  are estimated  estimated  age d i s t r i b u t i o n  clupecid  stcck,  of  the  and  a  parameters  of  and  Two t y p e s o f  clupeoid  of  a  to  stock.  obtain  the  i n t h e o p t i m i z a t i o n , and a p r e 1  i s included.  For the  recruitments are estimated, i n s t e a d stock-* r e c r u i t  function;  then  the  a s t o c k - r e c r u i t d i s t r i b u t i o n are estimated  using  t h e s t a t e r e c o n s t r u c t i o n . The n u m b e r s - a t - a g e i n y e a r estimated,  some  3 : the parameters of a s t o c k -  f o r year  the annual  parameters  In  d i f f e r s i n t h e e s t i m a t o r t h a t i s used. F o r  f i n a l harp s e a l r e s u l t s i n chapter function  data.  estimates  t h e p i n n i p e d s t o c k , t h e method i s t h e same used  recruit  real  which a f f e c t t h e i r r e l i a b i l i t y .  cf fishery  assessment  i s made t o g u a n t i f y t h e  least-squares  f i s h e r y a r e examined: a p i n n i p e d s t o c k Each  a stock  instead  of  including  a  1 are  also  pre-estimated  age  distribtuticn. The  information  characterized  content  of  a  given  data  set  is  by i t s s i z e , i t s d e g r e e o f e r r o r , a n d i t s l e v e l o f  contrast  ( i n abundance  and  exploitation  describe,  f c r example, the P a c i f i c  Halibut  rates).  One  could  (Area 2) d a t a s e t  as  81  huge  (18  ages sampled s i n c e  recently), MS  and p r o b a b l y  1 9 3 2 ) , w i t h enormous e r r o r s  moderate c o n t r a s t  (Hoag  and  McNaughton  1978, S o u t h w a r d MS 1 S 7 6 ) . T h i s s i m p l e c l a s s i f i c a t i o n h e l p s t o  give  form  reliability continua  to  a  in  assessments  can  search  be  for guantitative of  thought  parameter this  variances  "space".  of  as  increases,  increase  they  when  data.  The  the  magnitude  change a t d i f f e r e n t  will  decrease  errors  as  of  data  corresponding  to  a r e i n c r e a s e d , and s o o n . By estimates  various  points  we m i g h t o b t a i n i n some u n i n v e s t i g a t e d  The  Mcnte  Carle  approach  to  cf  both  abundance,  as  measured by b i a s e s and v a r i a n c e s . . T h e  simple,  The  information content mimicked  by  parameter  means  attributes  estimates  of  the  computer  model  and  data  forecasted basic  Replications  cf  fake  analysed  by t h e l e a s t - s q u a r e s  various  factors  assessed  t y comparing the r e s u l t s  to  the  data  idea  determining guantity)  representing not j u s t the  p o p u l a t i o n dynamics with f i s h i n g , b u t a l s o t h e process.  the  evaluate the  ( i . e . , data e r r o r s , c o n t r a s t , data a  of  circumstance..  reliability  quite  the  i s a  with  i n "information-  s p a c e " , we s h o u l d be a b l e t c a n t i c i p a t e t h e r e l i a b i l i t y estimates  the  quantity  t h e f o r e c a s t o f abundance, u s i n g models g e n e r a t i n g d a t a  attributes  are  three  points i n  q u a n t i f y i n g the variances of t h e p a r t i c u l a r parameter  is  about  d e f i n i n g dimensions i n space  and hew t h e y  We e x p e c t  generalities  catch-at-age  ( F i g u r e 6 ) . The p r o b l e m i s t o d e t e r m i n e  and  (until  are  estimator.  data  generated,  The  collection then are  contribution  of  b i a s and v a r i a n c e , o f t h e e s t i m a t e s , i s of  the  estimator  with  the  " t r u e " o r known v a l u e s i n t h e s i m u l a t i o n m o d e l . The a p p r o a c h h a s the  advantage t h a t t h e c o n t r i b u t i o n t o b i a s and v a r i a n c e o f t h e  82  Figure  6.  The d a t a s e t " i n f o r m a t i o n - s p a c e " . . Catch-atage d a t a s e t s c a n be c l a s s i f i e d by q u a n t i t y c f d a t a , d e g r e e o f e r r o r s , and l e v e l of contrast in abundance and e x p l o i t a t i o n r a t e s . The p o s i t i o n s c f t h e b o x e s i n d i c a t e the c o n d i t i o n s f o r t h e Monte C a r l o C a s e s AG.  84  estimates overall  can  be e v a l u a t e d f o r e a c h  v a r i a n c e can  collectively. somewhat of  the  be d e t e r m i n e d  However, a  circular.  We  parameters  by  drawback  if  analysis  that  the  in  the  model  is reliable  little  The  factors  rationale  assume we knew t h e t r u e v a l u e s i n  used  generating  is  nature  the  data  we know a b o u t them. I f problem,  but  i t i s u n r e l i a b l e i t i s p o s s i b l e t h a t t h e model g e n e r a t i n g  the  fake catches-at-age data are  actually  estimates  of  will  t h i s i s n o t an i m p o r t a n t  p r o d u c e i n f o r m a t i v e d a t a , when t h e  u n i n f o r m a t i v e . In t h i s case, the parameter  inappropriately  4.1  independently.  m o d e l l i n g a l l the  is  r e p l i c a t e s , i n c r d e r t c a s s e s s how the  factor  and  forecast  the E e l i a b l i t y  of the  Factors affecting r e l i a b i l i t y classes:  those  and  that i n h i b i t  Information  and  explcitaticr  variances. mortality  cf estimates  i s decreased  (4)  rates  Information unknown,  to  (1)  low  of  be  in  the p e r i o d over the  extraction  (6) s t r u c t u r a l  estimates is  impaired  l a c k o f f i t due  f i t due  t o changing  parameters are estimated  age  selection.  into  two  of the  data  estimator.  data g u a n t i t y ,  (3) l e w c o n t r a s t i n  during  errors  due  n a t u r a l m o r t a l i t y o v e r a g e s c r y e a r s o r b o t h , and lack  fall  i n f o r m a t i o n e x t r a c t i o n by t h e  error variation,  were t a k e n , and  will  Estimates,  t h a t reduce the i n f o r m a t i o n c o n t e n t  content  (2) h i g h d a t a  variances  Carlo  optimistic,  Factors Determining  those  Monte  real  stock  sizes  which the of by  data  data error  (5) n a t u r a l  to changes (7)  in  structural  When s t o c k - r e c r u i t  i n the o p t i m i z a t i o n , a d d i t i o n a l  factors  85  are  (8) l a c k o f f i t due t o  recruit  function/  variation the  (9)  i n recruitment,  an  incorrect  lack  of  and  (10)  form  of  the  f i t due t o n a t u r a l  stochastic  l a c k o f f i t due t o c h a n g e s i n  s t c c k - r e c r u i t parameters over  time.  In the remainder cf t h i s chapter, the r e s u l t s of seal  analysis  simulation  from  chapter  model t h a t  3  below  corresponds  Specifically, (A)  data  are  to t h i s single point  cf  factors  r e s u l t s provide  and  known, p e r f e c t  f i t ) to  mere  known, p e r f e c t  assessment variance  causing  cf  weighting,  with  parameter  a p p r o x i m a t e l o w e r bound.  will  a  moderate  cases examined  "information-space". the  (C)  effects  these  lack  estimation  of  With  structural  of  (B) d a t a plus  a  f i t . The  conditions  conditions. and no  harp  parameterize  weighting of the data,  realistic  reliability  and  structural  a range frcm o p t i m a l  natural mortality  fit,  the  e r r o r s , i n t h e a b s e n c e o f any o t h e r f a c t o r s ;  collection  mortality  in  we a r e i n t e r e s t e d i n q u a n t i f y i n g  n a t u r a l m o r t a l i t y unknown;  of  to  Each o f t h e t h r e e  errors plus  lack  used  generates large data s e t s ,  e r r o r s and tremendous c o n t r a s t .  stcck-  (with  and  no  natural lack  of  he a t a minimum, and t h e b i a s  estimates  should  represent  an  86  4,2 The C a s e S t u d y  Continued.  For t h e harp s e c t i o n 4,1  in  s e a l a n a l y s i s , some o f t h e f a c t o r s are probably unimportant.  ennumerated  There i s a r e l a t i v e l y  l a r g e s e t o f d a t a , a n d i t h a s a p p a r e n t l y been c o l l e c t e d wide due  range  cf  to changing  important  data  m o r t a l i t y , changing  errors,  age s e l e c t i o n , these  and s t o c h a s t i c  Before  collectively,  t w c more s p e c i f i c c a s e s  addresses  g u e s t i o n about d a t a e r r o r s :  By  will  effect  of  data  eliminating  can  in  be  even more s c . The s e c o n d .  The  object  mortality i s feasible  is  i n the  Case  to result  in  sources  of  a l l other  a p e r f e c t f i t and p e r f e c t known e x a c t l y ,  of  C a s e B, a s k s :  to find face  additional  C,  a l l important  cf  lead  does to  variation unreliable  out i f e s t i m a t i n g n a t u r a l realistic  f a c t o r s are assessed  model used i n e a c h e x p e r i m e n t Essentially  data  errors,  generated  study,  collectively..The  i s described next.  t h e models r e p r e s e n t t h e f i n a l  a n a l y s i s i n c h a p t e r 3. I n e a c h r e p l i c a t i o n were  the  factors  given that a l l ether conditions are optimal. In the t h i r d Case  A  are data e r r o r s of the  t h e d a t a p l u s n a t u r a l m o r t a l i t y unknown  predictions?  in  be t e s t e d i n d e p e n d e n t l y . I f t h e s e  r e s u l t s are unacceptable, the i n c l u s i o n will  be e x p l o r e d .  with n a t u r a l m o r t a l i t y  errcrs  variation  #2 s u f f i c i e n t  u n c e r t a i n t y i n t h e model, g u a r a n t e e i n g weighting of the data  more  sources of u n c e r t a i n t y are assessed  magnitude c o r r e s p o n d i n g t o W e i g h t i n g estimates?  but  w e i g h t i n g e r r o r s , unknown n a t u r a l  recruitment.  unreliable  a  s t o c k s i z e s and e x p l o i t a t i o n r a t e s . Lack o f f i t n a t u r a l m o r t a l i t y c o u l d be a p r o b l e m ,  are  a  over  in  r e s u l t s from t h e each  f o r a g e s 0 t o 25 and f o r 25 y e a r s .  case,  data  Recruitment  87  was m o d e l l e d in  by a B e v e r t o n - H o l t s t o c k - r e c r u i t  Eg. ( 3 - 1 ) .  mortality, in  section  to  those  The i n i t i a l , and  p>  s t a t e , age s e l e c t i o n f a c t o r s ,  were a l l t a k e n  3.5; t h e a n n u a l for  1952  to  components c f e x p l o i t a t i o n  correspond  least-sguares version  recruit  curve  The e s t i m a t o r was t h e same: t h e modified  constrained  respectively. without  include  to  5 except  [10~ ,10~ ], 7  5  The age d i s t r i b u t i o n  « ,  and  1 was  stock-  f o r year  p , and  [0,5],  f o r year  a  N  which  l+  [10  1.  ,10^ ] ,  6  assumed  known  error.  In random  to  and a p r e - e s t i m a t e d age d i s t r i b u t i o n  c o n s t r a i n t s a r e as i n Table  are  natural  presented  1S76.  from  as  the r e s u l t s  weighted  All  relationship,  a l l cases, variable  coefficients  observed  w i t h mean e q u a l t o  of v a r i a t i o n c  data  ij  the  "true"  (cv) a s i n W e i g h t i n g  ,  ^  were m o d e l l e d  e ~ N (  InCjj -  \  as a  lognormal  value #2,  (C)  and  i.e.:  , V:l  (4-.)  where  In  Case  mortality  A only  was assumed  the data e r r o r s  were t e s t e d ,  known  error.  i n v e r s e o f the data e r r o r Becruitment  was  without  modelled  constrained  "true"  stock  reconstruction For  case  to  be  history  without  stochastic  natural  was  w i t h i n [0.7,1.0]. In both is  f o r years C, e r r o r s  the  same  as  1952-1S76, shown i n the weights  Eg. ( 2 - 9 ) , b u t t h e c v  were  exactly.  c f f i t due t o age s e l e c t i o n  natural  were  the  the  exactly.  variation.  mortality  approximation Lack  Weights  v a r i a n c e s , asumed t o be known  was t h e same as C a s e A, e x c e p t and  while  Case B  estimated  A and B, t h e least-sguares  i n F i g u r e 4. were m o d e l l e d  using the  assumed  to  be  known  changing  with time  was  88  simulated  ty  making  the age s e l e c t i o n f a c t o r s f o r ages 1 to 4  n c r n a l randcm v a r i a b l e s v i t h a c o e f f i c i e n t of v a r i a t i o n of Age  s e l e c t i o n i s b e l i e v e d t o vary g r e a t e s t i n immature  likely  value  lcgnormal 15%,  animals,  i n a random manner, cr p o s s i b l y f o l l o w i n g trends  market  a  of  skins,.  Recruitment  random v a r i a b l e with a value  assumptions. numbers  for  was  so  used  the  is  coefficient  the age that  constant. . The  same  in  mimicked of  i n t e r m e d i a t e between the high and  In a l l r e p l i c a t i o n s the  10%.  the by  variation low  sequence  of  variablity of  random  s e l e c t i o n and recruitment random v a r i a b l e s  the  "true"  histcry  of  abundance  remained  " t r u e " h i s t c r y c f 1+ abundance i s almost e x a c t l y  the same as i n Cases & and  B and i s , t h e r e f o r e , not shown..  For each case the data g e n e r a t i o n and a n a l y s i s process replicated  25  times,  then  computing the means and parameters  and  the  the  results  variances  for  population  each  forecast.  were of  the  These  the true values w i l l be presented as " b i a s e s " ,  sampling  4.3  the  deviations  may  in  fact  was  summarized  r e s u l t s were  though  be  by  estimated  compared to the " t r u e " v a l u e s . . D e v i a t i o n s i n mean estimates  replications  a  due  from  with  25  t o random  e f f e c t s , r a t h e r than t r u e b i a s e s .  R e s u l t s and D i s c u s s i o n .  The  r e s u l t s are summarized i n Tables 9 to 12 , For Cases  and B they are very encouraging. cases of  have  The  A  parameter estimates i n both  a c c e p t a b l y s m a l l b i a s e s and almost a l l c o e f f i c i e n t s  v a r i a t i o n are l e s s than  10%. Those that exceed  10%  are,  as  89  Table  1 0 . Mcnte Carlo results for alpha, except v a r i a t i o n , are s c a l e d i n the b i a s e s are due  f o r Case A, A l l values the coefficient of by 1 0 . S l i g h t e r r o r s t c rounding. 6  90  TOTAL ABUNDANCE IN YEAB 1: MEAN TRUE VALUE VALUE 8 7 1 0 7 7 8 . 8654287.  "BIAS" -143509.  STD ERROR  5 2 8 4 3 2 . . .061  ANNUAL COMPONENTS Of FISHING MORTALITY: STD MEAN TEUE EEBOB "BIAS" VALUE VALUE YEAB 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25  .271 .209 .232 .281 .210 .313 .427 .373 .440 .096 .389 .607 ,.533 .344 .547 .511 .356 .447 .448 .199 .211 .303 ,324 .283 .233  .267 .208 ,230 .278 . 210 .310 ,424 .366 ,43 9 .097 ,386 .605 ,535 .344 .538 .506 . 352 .443 .447 . 201 .207 .303 ,321 .283 . 232  .004 .001 .002 .003 . 000 . 003 . 003 .007 .001 -.001 .003 .002 -.002 .000 .009 .005 .004 .004 .001 -.002 . 004 . 000 . 003 .000 .001  COEF VAE  ,0138 .0104 . 0106 .01 55 .0127 .0150 .0211 .0224 .0204 .0050 .0143 .0180 .0229 .0165 .0234 .0229 .0270 .0270 .0293 .0137 .0120 .0 234 .0206 .0194 .0187  COEF VAE ,051 .050 .046 .055 .060 .048 .050 .060 .046 .052 .037 .030 ,043 .048 .043 .045 .076 .060 .066 .069 .057 .077 .064 .069 .080  91  AGE  SELECTION FACTORS: TEUE MEAN VALUE VALUE AGE 1 2 3 4 5 6 7 8 9 1 0 1 1 12 13 1 4 15 16 17 1 8 19 20 21 22 23 24 25  ,042 .0 53 .045 .044 .043 .039 .038 .037 .031 .032 .027 .028 ,025 .025 .027 .023 .020 .019 .015 .014 .008 .002 .005 • .006 .002  .043 .052 ,045 .043 ,044 .039 .037 .037 .031 . 032 . 027 .028 .024 .025 .027 .023 .019 .018 .015 .014 ,008 .002 .005 .005 . 002  STOCK-BECEUI1 PAEAMETEBS: MEAN IBUE VALUE VALUE ALPHA BETA  1. 106 2.484  1.082 2.545  "BIAS"  STD EEBOB  COEF VAR  - . 000 .001 .000 .001 -.000 .000 .001 .000 .000 .000 . 000 .000 .000 .000 .000 .000 .000 . 000 .000 . 000 . 000 . 000 -.000 .000 -.000  .0021 ,0034 .0018 .0016 .0025 .0018 .0016 .0021 .0018 .00 19 .0016 .0016 .0016 .0013 .0015 .0012 .0013 .0014 .0009 .0010 .0009 .0002 .0004 .0005 .0001  .050 .064 .041 .036 .057 .045 .042 .057 .058 .058 .058 .058 .065 .054 ,055 .051 .064 .075 .058 .070 .114 .078 .078 .094 .083  STD "BIAS"  COEF ERROR  VAR  .0743 .1996  .067 .080  .025 1.0619  92  Table  11, Monte C a r l o results for alpha, except v a r i a t i o n , are sealed i n the b i a s e s are due  f o r Case B. a l l values the coefficient of by 10 . Slight errors to rounding. 6  93  TOTAL MEAN  ABUNDANCE IN YEAE 1: TEUE VALUE VALUE 8780185. 8854287.  STD "BIAS" -74102.  COEF EEBOE 722905.  ANNUAL COMIONENTS CF FISHING MQETAIITY: STD TEUE MEAN EBEOB "BIAS" VALUE VALUE YEAB 1 2 3 4 • 5 6 7 8 S 1 0 11 12 13 14 15 16 1 7 18 19 20 2 1 22 23 24 25  r  .269 .208 .230 .280 .210 .312 .426 .373 .439 .096 .389 .607 .534 ,345 .549 .513 .358 .450 .452 .202 .214 .3 08 .329 .288 .238  .267 ,208 .230 .278 .210 .310 ,424 ,366 .439 .097 .386 ,60 5 .535 .344 .538 , 506 .352 .443 .447 .201 ,207 . 303 .321 . 283 .232  .002 .000 .000 .002 -.000 .002 .002 .007 . 000 -.001 .003 . 002 -.001 .001 .011 .007 .006 .007 . 005 . 001 .007 . 005 .008 .005 .006  .02 54 .0188 .0177 .0226 .0168 .0181 .0 182 ,02 37 .0200 .0044 .0140 .0172 .0221 .0159 .0227 .0230 .0272 .0313 .0353 .0166 .0180 .0315 .0328 .0291 .0313  VAE ,082  COEF VAE .094 .090 .077 .081 .080 ,058 .043 ,064 .046 .046 .036 ,028 .041 .046 .041 .045 .076 .070 .078 .082 .084 .102 . 100 . 101 . 132  94  AGE SELECTION FACTORS: TEUE MEAN VALUE VALUE AGE 1 2 3 4 6 7 8 9 10 1 1 12 13 1 4 1 5 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25  .042 .053 .045 .044 .044 .039 .038 .037 .031 .032 .027 ,028 .025 .025 .027 .024 .020 ,019 .015 .014 .008 .002 .005 .006 .002  .043 .052 .045 .043 .044 , 039 .037 .037 .031 . 032 .027 .028 .024 .025 .027 ,023 .019 ,018 .015 .014 .008 . 002 .005 .005 .002  STCCK-EECEUIT PABAMEIEES: MEAN TEUE VALUE VALUE ALPHA BETA  1. 104 2.479  SURVIVAL  THROUGH KATURAL MEAN TEUE VALUE VALUE .900  1. 082 2. 546  .901  "BIAS"  STD ERROR  COEF VAE  -. 000 .001 .000 .001 -.000 .001 .001 . 000 .000 . 000 .001 . 000 .000 .000 .001 . 000 . 000 . 001 .000 .000 . 000 .000 -.000 .000 -.000  .0022 .0034 .0018 .0016 .0025 .0018 .0016 .0021 .0018 .0019 .0016 .0017 .0016 .0014 .0015 .0012 .0013 .0015 .0009 .0010 .0009 .0002 .0004 .0006 .0002  .051 .063 .041 .037 .057 .046 .042 .055 .058 .058 .058 ,059 .066 .057 .057 .053 .065 .078 .061 .072 .119 .079 .083 .100 .091  "BIAS"  STD ERROR  COEF VAR  .0998 .2424  .090 .098  "BIAS"  STD EEEOE  COEF VAE  -,001  .0073  .008  .023 -0.0665 MORTALITY:  95  Table 12. Monte C a r l e results for alpha, except v a r i a t i o n , are s c a l e d i n the b i a s e s are due  f o r Case C, A l l values the coefficient of by 1 0 . S l i g h t e r r o r s to rounding.  96  TOTAL  ABUNDANCE IN YEAR 1: MEAN TRUE VALUE VALUE 8758028,  8854287,  "BIAS" -96259.  STD EBBOE 1150673,  ANNUAL COMPONENTS OF IISHING MORTALITY: STD TRUE MEAN ERBOR "BIAS" VALUE VALUE YEAR 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 15 16 17 18 19 20 21 22 23 24 25  .266 ,201 .226 .283 .204 .287 .408 .376 ,413 .091 .362 .593 .545 .324 .554 .496 .341 .432 .450 .194 .208 .299 .318 .268 .221  .267 . 208 .230 . 278 .210 .310 .424 , 366 .439 .097 .386 .605 .535 .344 .538 .506 .352 .443 .447 , 201 .207 .303 .321 .283 .232  -.001 -.007 -.004 .005 -.006 -. 023 -.016 .010 -.026 -.006 -.024 -.012 .010 -. 020 .016 -.010 -.011 -.011 .003 -. 007 .001 ^.004 -.003 -.015 -.011  .0342 .0290 .0257 .0386 .0241 .0241 .0282 ,0327 .0226 .0066 .0185 .02 30 .0314 .0178 .0194 .0276 .0268 .0329 .0345 .0175 .0189 .0315 ,0355 .0334 .0313  COEF VAR .131  COEF VAR .128 .144 . 113 .136 . 118 .084 .069 .087 .055 .073 .051 .039 .058 .055 .035 .056 .079 .076 .077 ,091 .091 .105 .111 .125 .142  97  AGE SELECTION FACTOES: MEAN TEUE AGE VALUE VALUE  "BIAS"  STD EEBOE  COEF VAR  .041 .048 .044 .043 .043 .039 .038 .036 .030 .031 .026 ,027 .024 .024 .026 .023 .019 .018 .015 .014 .006 .002 .004 .005 .001  -.001 -.004 -.001 . 000 -.000 .000 .001 - . 000 -.000 -.001 -.000 - . 001 -.001 -.001 -.001 -.000 -.000 -.000 -.000 - . 000 -,001 - . 000 -.001 -.001 - . 000  .0028 .0029 .0028 .0022 .0025 .0023 .0021 .0025 .0022 .0021 .0017 .0021 .0017 .0018 .0017 .0016 .0014 ,0014 .0010 .0011 .0008 .0002 .0006 .0006 .0002  .068 .060 .065 .052 .058 .059 .057 .070 .073 .067 .064 .077 .071 .073 .066 .069 .074 .078 .068 ,083 . 126 .102 . 132 .127 .115  "BIAS"  STD EEEOB  COEF VAR  .2029 .43 26  .148 . 253  STD ERROR  COEF VAR  .0102  .011  1 2 3 4 c 6 7 8 S 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  .043 . 052 .045 ,04 3 .044 .039 .037 .037 .031 .032 ,027 .028 .024 .025 .027 .023 .019 .018 .015 .014 .008 .002 .005 . 005 .002  STOCK-EECEUIT PAEAMETEES: MEAN TEUE VALUE VALUE ALPHA BETA  1.367 1,708  1.082 2. 546  . 285 I.8377  SURVIVAL THROUGH KATURAL MORTALITY: MEAN TRUE VALUE VALUE "BIAS" .900  .901  -,001  98  Table  13. F o r e c a s t e r r o r s f o r Cases A-C. fill are given as a percentage of the value.  values "true"  CASE C  CASE B  CASE A COEF VAR  AGE  "BIAS"  COEF VAR  AGE  "BIAS"  COEF VAR  AGE  "BIAS"  0 1 2 3 4 5 6 7 8 9  -3.9 -0.1 -0.1 -0.5 -0.3 -0.9 -0.2 -0.4 -1.0 -1.2  6. 8. 8. 8. 8. 6. 6. 10. 9. 8.  0 1 2 3 4 5 6 7 8 9  -5.0 -1.6 -1.7 -2.1 -1.7 -2.2 -1.5 -1.9 -2.4 -2.5  9. 13. 14. 15. 14. 12. 12. 16. 16. 13.  0 1 2 3 4 5 6 7 8 9  11.9 17.0 22.6 -1.7 0.8 14.4 -1.4 7.6 3.1 -19.7  12. 17. 19. 16. 16. 17. 14. 19. 19. 13.  10 11 12 13 14 15 16 17 18 19  -1.7 -2.7 -0.9 -0.3 -1.3 -1.5 -1.1 -1.2 -2.3 -1.7  9. 9. 7. 10. 9. 7. 6. 8. 8. 8.  10 11 12 13 14 15 16 17 18 19  -3.1 -4.2 -2.2 -1.6 -2.6 -2.8 -2.2 -2.5 -3.6 -2.8  14. 13. 13. 15. 14. 12. 13. 13. 13. 14.  10 11 12 13 14 15 16 17 18 19  -7.4 3.5 -31.4 10.9 -2.3 -25.3 21.9 -9.5 3.8 -7.6  16. 16. 11. 20. 17. 12. 20. 15. 17. 15.  20 21 22 23 24 25  -1.8 -1.5 -1.8 -1.7 -1.7 -2.0  7. 7. 7. 7. 7. 7.  20 21 22 23 24 25  -3.1 -2.8 -3.2 -3.0 -3.0 -3.4  13. 14. 13. 14. 14. 14.  20 21 22 23 24 25  -13.7 -9.9 17.5 -5.4 -0.4 -3.0  15. 16. 19. 17. 18. 18.  1+  -0.8  7.  1+  -2.2  13.  1+  1.3  16.  100  would  te  expected,  those  with  the  l e a s t supporting data:  a n n u a l components of e x p l o i t a t i o n f o r l a t e r selection  f a c t o r s f o r c i d e r ages. For  h o t h c a s e s show a c o n s i s t e n t , Most r e m a r k a b l e i s natural  mortality  hut  that  is  the  years  l e s s than  variation 1%.  the  population  small, negative  the  and  bias.  in  However, t h e  estimates  of  e f f e c t of  not  m o r t a l i t y i s t o almost double v a r i a t i o n  prediction  1+  h a v e an a p p a r e n t estimated  a b u n d a n c e , f r o m 7% paradcx:  almost  with  l e a d s to t w i c e the even on  very small  the  an the  uncertainty  5,  increase  i n the  optimal  data sets. For  Eut  but  and  variances  (b , t h e  larger  variation  for natural  than  decrease  parameter  i n the  50%  k n o w i n g Sn  of  stochastic  the  f o r low  increase  interpreted of  the  in  Case  mortality  in  test,  with  i s just  1%.  reliablity.  For  B e v e r t c n - H o l t c u r v e , 95% estimate,  variation  to  p o p u l a t i o n s would i n c r e a s e  the  more s o u r c e s o f  Presumably levels  a  p,  in  contrast  results  are  Except f o r are  for  only  the  stock-  uncertainty the  leads  productivity  confidence l i m i t s further  a, and  ov  c o e f f i c i e n t of  increase  characteristic  uncertainty  (over  48%.  the  But  In  results in  estimates  Again,  effect  good..  high  caution.  parameter B.  tc  is  unknown  prediction  quantity,  i s f r o m 43%  realistic  parameters, including  to a large  ±  not  must be  slightly  recruit  seen a g a i n t h a t  1+  we  Apparently,  otherwise  mcst  encouraging  an  is  c f a b o u t 5%  the  i t as  predictions.  of t h e  the  mortality  prediction  t h e r e the  C a s e C,  natural  m o r t a l i t y have a l a r g e  r e l a t i v e variance  case)  in  In o t h e r words,  including  about  errors i n natural  i t w i l l be  13%.  though  certainty,  p r e d i c t i o n , when t h e  chapter  even  to  age  forecasts,  knowing n a t u r a l of  the  of p  to  most the  are in fish point  101  where e s t i m a t i n g s t c c k - r e c r u i t  parameters  in  the o p t i m i z a t i o n  would be g u i t e u s e l e s s . With  regard  gualitatively  to  the  forecasts,  Case  C  exhibits  d i f f e r e n t behavior. Whereas the b i a s  some  f o r the  1+  p o p u l a t i o n , -1.351, i s even l e s s than i n Case B, the b i a s e s i n the age  structure  are  both  much  larger  and  more i n c o n s i s t e n t ,  ranging frcm -31% t o +23%. T h i s r e s u l t has important bearing  cn  the r e l a t i o n between f o r e c a s t s of age s t r u c t u r e and f o r e c a s t s of total  abundance.  estimates  cf  population  While  age  we  cannot  structure,  the  place  much  estimate  confidence on  for  the  total  i s probably unbiased. T h i s apparently strange  result  can be e x p l a i n e d g u i t e simply:the v a r i a n c e o f a random is  variable  r e l a t e d t c the variance o f the mean c f n o f t h e same, i n the  p r o p o r t i o n n,. On  the  whole,  these  results  are  what  we  might  have  expected.  By f i s h e r i e s standards, a value f o r the c o e f f i c i e n t of  variation  c f a f o r e c a s t i n the range c f 7% t o 16% i s r e l a t i v e l y  good, even s u s p i c i o u s l y s m a l l . From a l a r g e data s e t contrast  and  moderate  e r r o r s , we cught  r e s u l t s . Frcm a s m a l l data errors,  the  s e t , with  with  to get r e l a t i v e l y  peer  contrast  and  high good bad  r e s u l t s should be r e l a t i v e l y d i s m a l . But what i s a  s u i t a b l e value corresponding t c "dismal"? A r e l a t i v e v a r i a n c e of 16% means an approximate S5% c o n f i d e n c e i n t e r v a l o f ± 32%. I f a forecast  with  a  confidence  interval  greater  than  ± 50% i s  bordering on i n u t i l i t y , then a r e l a t i v e v a r i a n c e i s dismal i f i t is  g r e a t e r than 25%. At any r a t e ,  though,  in  essence,  the  horrifying  implication,  the wisdom o f o l d f i s h e r i e s b i o m e t r i c i a n s ,  i s t h a t the f o r e c a s t s based  on most data s e t s , i n s o f a r  as  they  102  are  s m a l l e r , c r with  of harp s e a l s , w i l l c o u l d be  with the  increase.,  exceedingly  improvement  passage  The  only  analyst's patience. bigger  be  samples  cf  Designing  as  better  How  reliability  that  have not  three  guantity  data  e r r o r s can  sampling  of  i t  the  management's  sources.  data  process  be  designs.  will i s the  reduced The  with  resource's  resources.  manipulating  Third,  harvest  rates.  with a view to producing  an  i n o p t i m i z i n g y i e l d , i s termed a c t i v e  aid  waiters  Smith  these  1975;  Halters  results  sequential  and  informative  Hilborn  1976,  1S79). bear  on  the c o n t r o v e r s y  of the l e a s t - s g u a r e s harp  frcm  population  seal  about  predictions  a n a l y s i s ? On  the  one  the  versus hand  we  i n c l u d e d t i e p o s s i b i l i t y of trends i n n a t u r a l m o r t a l i t y  and  have assumed t h a t W e i g h t i n g  the  catch-at-age  recruitment  data. and  underestimated. a b u n d a n c e as  A l s o , the changing  If  the  #2  accurately reflects  variablity age  real  considerations  of the  lead  prediction  cf the  mortality  was  1+  to  population  will  be  decrease  population,  assumed  i s identical  data a  used  unknown  has  errors in  in  selectivity  much a s t h e r e s u l t s f r o m t h e  information content  estimate  that  policies  1S78;  do  the  be i n t r o d u c e d by  c o n t r o l (see  Silvert  those  can  harvest  data,  adaptive 1S78;  wishes  i s p o s s i b l e frcm  time  Second, the  and  contrast  catch  d i s m a l . One  a a n i p u l a b l e element i n t h i s  management depends h e r e on more  c o n t r a s t o r worse e r r o r s , t h a n  otherwise.  Nevertheless, First,  poorer  not  modelling could  be  varied  in  a n a l y s i s i n d i c a t e , the  overestimated,. in  A l l these  reliability  On  the  other  in  C a s e . C,  t o L e t t e t a l ' s . I f we  hand, even  of  the  natural  though  the  assume Sn i s known,  103  the  reliability  i n the  1+  These c o m p l i c a t i o n s the  deficiencies  prediction i s increased s u b s t a n t i a l l y . are  impossible  to  quantify.  Ignoring  i n the  analysis discussed  i n s e c t i o n 3.7,  r e s u l t s here i n d i c a t e the  present population  i s probably  1.6  million  million. with  3.0  sequential  at-age data,  sequential  w c u l d be  that  survival  early  the  are  made  s e a l i n g d y n a m i c s . An  rate  and  index  survival  dependent  methcd,  (e.g..  evaluation  cf  though  B i c k e r MS  assumes t h a t and  rates pup  pup  from  production the  errors catch-  using  the  results  underlying  production  d e r i v a t i o n of the  pup  in  point i s  that  from a way  the as  assumptions. frcm  then c o n c l u d e s i t i s c h a n g i n g i n the  in  t e c h n i q u e i s much  1971), there  method,  to  uncertainties  uncertainties inherent  S e r g e a n t MS  i s constant  density-  reconstruction  such  a l t h o u g h the  the  model  functions,  productions  related  of  complicated  e x a m i n a t i o n of the  reflect  1971,  accuracy of  its  method  e r r o r s i n the  resulting state on  i n d e x method. Y e t ,  a n a l y s i s cf the  next  al's  mean-age-at-whelping  adjustments i n the  critically  performed  a n a l y s i s c o m b i n e d w i t h a more  model t c a g r e a t e x t e n t  discussed nc  1950s and  2.283  informative.  guarantee o v e r a l l consistency..The the  et  exploitation  n a t u r a l m o r t a l i t y , the  finally, a l l  very  population  dependent pregnancy  in  Lett  a s s u m p t i o n o f no  forecasts  m c d e l c f s e a l and  are  terminal  p l u s the  Mohn e t a l ' s  and,  analysis.  been  from s u r v i v a l i n d i c e s . I n v e s t i g a t i o n i n t o process,  within  t h a t i t i s most l i k e l y  e x p e r i m e n t s have n o t  data sampled s i n c e the  estimated  reveals  and  population  reconstructing  fertility  this  million  S i m i l a r numerical  invclves  in  to  the  nor For one long  any  exists critical  example, year run.  i t  to  the  It  also  104  assumes t h a t f i s h i n g i n t e n s i t y i s constant over this  although  i s not the case comparing the p e r i o d s before and a f t e r  i n t r o d u c t i o n o f e f f e c t i v e quotas these  time,  contradictions  entirely  i n 1972.  The  can  of  to the accuracy of the assessment are not  obvious. Ike s i t u a t i o n with r e s p e c t  predictions  implications  the  best  u n c e r t a i n t i e s themselves  be  described  by  to  Mohn  et  al's  saying t h a t i t i s the  t h a t are u n c e r t a i n .  105  C h a p t e r 5. , NUMERICAL EXPERIMENTS WITH LEAST-SOUABES CATCH-AT-AGE ANALYSIS: A CLUEEOID FISHEBY  In  this  chapter  the  initiated  in  chapter  4  examined,  with  a  E,  guantity  tc of  moderate  guantity  and  B.  numerical  concluded.  stock.  single  The  examined  about  similar very  f o r C a s e s D and  Those f o r Case D a r e  to  two D  except are,  the  again,  c a s e s , F and and  E  in  G, data  contrast. E are  analagous to Cases A  natural mortality i s  a l s c known e x a c t l y ; and  structural f i t i s perfect.  Case E the  conditions  e x c e p t Sn i s unknown. C a s e s F  with  low  conditions  fcr  comparison of how  the  guantity, state  to  contrast, F  a  optimal, also  are o p t i m a l ,  bound  on  b i a s and  g u a n t i t y , low  the  to  A  w h e r e a s f o r G, D,  and  variance  c c n t r a s t s t a t e t c a low low  of  correspond  t h e r e s u l t s f r o m C a s e s A,  lower high  inverse  known  variances,  are  the  Cases D  the  G,  are  of  exactly;  and  weights  optimal:  model  of t h e s e ,  Errors  final  lew  are  "information-space",  75%.  i s investigated,  conditions  two  e x a m i n e d i n C a s e s A-C,  In the  with  more c a s e s  a simulation  in  ccntrast i s high.  e r r o r s , but  experiments  Four  first  point  d a t a i s r e d u c e d by  and  The  a  that  another point  and  clupeoid  represent  identical  are  of  d a t a g e n e r a t e d f o r e a c h by  hypothetical  and  series  and  error  B:  In  the  Sn i s unknown. A F  will  indicate  changes from a  guantity,  contrast  data  high  high  contrast  s t a t e . . W e have s e e n  how  106  n a t u r a l m o r t a l i t y unknown i n c r e a s e s t h e r e l a t i v e 1* p r e d i c t e d a b u n d a n c e c f Case  A  S i m i l a r c o m p a r i s o n s , E w i t h D and understanding prediction  5.1  of  the  about  691  (in  the  Case  B).  G w i t h F, o u g h t t o f u r t h e r o u r o f n a t u r a l m o r t a l i t y unknown  Clupecid  Data  Quantity.  s p e c i e s , s u c h as  experience  erratic  sardines or h e r r i n g , r e c r u i t m e n t , and  are  suffer  highly  relatively  h i g h r a t e s cf n a t u r a l m o r t a l i t y . These f e a t u r e s a r e embodied the  data  simulation  comprised  of  about f i v e The  16 age  years only  elements are the  are  must  annual  and  completely  relative  s p e c i f y how  selection plus the  survival  factors  of  model s t o c k i s  fish  the  is  components  of  and  exploitation.  a r e g i v e n by  Eg.(2observed egual  variances  Table  (cv) a s shown i n  the t r u e c a t c h e s  are generated,  numbers-at-age i n  natural  annual  mortality;  components o f  we year  the  age  exploitation;  stock-recruit distribution.  lognormal, breeding  than  Observed.  s t o c h a s t i c . The  s t a t e , or the  the  older  randem v a r i a b l e s w i t h means  through  parameters of the  Eecruitment function  and  in  between the h i g h c o n r a s t  strcngly  g u a n t i f y : the i n i t i a l  1; t h e r a t e c f  though  populaticn-at-age  lcgnormal  t o t h e t r u e c a t c h and To  differing  recruitment i s  catches-at-age  o u t l i n e d b e l o w . The  g r o u p s , a g e s 0-15,  true catch-at-age  18), except  14.  model  a r e r a r e under heavy e x p l o i t a t i o n .  lew c o n t r a s t c a s e s The  on  accuracy.  F o u r C a s e s c f lew  fecund,  effects  ty  variance of  with  stock,  mean given  recruitment, by  the B i c k e r  as  a  stock-  107  Table 14. Data error Cases D?G.  coefficients  of  variation for  108  AGE  COEF VAR  0  1  2-8  9-11  12-13  14  .4  .3  .25  .2  .25  .3  15  .  .4  109  recruit  model.  The  breeding  population. In ether  w o r d s , we  where the  c*  and  variance  p> a r e  defined  parameter. a  eguilibrium  population  (Here  and  Eg.(A-12),  t h e 2*  the  2+  have:  are  N-,  is  0  fixed  (Noo)  and  in  to  but  be  terms  thought of  similar  derived with the breeding  population, N  ro  the  O" i s 7  of  as  unfished  the r e c r u i t m e n t r a t e at h a l f  u n f i s h e d e g u i l i b r i u m (A), using eguations and  as  p a r a m e t e r s o f t h e B i c k e r f u n c t i o n and  N; , /Sn.) +l  is  -pp. «P,-e  =  . h(P^  stock  i s s e t a t 10  and  6  to  the  Eg. (A—11)  stock defined  r\ a t 1.0,  giving  as  values  -6  for  cx and  ^=  1.0,  £>  cf  is  capable  l e v e l s l e s s than  0.914  relative  recruits  in  of the  o f more t h a n  the b r e e d i n g  a particular  year  (N;  Q  mean f o r t h a t b r e e d i n g  random e f f e c t s  so t h a t t h e The (or  for recruitment  "true" history  survival  respectively.  i s 0.4  stock  With 2+  ) will  stcck  usually f a l l  The  and  same  used i n e a c h  the  sequence  replication,  constant.  natural mortality is  of  within ±  (h (P-, ) ) . T h i s m i m i c s  variabilty. was  ( o r O* = 0. 3 8 5 ) .  (P-, ) , t h e number  cf the s t c c k remained  r a t e through  M = - l n Sn = 0 . 4 ) ,  years.  ,  d u p l i c a t i n g i t s numbers a t  variance cf recruitment  s t r o n g i n f l u e n c e of e n v i r o n m e n t a l of  x 10  a b o u t Nco/2.  This i m p l i e s t h a t , given  80%  and  t h e r e p r o d u c t i v e p o t e n t i a l c f t h e s t o c k i s h i g h . The  population  The  1.358  assumed c o n s t a n t  is  low,  0.67  over ages  and  110  The  numbers-at-age  unfished be  in  year  1  represent  c o n d i t i o n s . A t e q u i l i b r i u m , t h e numbers one y e a r  t h e same a s t h e n e x t , i n t h e t o t a l  age  group, i f s t c c h a s t i c  Then  "equilibrium"  recruitment  variation  would  exactly  population  i n recruitment balance  and  for  were  excluded.  in  partially  year  1  and  e x p l o i t e d by  the  for  fishery  at  10  years. 1  age  Harvesting A cohort i s  and  i s  by age 3 , . I n t h e e s t i m a t i o n , t h e age s e l e c t i o n  exploited  f o r age 1 5 i s f i x e d age  continues  groups,  a t 1 . 0 . Catch  each  l o s s e s due t o n a t u r a l  m o r t a l i t y and t h e t r u n c a t i o n o f t h e model a t a g e 1 5 . begins  would  data are generated  fully factor  f o r a l l 16  1 6 0 o b s e r v a t i o n s , compared w i t h 6 5 0 f o r Cases  giving  A-C. I n C a s e s D and E, w i t h h i g h c o n t r a s t , t h e a n n u a l  r a n g e d f r o m 0 . 1 t o 0 . 9 , c a u s i n g t h e 1+ a b u n d a n c e  of e x p l o i t a t i o n t o d e c l i n e frcm ccntrast,  components  t o about  i N  w  . I n Cases F  and  G,  with  low  r e m a i n e d c o n s t a n t a t 0 . 1 . The 1+ p o p u l a t i o n d i d  they  j  n o t d r o p much b e l o w  ^ N o o . Beth  " t r u e " h i s t o r i e s o f 1+  abundance  a r e shown i n F i g u r e 7 , The Table state  estimates  r e c o n s t r u c t i o n was t h e n  o  distribution  = 0 ,  and  cf  0-  v  was  of  and  (b o f t h e  r e g r e s s i o n of l n (N;  logarithmically  then c a l c u l a t e d  a  The  /P- )  0  (  transformed.  from Eg. ( 2 - 2 0 )  with  An  1= 10,  x=2.  I n each case, parameter  used t o e s t i m a t e  viaa linear  E ; , i . e . , t h e B i c k e r model  estimate t  constrained t c the i n t e r v a l s given i n  1 5 . . A n n u a l r e c r u i t m e n t s were e s t i m a t e d a s p a r a m e t e r s .  stock-recruit on  were  estimates  25 replications  were made a n d  the  resulting  were s u m m a r i s e d a s b e f o r e . The " t r u e " v a l u e s  t h e numbers-at-age i n year  1 , t h e age s e l e c t i o n f a c t o r s , t h e  111  Figure 7 . "True" h i s t o r i e s C a r l o C a s e s E-G.  of  abundance  for  Monte  113  Table 15.  Least-sguares catch-at-age parameter c o n s t r a i n t s f o r Cases  analysis D-G.,  A l l numbers-at-age i n year 1: [ 1, 750,000 ] A l l annual ccmponents of e x p l o i t a t i o n : [ .00001, 1.0 ] A l l age s e l e c t i o n f a c t o r s : [ .00001, 1.0 ] A l l annual recruitments: [ 1, 750,000 ] Rate o f s u r v i v a l through n a t u r a l m o r t a l i t y [ 0.5, 0.9 ]  115  annual  components o f e x p l o i t a t i o n ,  and  with t h e Mccte C a r l o r e s u l t s i n T a b l e s  recruitment  are  given  16-20.  5.2 B e s u l t s a n d D i s c u s s i o n .  Table  16  shows  the  results  f o r t h e low g u a n t i t y , high  c o n t r a s t case under o p t i m a l e s t i m a t i o n c o n d i t i o n s .  It  exhibits  some p r o p e r t i e s common t o a l l f o u r e a s e l , p r o p e r t i e s due b o t h t o the  estimator,  discuss  these  and  to  cemmon  the  reduction  features  of  i n data.  the  First,  results,  I  then  will some  particulars. In  Table  abundance data  16, t h e c o e f f i c i e n t s o f v a r i a t i o n  p a r a m e t e r s show c l e a r l y hew a d e c r e a s e  decreases  reliability  cf  y o u n g e s t ages a r e s m a l l e s t , t h o s e  for  the  w i t h t h a t f o r age 15 b e i n g The e n l y cn  v a r i a n c e o f N. . the  observation  exploitation rate or will is  the  of the i n i t i a l  error  be p c o r . fixed,  in  supporting  the estimate.  Those f o r  cider  are  i s l a r g e because t h e estimate  depends  cf  of the  C,  and  l5  < s  i n observing C , Vi  Since the r e l a t i v e  the large variance  the estimate of y  (  or a  variance cf y  for N  (  are poor,  on  datum,  residual  will  be z e r o , u n l e s s t h e  15  3  must be due t o t h e l a r g e  )i(5  depends  catch-at-age  o f N,  i s s m a l l and a,  o f an a b u n d a n c e  single  ) 5  i s l a r g e , the estimate  5  o b s e r v a t i o n e r r o r s . When t h e e s t i m a t e a  larger,  enormous c o m p a r e d t o t h e o t h e r s .  (y a ) . I f t h e e s t i m a t e s (  ages  parameter  the  parameter  corresponding  i s bounded  by  a  constraint. The  estimates  f o r the  annual  components  of  fishing  116  Table 16. Monte C a r l o r e s u l t s for beta, except v a r i a t i o n , are s c a l e d i n the b i a s e s are due  f o r Case D. A l l values the ^coefficient of by 10 . Slight errors to rounding.  117  ABUNDANCE AGE  1 2 3 a  c  6 7 8 9 10 11 12 13 14 15 ANNUAL YEAB  1 2  3  4 5 6 7 8 9 10  I N YEAB 1 : TEUE MEAN VALUE VALUE  330500, 329570. 212640. 22154 1. 14.6326, 148503. 97988. . 99545. 66670. 66727, 44541. 44728. 29682. 29982, 20367. 20098. 13115. 13472. 9076. 9030. 6129. 6053, 4028. 4058. 2836. 2720. 1764. 1823. 1333. 1222. COMPONENTS MEAN VALUE  .107 .262 .517 ,516 .836 . 9 38 .938 .931 .109 .109  "BIAS"  STD EBEOB  COEF VAR  -930. -8901. -2177. -1557. -57. -187. -300. 269. -357. 46. 76. -30. . 116. -59. 111.  21460. 17006. 11068. 5695. 3828. 3853, 23 8 0 , 1671. 1220. 1020.. 766. 583. 656, 319. 473.  .065 .080 .076 .058 .057 .087 .080 .082 .093 .112 .125 . 145 .231 .181 ,355  OF F I S H I N G! M G E T A L I T Y STD TEUE ERROR "BIAS" VALUE  COEF VAE  .0060 .0141 .0220 .0239 .0369 .0316 .0341 .0421 .0384 .0430  .056 .054 .043 .046 .044 .034 .036 .045 .354 ,395  . 100 ,250 .500 ,500 .800 .900 .900 ,900 . 100 . 100  .007 .012 .017 .016 .036 .038 .038 .031 . 009 . 009  118  AGE  SELECTION FACTORS: TEUE MEAN VALUE VALUE AGE 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4  iNNOAL YEAB 1 2 3 4 5 6 7 8 9 10  .000 .245 .721 .957 .964 .957 .962 .957 .958 .960 .959 .961 .959 = .963 .S51  .000 . 250 .750 1.000 1.000  1,000  1.000 1, 000 1.000  r.000  1.000 1,000 1.000 1.000 1.000  RECRUITMENT: MEAN TEUE VALUE VALUE 422717. 318743. 378108. 308122.. 449874. 231065. 295564. 159433. . 177756. 121872.  408800. 319807. 376651. 305835. 442427. 226550. 294356. 157440, 181090. 101033.  "BIAS"  STD EBEOR  COEF VAR  ^.000 -.005 -.029 - . 043 -.036 -.043 -. 038 -.043 -.042 -.040 -.041 - . 039 - . 041 - . 037 -. 049  .0000 ,0258 .0398 .0343 ,0356 .0330 ,0325 .0379 .0317 ,0352 .03 07 .0337 .0368 .0375 .0349  . 179 , 105 .055 .036 ,037 .034 .034 .040 .033 .037 .032 ,035 .038 .039 .037  "BIAS"  STD EEROR  COEF VAR  13917. -1064, 1457. 2287. 7448. 4515. 1209. 1993. -3334. 20839.  33436. 28674. 43033. 47113. 58069. 45160. 69068, 54695. 72680. 88760.  .079 .090 .114 .153 ,129 .195 .234 .343 .409 .728  STOC^-BECEUIT DISTRIBUTION MEAN TEUE VALUE VALUE  'AR A M E T E B S :  ALEHA BETA SIGMA  I, 5 6 3 7 0. 293 1.0091  .794 .621' .376  1. 358 .914 .385  "BIAS"  , . .  .  STD ERROR  COEF VAE  .2375 .3999 .1047  .299 .644 .278  119  Table  17.  Monte C a r l o results for beta, except v a r i a t i o n , are scaled i n t h e b i a s e s a r e due  f o r Case E . . A l l v a l u e s the coefficient of by 10 . Slight errors to rounding.  120  AEUNEANCE IN YEAE 1: TBUE MEAN VALUE VALUE AGE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  330500. 404084. 255742. . 22154 1. 174100. 148503. 1 16833. . 99545. 66727. 79196. 44728. 52601. 35096. 29982. 20098. . 24239. 15690. 13472. 9030. 10838. . 7267. 605 3, . 4058.. 4744. 2720, 3360, 1823. 2133. 1222. 1581.  "BIAS" 73584. 34201. 25597. 17288. 12469. 7873. 5114, 4141. 2218. 1808. 1214. 686. 640. 310. 359.  STD EBBOfi 144427. 85139. 54525. 36431. 24033. 15848. 10246. 7731. 5329. 3668.. 2392. 1461. 1314. 832. 750.  COEF VAB .357 .333 .313 ,312 .303 .301 .292 .319 ,340 .338 .329 .308 .391 .390 .475  ANNUAL COMPONENTS OE FISHING MOETA1ITY STD TBUE MEAN EEROB "BIAS" VALUE VALUE YEAB  COEF VAB  .0373 .0618 .0651 .0452 .0477 .0353 .0375 .0460 .0397 .0573  .369 .249 . 131 .090 .058 .038 .040 .050 .361 .485  1 2 3 4 5 6 7 8 9 10  .101 .248 .497 .502 .826 .934 .934 .929 .110 .118  . 100 .250 .500 ,500 .800 . 900 .900 .900 .100 . 100  .001 -.002 -.003 .002 .026 ,034 .034 . 029 .010 .018  121  AGE  SELECTION FACTORS: TEUE MEAN VALUE VALUE AGE .000 .237 .710 .954 . 962 .955 .960 .955 .956 .957 .956 .959 = . S56 .961 .949  "EIAS"  STD ERBOfi  COEF VAB  -.000 -.013 -.040 -. 046 -.038 045 -.040 -.045 -.044 -.043 -.044 -.041 -.044 -.039 -.051  .0000 .0322 .0564 .0374 .0378 .0353 .0351 .04 06 .0340 .0372 .0338 .0363 .0400 .0401 .0355  .279 .136 .079 .039 .039 .037 .037 .042 ,036 .039 .035 .038 .042 .042 .037  "BIAS"  STD EBBOB  COEF VAB  134678. 67974. 70085. 51 738. 73543. 38587. . 38700. 17000. 4695. 21540.  222330. 134655. 150659. 115859. 151655. 92661. 126663. 75350. 85838. 89406.  .409 .347 .337 .324 .294 .349 .380 .432 .462 .729  STD EEROB  COEF VAB  .1131  .174  STOCK-EECBDIT DISTEIEUTION PABAMETEBS: MEAN TEUE STD VALUE VALUE "BIAS" EEBOB  COEF VAB  0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 ANNUAL YEAB  EECEUITMENT: MEAN TBUE VALUE VALUE  1 2 3 4 5 6 7 8 9 10 IUBVIVAL  408800. 543478. 31S807. 387781. 446736. 376651. 357574. 305835. 442427. 515970. 265138. 226550. 294356. 333056. 174440. . 157440. 181090. 185786. 122572. 101033.  THBOUGH KATURAL MOBTALITY: TBUE MEAN "BIAS" VALUE VALUE .648  ALPHA EETA SIGMA  .000 ,250 .750 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000  .818 .605 .384  .670  1.358 .914 .385  -.022  -0.5403 -0.309 -0.0012  .2352 .5030 .1114  .288 .831 .290  122  Table  18. Monte C a r l o results for beta, except v a r i a t i o n , are s c a l e d i n the b i a s e s are due  f o r Case F, A l l values the coefficient of by 10 ,. S l i g h t e r r o r s t c rounding,. fe  123  ABUNEANCE AGE 1 2 3 4 6 7 8 9 10 1 1 12 13 14 15  ANNUAL YEAR 1 2 3 4 5 6 7 8 9 10  I N YEAR MEAN VALUE  1: TRUE VALUE  276684. 330500. . 221541. 178859. 148503.. 122235. 99545. 79901. 66727. 53231. 44728, 34594. 22590. . 29.982, 20098. 15072. 13472. 9318. 9030. 6176. 4034. 6053. 4058. 2578,. 2720. 1659. 1823. 999. 122 2. 706.  COMPONENTS MEAN VALUE .440 .433 .429 .428 .439 .440 .441 .448 .477 .535  "BIAS" -53816. -42682. -26268. -19644. -13496. -10134. -7392. . -5026. -4154. -2854. -2019. -1480. -1061. x824. -516.  OF F I S H I N G M O R T A L I T Y TRUE "BIAS" VALUE . 100 . 100 ,100 .100 . 100 .100 . 100 .100 . 100 . 100  . 340 .333 .329 .328 . 339 .340 . 341 . 348 .377 . 435  STD ERROR  COEF VAR  51058. 32296. 22261.. 14819. 106 5 7 . 7069. 6145. 4229. 2586. 1867. 1523.. 1264. 873. 627. 576.  . 185 .181 . 182 .185 .200 .204 .272 .281 .278 .302 .377 .490 .526 .628 .816  STD ERROR  COEF VAR  .3656 .3588 .3465 .3407 .3374 .32 84 .3157 ,3037 .3081 .3608  .830 .828 .808 .795 .769 .746 .716 .678 .646 .675  124  AGE SELECTION FACTOES: TBUE MEAN VALUE VALUE AGE 0 1 2 3 4 c 6 7 8 S 10 1 1 1 2 13 14  .000 . 150 .418 .553 .577 .576 .575 .581 .589 .614 .622 .651 .670 .709 .773  .000 .250 ,750 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000  LNNUAL RECEOITMENT: TBUE MEAN VALUE VALUE YEAB 1 2 4 5 6 7 8 9 10  "BIAS"  STD EBBOB  COEF VAB  -.000 - . 100 -.332 - . 447 -.423 - . 424 - . 425 -.419 -.411 - . 386 - . 378 ^.349 - . 330 - . 291 -.227  .0000 . 1033 .2771 .3606 .3741 .3592 .3482 .3473 .3455 .3480 .3244 .3224 .2914 .2589 .1874  .739 .687 .663 .653 .648 .623 i605 .597 .586 .567 .521 .495 .435 .365 .242  "BIAS"  STD EEEOB  COEF VAB  -59101. 408800. 349699, 319807. -59886. 259920. -74261. 3 2 7 0 2 3 . . 401284. -80979. 409325. 328345. 655778.. -122941. 532837. -93641. 485016. 391374. -141024, . 506258. . 6 4 7 2 8 2 . -111059. 513472, 402413. 585546. . - 1 6 9 1 1 6 . 416430. -40989. 278280, 237291.  STOCK- E E C B O i l DISTRIBUTION PABAMETEBS TEUE MEAN "BIAS" VALUE VALUE ALPHA BETA SIGMA  1.584 1.080 .373  1.358 . 914 .385  .2264 . 167 -0.0120  82460. . 69567. 95346. 102937. 171455. 168574. 229666. 232939. 231404. 217360.  .236 .268 .292 .314 .322 .431 .454 .579 .556 .916  STD EBBOB  COEF VAB  1. 1798 1. 1444 i0972  .745 1.059 .261  125  TaJ3l€  19. .Monte C a r l o results for beta, except v a r i a t i o n , are s c a l e d i n the b i a s e s are due  f o r Case G, a l l values the ^coefficient of by 10 . S l i g h t e r r o r s t c rounding.  ABUNIANCE IN YEAB 1: TEUE MEAN VALOE VALUE AGE 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15  232315. 150994. 105384. 70374. 46895. 31664. 20557. 13903. 8966. 5963. 3910. 2707. 1843. 1181. 851.  330500. 22154 1. . 148503. 99545, 66727. 44728. . 29982. 20098. 13472. 9030. 6053. . 4058. 2720. . . 1823. 1222.  "EIAS" -98185. -70547. -43119. -29171. -19832. -13064. -9425. -6195. -4506. -3067. -2143. -1351. -877. -642. -371.  STD EEECE  COEF VAE  213940. .921 120054. .795 .777 81918. ,7 84 55151, .758 35533. .805 25478. .771 15858. .763 10605. 7024. .783 4357. .731 2834. .725 .900 2438. .925 1705. 1227. 1.038 933. 1.097  ANNUAL COMPONENTS OF FISHING MCETALITY STD TEUE MEAN EEEOR "BIAS" VALUE VALUE YEAB  COEF VAE  .2896 .27 66 ,2649 .2648 .2637 .2358 .2198 .2275 .2645 .3663  .766 .758 .749 ,769 .781 .746 .731 .770 .846 1.000  1 2 3 4 5 6 7 8 9 10  .378 .365 .353 .344 .338 .316 .301 .296 .313 .366  . 100 . 100 . 100 .100 . 100 . 100 , 100 . 100 . 100 . 100  . 278 . 265 . 253 . 244 . 238 .216 . 201 . 196 .213 . 266  127  AGE  SELECTION FACTORS: TEUE MEAN VALUE VALUE AGE 0 1 2 3 4 5 6 7 8 9 1 0 1 1 12 13 14  .000 .203 .546 .710 .726 .735 .733 .733 .730 .752 .758 .781 .792 .817 . 847  .000 .250 .750 1. 000 1.000 1.000 1. 000 1.000 1. 000 1.000 1.000 1.000 1.000 1.00 0 1.000  INNUAL RECRUITMENT: TBUE MEAN VALUE VALUE YEA E 1 2 4 5 6 7 8 9 10 SURVIVAL  274074. 193475. 229449. 227219. 370028. 272615. 367546. 337029. 377097. 245441.  408800.. 319807. 401284. 409325. 655778. 485016. 647282. 513472. 585546. . 278280.  COEF VAR  -.000 -.047 -. 204 -.290 -.274 -.265 -.267 -. 267 - . 270 -. 248 -. 242 -.219 -. 208 -. 183 -. 153  ,0000 .0644 .1787 .2371 .2394 .2375 .2333 .2330 .2240 .22 18 .2057 .2122 . 1931 .1722 .1074  .379 .318 .327 .334 .330 ,323 .318 .318 ,307 .295 .272 .272 .244 .211 .127  "BIAS"  STD ERROR  COEF VAR  -134726. -126332. -171835. -182106. -285750. -212401. -279736. -176443. -208449. -32839.  262558. 168006. 175226. 146881. 209296. 126184. 172338.. 224300. 243872. 228454.  .958 .868 .764 .646 .566 .463 .469 .666 .647 .931  STD ERROR  COEF VAR  .. 1455  . 186  STD ERROR  COEF VAR  .2362 1.6357 .0967  .410 3.280 .239  THROUGH NATURAL MORTALITY: TRUE MEAN "BIAS" VALUE VALUE .783  ,670  STOCK-EECRUIT DISTRIBUTION MEAN TRUE VALUE VALUE ALPHA BETA SIGMA  "BIAS"  STD ERROR  .577 .499 .405  1.358 .914 .385  . 113  PARAMETERS: "BIAS" -0.7811 -0.415 .0197  128  Table 20. F o r e c a s t e r r o r s f o r Cases D-G. A l l values are given as a percentage of the "true" value.  CASE D  AGE  "BIAS"  CASE E COEF VAR  AGE  "BIAS"  COEF VAR  0 1 2 3 4 5 6 7 8 9  -37.4 20.6 -1.7 1.9 1.2 4.5 10.1 4.9 7.7 4.8  32. 88. 41. 37. 29. 31. 37. 37. 43. 40.  0 1 2 3 4 5 6 7 8 9  -44.9 18.3 -6.1 -3.6 -3.1 -1.1 5.1 0.0 1.8 -0.9  26. 90. 41. 40. 44. 46. 54. 50. 51. 49.  10 11 12 13 14 15  12.1 10.6 7.1 9.5 6.6 7.8  40. 40. 37. 37. 37. 41.  10 11 12 13 14 15  5.3 5.9 1.2 3.5 1.3 0.8  50. 55. 48. 50. 51. 46.  1+  6.1  43.  1+  2.0  48.  CASE F  AGE  "BIAS"  CASE G COEF VAR  AGE  "BIAS"  COEF VAR  0 1 2 3 4 5 6 7 8 9  -35.5 -14.7 -29.4 -23.7 -26.5 -26.5 -29.5 -33.0 -34.1 -36.9  40. 78. 40. 50. 43. 47. 41. 42. 42. 41.  0 1 2 3 4 5 6 7 8 9  -44.9 8.9 -0.9 18.3 21.2 28.6 27.1 25.4 26.5 27.5  44. 108. 80. 105. 104. 112. 109. 112. 112. 117.  10 11 12 13 14 15  -37.3 -43.0 -48.3 -50.8 -55.3 -58.9  42. 39. 39. 41. 41. 43.  10 11 12 13 14 15  32.2 24.3 18.9 17.1 13.6 12.2  120. 114. 109. 109. 108. 111.  1+  -24.8  46.  1+  11.8  91.  130  m o r t a l i t y i n years supporting  9 and  10 are  also  affected  by  a  lack  data: the r e l a t i v e v a r i a n c e s are almost ten times  great as f c r the other y*s. The  conseguence of a  poor  d r a s t i c , Eecruitment  i n the l a s t year  (N  x o  (N  ) . I f the r e l a t i v e  (T  (y^) i s  ) depends on a  i n the same manner as the number-^at-age i n year  o l d e s t age  v a r i a n c e of y  single  1 f o r the  i s l a r g e , that  of B ' w i l l a l s o be l a r g e , with the r e s u l t t h a t the estimate recruitment  in  the l a s t year  alsc  be  variance c f N great  as  lo o  very  poor.  (N  V ( 0  f o r the other recruitment  ) w i l l have  component estimate  For example, i n Case D the  and the p r e d i c t i o n of age  S i m i l a r l y , the estimate  of  a  variance,  exploitation  (y  )  1 (N  (M  a  if  is  c f recruitment i s supported  that  as This  the is  ) are twice  of  the  l a r g e , except,  is  here,  selection  because Though  of  selection  effect  y . x  f a c t o r s f o r o l d e r ages are not as  annual components c f f i s h i n g  to f i x the estimate and  the  determnined  they  are  the  tightly  amount  unreliable  mortality for later bound  by  their  are  of supporting data i s low,  a c c u r a t e l y . The  constrained  decreases  to  years. upper  upper c o n s t r a i n t ( r e p r e s e n t i n g v a l u a b l e p r i o r information)  15  year  s t c c k - r e c r u i t f u n c t i o n , embodying enough i n f o r m a t i o n to  constraint.  a  as  annual  by more data and the  c o u n t e r a c t the e f f e c t of u n r e l i a b l e estimates The  relative  i n the next to l a s t  w i l l net be as l a r g e . In Cases ft-C, recruitment frcm  age 1  parameters and p r e d i c t i o n s .  f c r recruitment  large  of  w i l l o f t e n be nonsense or bounded  only by i t s c o n s t r a i n t s , I t f o l l o w s that the f o r e c a s t f o r will  as  estimate  of the annual component of e x p l o i t a t i o n i n the l a s t year  datum  of  be  a's are s c a l e d no  helps  relative  greater than a  at c i d e r ages (e.g., by o l d e r f i s h  v5  the  to  . . I f age migrating  131  o u t o f t h e f i s h e r y , by a v o i d a n c e b e h a v i o r ) , i t i s either  increase  of t h e f i x e d  important  t h e u p p e r c o n s t r a i n t f r o m 1.0 o r l o w e r  t h e age  term.  In each o f the four cases, the biases f o r a l l t h e of  to  y's  are  t h e same s i g n , a s a r e t i e b i a s e s f o r a l l t h e a ' s . I n C a s e  those  f o r t h e y's a r e p o s i t i v e ,  negative. the  This  consistency  while  f o r the  a's  r e f l e c t s the indeterminacy  y's a n d t h e a ' s , f o r w h i c h r e a s o n  one  those  i t was  necessary  D,  are  between to f i x  c f t h e p a r a m e t e r s . The r e s u l t f o r C a s e s D and E i s t h a t t h e  exploitation rates  (y. a. ) a r e r e l a t i v e l y  unbiased. . In  Cases  F  J  and  G,  t h e same c o n s i s t e n c y i n b i a s i s s e e n , b u t t h e r e s u l t i n g  explcitaticn  rates are overestimated.  order" indeterminacy F,  the s t o c k - r e c r u i t d i s t r i b u t i o n  each  this  c c u l d w e l l be an a r t i f a c t o f t h e  replication  in  random numbers were u s e d t o recruitment. relative  are  estimates relative  For  seme  each  case,  simulate  reason,  f i t using  procedure.  stochastic  cx ,  of  variation  in  (b a l w a y s h a s a  regardless  of  whether  c r l o w c o n t r a s t i n a b u n d a n c e . The b i a s e s i n <x and  usually of  test  bias,  t h e same s e q u e n c e o f 10  the estimate  v a r i a n c e much l a r g e r t h a n  there i s high  cr  parameters,  s t a t e r e c c n s t r u c t i c n , t h e r e seems t o be no s y s t e m a t i c  although  p>  and a b u n d a n c e f o r Case  G.. For  For  between e x p l o i t a t i o n  "higher  and among e x p l o i t a t i o n , a b u n d a n c e , and n a t u r a l m o r t a l i t y f o r  Case  the  This r e f l e c t s the  0*  large, are  w h e r e a s t h e b i a s f o r cr i s s m a l l . Good  obtained  in  these  studies  variance i n recruitment i s constant.  because  I n nature  this  the may  may n o t be t h e c a s e . Now  consider  seme p a r t i c u l a r r e s u l t s . I n C a s e D, w i t h l o w  132  quantity of  and h i g h  contrast  data,  the parameter estimates  with  high  guantity  the c o e f f i c i e n t s  a r e n o t much g r e a t e r  and h i g h c o n t r a s t  data,  of  variation  t h a n i n C a s e ft,  except f o r the  stock-  r e c r u i t p a r a m e t e r s , w h i c h i s t o be e x p e c t e d s i n c e C a s e A d i d n o t include stochastic variation. parameters  are  well  The  separated.  abundance  and  Although there  b i a s i n t h e y's and a ' s , t h e o v e r a l l b i a s f o r rates  most  significant  f o r e c a s t . I n Case  predicted  recruitment  where mean  recruitment  distribution 32%.  of  importantly,  A  the are  bias  is  predicted  with <x  and  more p  years in  Case  are  variance  from  the  stock-recruit  data,  errors  in  D  would  d e c r e a s e . . More  and  though r e l a t i v e l y  i s 4 8 % , an i n c r e a s e  2%, a d e c r e a s e o f 4%.  contrast  appreciably increase A i s very In  the  coefficient  over Case D  This  unbiased. In  variance  alter  of  Case  of the predicted of  5%,  and  the  i s a b o u t t h e same m a g n i t u d e  d a t a s e t s , n a t u r a l m o r t a l i t y unknown d o e s the r e l i a b i l i t y  1+  forecast  o f change a s f r c m C a s e A t o C a s e B, w h i c h seems t o i n d i c a t e high  D,  of  bias  Sn i s unknown, t h e r e l a t i v e  1+ p o p u l a t i o n  for  of  r e p e c t i v e l y . I n Case  - 1 % a n d 7%, f o r C a s e D, 6% and 4 3 % . The  D i s much l e s s r e l i a b l e ,  is  relative  the d i f f e r e n c e s i n the r e s u l t s f o r the predicted  variation  where  and  - 3 % a n d 6%,  a r e huge: f o r C a s e A, t h e  bias  exploitation  d i f f e r e n c e b e t w e e n A and D c o n c e r n s  population  E,  the  consistent  and compared t o t h e " t r u e " mean, t h e y a r e - 3 7 % and  Presumably,  estimates  in  i s a  (y. a. ) i s s m a l l . The  the  exploitation  o f t h e 1+ f o r e c a s t  that not  (though t h e  a p p e a r s d r a m a t i c i n Case B s i n c e t h e p r e d i c t i o n i n Case good) . Case  E,  the  effects  of  the  indeterminacy  between  133  a b u n d a n c e , e x p l o i t a t i o n , and only  has  increases D,  but  natural  at-age  mortality  i n most o f t h e  there  unknown  resulted  visible.  recruitment  p o s i t i v e b i a s i n the parameters,  and  a  Net  in substantial  parameter v a r i a n c e s , compared  i s a definite  and  n a t u r a l m o r t a l i t y are  to  Case  initial  numbers-  negative  bias in  through  natural  survival. I n C a s e s B and mortality  was  C,  the  estimated  rate  of  survival  a l m o s t w i t h c e r t a i n t y . . I n C a s e s E and  the  r e s u l t s are  is  s m a l l , but  95%  c o n f i d e n c e . i n t e r v a l of [0.54, 0.76]. In  contrast, greater  much l e s s e n c o u r a g i n g . The the  the  estimate  inability  exploitation, For  both  negative  bias, strong  positive completely  and  low  of  Alsc,  guantity,  the  25% E,  and  and  bias.  about  17%.  with high  about the not  In  low  the  variances  variance  Case  low  i n d i c a t e s an  even  of the  abundance,  E.  contrast cases, show  rates Sn  a  the  strong  (y. a. ) show also  shows  abundance  and  parameter estimates  are  o f t e n ten t i m e s as g r e a t .  The  1+  f o r e c a s t f o r Case F a r e  c o n t r a s t , s t r a n g e l y enough t h e r e l a t i v e  same. When Sn i s known, t h e  as t h e r e d u c t i o n  i n data  a  i t is  W h i l e t h e b i a s i s much g r e a t e r t h a n i n C a s e s D  seem t o h a v e s u c h a d r a m a t i c  an  (  contrast cases,  e f f e c t s of  of the  E,  G,  among  G,  low  approximate with  parameters  In the  to s e p a r a t e  the  relative  46%.  This  exploitation  much l a r g e r t h a n i n C a s e s D and biases  Case  indeterminacy  recruitment  positive  impossible  mortality.  i s also biased.  the  while  bias  g i v i n g an  n a t u r a l s u r v i v a l than i n Case  of  abundance  b i a s f o r Sn i n C a s e E  e r r o r i s 0.11,  to r e s c l v e the  and  initial  equally  standard  G,  reduction  and  variance  i n contrast  -  is  does  e f f e c t on t h e  prediction errors  q u a n t i t y . I t c o u l d be  that fluctuations  134  in  recruitment  cause enough v a r i a b i l i t y  o f f s e t the e f f e c t s of constant i n recruitment, of  information  i n the age s t r u c t u r e to  e x p l o i t a t i o n , and  that  r a t h e r than i n 1+ abundance, i s a b e t t e r measure content f o r c l u p e o i d  Case G where Sn i s unknown, the  f i s h e r i e s , At any r a t e , i n  relative  variance  of  p r e d i c t i o n i s 91%. The e f f e c t of low c o n t r a s t and data when  Sn  is  unreliable.  contrast  unknown,  is  to  render  any  prediction  the  1+  quantity, totally  135  C h a p t e r 6.  The  primary concern  how o u r u n d e r s t a n d i n g the  quality  cf  CONCLUSION  o f t h i s t h e s i s h a s been t o  of f i s h  investigate  p o p u l a t i o n d y n a m i c s i s a f f e c t e d by  catch-at-age  d a t a , o r , more p r e c i s e l y ,  by t h e  " i n f o r m a t i o n s t a t e " , a s r e f l e c t e d i n a few s i m p l e a t t r i b u t e s  of  the data. But understanding i s not the u l t i m a t e o b j e c t i v e of the management of  process.  seme y i e l d ,  certain  The u l t i m a t e o b j e c t i v e i s t h e m a x i m i z a t i o n  such as  ecological,  biomass  or  net  economic,  or  social  a c c o u n t o f t h e management information  state  g o a l s , and hew information  i s  process  related  management  revenue,  would  investigate  (1979)  data  analysis  p o p u l a t i o n . Each y e a r i n h i s  estimated  the  He  then  control  these the  then  "fished"  calculated  an  he  optimal  first  effort  the population, thereby generating  d a t a t o be u s e d t h e  way,  compare v a r i o u s h a r v e s t p o l i c i e s  could  procedure  p r o d u c t i o n model frcm  c a t c h and e f f o r t  procedures,  the  affect  simulations,  parameters of t h e Schaefer  p a s t c a t c h e s and e f f o r t s ,  he  how  r o u t i n e t o a s t o c h a s t i c m o d e l o f an age-<-  structured  control.  full  ingeniously contrived to simulate the f u l l  p r o c e s s on a c o m p u t e r , by c o u p l i n g a h a r v e s t a  A  t o achieve  consequently  of  state,  Hilborn  and  lieu  constraints.  t o our a b i l i t y  decisions  in  u s i n g as an i n d e x o f  following  performance  year.  In  this  and e s t i m a t i o n  the  total  catch  136  over  t i e simulated  dene  with  d u r a t i o n . P r e s u m a b l y , t h e same t h i n g c o u l d  e s t i m a t o r s t h a t use  catches-at-age  p l u s e f f o r t d a t a . H i l b c r n p c i n t e d out model,  management  estimates  due  to  exploitation  often  failed  insufficient  rates, results  that,  catches-at-age  with  because  contrast  or  of in  the  Schaefer  poor  parameter  abundance  analagous to those  be  and/or  o f C a s e s F and  G  above. Fish first  pcpulaticn analysts  is  scientific,  guantifying  t h e two  evidence  that  to achieve  suppose  that  (personal  ccmmunicaticn)  inability  of  an  estimator  parameters of the  Schaefer  increase,  carrying  the  coefficient)  is of  The  does not  the  to  Perhaps analyses:  there a  qucta  there  separate  or  may  similar  p a r a m e t e r s , t h i s may catch  n o t be able  the  capacity,  show  the  Hilborn  that  the  i d e n t i f y the  three  intrinsic  and  impair  its  the  rate  not  be  be  ratio  to  (or each o f t h e  may  not  the e f f e c t s  will be  be  of  to This two  enough i n f o r m a t i o n t o  enough t o e s t i m a t e t h e i r result  of  catchability  capacity  d e p e n d s o n l y cn t h e  accurately  of that  Whereas i n  the case.  to  accurately  model ( i . e . ,  optimal e f f o r t  while  accurately  stock,  synonomous, t h e r e i s r e c e n t  been  necessarily  e a c h o f t h e two cases  the  The  second i s pragmatic,  may  t h e p a r a m e t e r s . W h i l e t h e r e may  certain  problems.  optimal e q u i l i b r i u m l e v e l of f i s h i n g e f f o r t .  because the  estimate  two  optimal yields.  this has  face  understanding  h a v e b e e n t h o u g h t t o be  to  identify  of  demographic processes.  of d e s i g n i n g c c n t r c l s past  really  found f o r  enough  three),  in  ratio. catch-at-age  information  to  of abundance from the m o r t a l i t y  n o t be c r i t i c a l  in identifying  e f f o r t r e g u l a t i o n . In view o f our  the often  optimal dismal  137  a t t e m p t s a t a s s e s s m e n t , t h i s i s a welcome b a s i s f o r o p t i m i s m . .  6.1 Summary.  (1) The a n a l y s i s o f c a t c h - a t - a g e inherent  (2)  indeterminacy  data among  abundance,  m o r t a l i t y , and e x p l o i t a t i o n  rate.  Cohort  an  analysis  natural  reguires  mortality  cohorts. estimate  The  and  independent  limits  least-sguares  natural  involves resolving the natural  estimate  information  to  of  within  approach can, i n p r i n c i p l e ,  mortality,  and  exploits  comparisons  between c c h c r t s . . (3) The  least-sguares  catches-at-age. and  method  Effort  catch-at-age  data,  error  data  variances  also,  d a t a c a n be used t o e s t i m a t e  n a t u r a l m o r t a l i t y by  Erior be  statistically  can  in  and  the  other  reproduction  simultaneously tagging  a  utilizes  statistics, be  analysed  consistent  manner. .  catchability  least-sguares  i n f o r m a t i o n about p o p u l a t i o n  than  parameter  technigue. values can  i n c l u d e d as c o n s t r a i n t s .  (4) The r e l i a b i l t y variance,  o f a s s e s s m e n t i s measured  and c o v a r i a n c e  by  the  o f t h e p a r a m e t e r e s t i m a t e s and  f o r e c a s t . I t depends on t h e i n f o r m a t i o n c o n t e n t data  and  evaluated (5) D a t a  structural  f i t of  by Monte C a r l o  information  bias,  the  model,  of  and  the  c a n be  methods.  content  i s  g u a n t i t y , the magnitude o f data  characterized  by  data  e r r o r s , and t h e l e v e l o f  138  contrast  i n abundance and e x p l o i t a t i o n r a t e s  from which  the data were taken. (6) N a t u r a l data  m o r t a l i t y can be estimated with  guantity  is  high, c o n t r a s t  i s high,  e r r o r s are moderate ( r e l a t i v e v a r i a n c e s 25%), (7) The  when  catch-at-age  on the order  of  and s t r u c t u r a l f i t of the model i s good. c o e f f i c i e n t of v a r i a t i o n f o r f o r e c a s t s of abundance  ranged  from 7% i n a high c o n t r a s t ,  data g u a n t i t y to  precision  9 1%  moderate e r r o r ,  case i n which n a t u r a l m o r t a l i t y  high  was known,  i n a low c o n t r a s t , moderate e r r o r , low g u a n t i t y  case i n which' n a t u r a l m o r t a l i t y (8) To a c c u r a t e l y  was unknown.  determine optimal harvest  not be necessary  to  fully  resclve  p o l i c i e s , i t may  the  among abundance, e x p l o i t a t i o n , and n a t u r a l  indeterminacy mortality.  139  LITEEATURE CITED.  Agger,P., I.Boetius, and H.Lassen. MS 1971. On e r r o r s i n t h e v i r t u a l p o p u l a t i o n a n a l y s i s . I C E S C.M. 1971..Doc. No. H:16 A l l e n , K . R , 1973. The i n f l u e n c e o f random f l u c t u a t i o n s i n the stock-recruitment r e l a t i o n s h i p on t h e e c o n o m i c r e t u r n f r o m s a l r a c n f i s h e r i e s . C o n s . I n t . E x p l o r . Mer Rapp. 164: 350-359 A l l e n , K , R . 1977. W h a l e s , i n F i s h P o p u l a t i o n Dynamics, J.A.Gulland ( e d . ) , J o h n W i l e y and S o n s , New Y o r k , Bard,Y. New  1974. Ycrk,  Nonlinear 341p.  335-358. 372p.  Parameter E s t i m a t i o n . Academic  Press,  Doubleday,W.G. 1S76. A l e a s t - s g u a r e s a p p r o a c h t o a n a l y z i n g c a t c h a t age d a t a . I n t . Comm, N o r t h w e s t A t l . F i s h . . Res. . B u l l . . 12: 6 9-8 1 D r a p e r , N . R . and H . S m i t h , J r . 1966. A p p l i e d J o h n W i l e y and S o n s , New Y o r k , 407p.  Regression  Analysis.  Fry,F.E.J. 1949, S t a t i s t i c s o f a l a k e t r o u t f i s h e r y , 5: 27-67  Biometrics  Gulland,J.A. 1955.. E s t i m a t i o n of commercial fish populations. 18(9).  growth Fish.  and mortality in Invest,, L o n d . (2) ,  G u l l a n d , J . A . MS 1965. E s t i m a t i o n o f m o r t a l i t y r a t e s . Annex to Rep.. A r c t i c Fish. W o r k i n g G r c u p , I n t . C c u n c . E x p l o r . Sea CM. 1S65 (3) : 9p. H i l h o r n , Bay. 1979. C o m p a r i s o n o f f i s h e r y c o n t r o l s y s t e m s utilize catch and e f f o r t d a t a . J . F i s h . Res. Board 36 (1 2 ) : 1477-148S  that Can.,  Hcag,S.H.,and E . J . M c N a u g h t o n . MS 1978. Abundance and fishing mortality c f P a c i f i c h a l i b u t , c o h o r t a n a l y s i s , 1935-1976. I n t e r n a t i o n a l P a c i f i c H a l i b u t Commission, S c i e n t i f i c Eeport No.65 Hilbcrn,E..1980. A comparison of f i s h e r i e s c o n t r o l systems that utilize catch and effort data. Canadian Journal of F i s h e r i e s and A g u a t i c S c i e n c e s , (submitted) J o n e s , R . 1961. The a s s e s s m e n t o f l o n g - t e r m e f f e c t s o f c h a n g e s i n g e a r s e l e c t i v i t y and f i s h i n g e f f o r t . Mar. Bes. (Scotland) 1961 ( 2 ) : 1-19 Jones,R.  1S68. A p p e n d i x t o t h e r e p o r t o f t h e N o r t h - W e s t  Working  140  G r o u p , I n t . Counc. E x p l o r .  Sea  2p.  L e t t , P . F . and T . B e n j a m i n s e n . 1977. A s t o c h a s t i c model for the management of the northwestern Atlantic harp seal ( P a g o p h i l u s g r c e n l a n d i c u s ) p o p u l a t i o n . J , F i s h . Ees.. Board Can., 34: 1155-1 187 Lett,P.F., S.K.Mohn, and E . F . G r a y . MS 1978. Density-dependent processes and management strategy for the northwest Atlantic harp seal p o p u l a t i o n . ICNAF Bes. Doc. 78/XI/84, S e r i a l No. 5299 Mohn,E.K., P . F . L e t t , and E . E e c k . . MS 1978. Some new analysis relevant to the 197S assessment of harp s e a l s . ICNAF Working P a p e r 78/XI/65 Murphy,G.I. 1965. A s o l u t i o n t o t h e Ees. B e a r d Can., 22: 191-202  catch  eguation.  J.  Fish.  Peterman,E.M. 1978. Testing for density-dependent marine s u r v i v a l i n P a c i f i c salmonids. . J. F i s h . Ees. Board Can., 35 (1 1 ) : 1434-1450 1  Pope,J.G. 1972. An investigation cf the accuracy of v i r t u a l population analysis using cohort a n a l y s i s . I n t . . Comm. N o r t h w e s t A t l . F i s h , Ees, B u l l . 9: 65-74 Pope,J.G.. MS 1974. A possible a l t e r n a t i v e method t o v i r t u a l population analysis for the calculation of fishing mortality frcm catch a t age d a t a . ICNAF B e s . Doc. 74/20, S e r i a l No. 3166 B i c k e r , W , E . 1948. Methods o f e s t i m a t i n g v i t a l s t a t i s t i c s o f p o p u l a t i o n s . . I n d i a n a U n i v . P u b l . S c i , S e r . 15: 101p. B i c k e r , W . E . MS 1971. Comments on t h e h e r d and p r o p o s a l s f c r t h e 1972 Meeting cf Panel A experts, 1S71 (unpublished manuscript)  fish  West Atlantic harp seal h a r v e s t . P r e s e n t e d t i ICNAF C h a r l o t t e n l u n d , 23-24 S e p t .  B i c k e r , W . E. 1975. C o m p u t a t i o n and interpretaton of biological s t a t i s t i c s c f f i s h p o p u l a t i o n s . F i s h . . E e s . , B d . Canada B u l l . 191. . 382pp. Bothschild,E.J. 1977, Fishing D y n a m i c s , 96-115. J . A . G u l l a n d New Y o r k , 372p.  effort. In Fish (ed.) , J o h n W i l e y  Sergeant,D.E. MS 1971, C a l c u l a t i o n o f p r o d u c t i o n i n t h e W e s t e r n N c r t h A t l a n t i c . . I C N A F E e s . Doc  Population and Sons,  of harp 71/7  seals  S i l v e r t , W . 1978. The p r i c e of k n o w l e d g e : f i s h e r i e s management as a r e s e a r c h t o o l . J , F i s h , Bes. E o a r d Can,, 35 ( 2 ) : 208-212 Smith,A.D.M. 1979. A d a p t i v e Management of Benewable Besources With U n c e r t a i n Dynamics. . Ph.D. T h e s i s , I n s t i t u t e of Animal  141  Resource  Ecology, University o f B r i t i s h  S o u t h w a r d , G . H . . HS 1976. S a m p l i n g l a n d i n g s composition. International Pacific S c i e n t i f i c R e p o r t No.58  Columbia of halibut f o r age Halibut Commission,  W a l t e r s , C . J . 1S75. O p t i m a l h a r v e s t strategies f o r salmon i n relation to environmental variability and uncertain p r o d u c t i o n parameters. J . F i s h . Res. Board Can., 3 2 ( 1 0 ) : 1777-1784 Walters,C.J. BS 1S76. An a l t e r n a t i v e a n a l y s i s o f s t o c k c h a n g e s i n t h e northwestern A t l a n t i c harp s e a l . I n s t i t u t e of Animal Resource Ecology, University of British Columbia (unpublished manuscript) W a l t e r s , C . J . and Ray H i l b o r n . 1976. A d a p t i v e c o n t r o l o f f i s h i n g s y s t e m s . J . F i s h . R e s . B o a r d C a n . , 33 ( 1 ) : 145-159 W a l t e r s , C . J . and Ray H i l b o r n . 1S78. E c o l o g i c a l o p t i m i z a t i o n a d a p t i v e management. Ann. Rev. E c o l . S y s t . 9:157-188  and  142  APPENDIX A Reparameterization The  problem i s to  and t h e p r o p o r t i o n i n terms  (A) of  of  of  find the  and  a  an a g e - s t r u c t u r e d  for  the Beverton-Holt Age-Structure  the  unfished equilibrium stock size  stock recruited at 8 .  Model  In  t h i s way,  s t o c k - r e c r u i t model  half  the unfished  we c a n i n t e r p r e t  (Nj equilibrium  the  parameters  i n a more b i o l o g i c a l l y m e a n i n g f u l  fashion. Assume r e c r u i t m e n t stock  (P)  i s the  in recruitment  1+  (R)  o c c u r s a t age  population.  is negligible.  We  Also,  0 , and t h a t t h e assume t h a t  breeding  stochastic  have:  P  R = aP + 6 J p =  the unfished  (A-l)  I N . .  j=i At  variation  J  equilibrium:  R = V NOO CO  OO  (A-2)  (  and  Therefore: 1  is  the  to natural Eq.  (A-2)  proportion mortality into  Eq.  of  the  and t h e  (A-l),  we  1+  population  truncation get:  (A-3)  of  d i s a p p e a r i n g each y e a r  t h e model  at  age  J  .  due Substituting  143  V.  N  «>  aN  °°  a  3  +  fl  The  half  equilibrium rate  of  (A-4)  recruitment  is defined  as:  00  a  -FT +  Therefore:  X =  (A-5) a  Substituting  Eq.  (A-4)  into  Eq.  ~Y  (A-5),  + 6  we  obtain:  1  a  2  a  2V  and  so.  \ =  Thus,  combining  i n terms A , we  of  have:  Eq.  a , 3,  (A-4) Sn,  and  and J  (A-6)  Eq. .  (A-6)  For  a  with and  Eq. 3  (A-3)  gives  i n terms  of  N  and and  X  I  JL  a  (A-7)  " A  V  2 JL 3= A " V  For  the  Ricker stock-recruit function,  defined  R = aPe  we  (A-8)  by:  -BP  have:  a  In  (A-9)  (A-10)  and  a  r  (A-ll)  v 2  in  ,  In  (A-12)  

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