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Application of Parker-Larkin equation to growth of fishes and other aquatic organisms Kilambi, Varadaraja Ayyangar 1961

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APPLICATION OP PARKER-LARKIN EQUATION TO GROWTH OF PISHES ANB OTHER AQUATIC ORGANISMS  by KILAMBI VARADARAJA ATTANGAR B.Sc.  (Honours); Andhra U n i v e r s i t y , W a l t a i r  ( I n d i a ) , 1954  Andhra U n i v e r s i t y , W a l t a i r  ( I n d i a ) , 1955  M.Sc.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER  OF  SCIENCE  i n the Department of Zoology  /We accept t h i s t h e s i s as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1961  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  ZOOLOGY  The University of British Columbia, Vancouver 8, Canada. Date  June 30 , 1961  i  ABSTRACT Any mathematical f o r m u l a t i o n f o r d e p i c t i n g the growth of organisms must y i e l d an e m p i r i c a l f i t t h a t i s reasonably  good.  I t s v a l i d i t y i s enhanced i f the equation y i e l d s i n f o r m a t i o n o f biological interest.  This i n v e s t i g a t i o n i s aimed a t a p p l y i n g  the P a r k e r - L a r k i n (1959) growth equation to a number of aquatic organisms t o d e s c r i b e the problems encountered i n making use of t h i s technique.  The data  a l s o analysed by the Von B e r t a l a n f f y  growth equation to b r i n g out the s i m i l a r i t i e s of the- constants of both the equations. The brill,  data p e r t a i n i n g to three species of marine f i s h ,  h a l i b u t and h e r r i n g , four species of freshwater  fish,  rainbow t r o u t , c u t t h r o a t t r o u t and sturgeon and to a l a m e l l i branch species s c a l l o p s , have been analysed. I't i s p o i n t e d out t h a t the exponent of the l e n g t h weight r e l a t i o n s h i p should not be taken as 3. that the length-weight  I t i s shown  r e l a t i o n s h i p of rainbow t r o u t v a r i e s  depending on sex, maturity and s i z e .  In many species the  Parker-SsLarkin growth equation p r e d i c t e d the lengths a t v a r i o u s ages a c c u r a t e l y . overestimated  Von B e r t a l a n f f y s equation p r o g r e s s i v e l y 1  the s i z e s .  In white sturgeon the growth increments f i r s t and then become equal.  decrease a t  In such a s i t u a t i o n i t i s sugges-  ted t h a t the data be s p l i t i n t o two stanzas f o r a n a l y s i s since the a n a l y s i s without  s p l i t t i n g underestimates  e a r l y years and overestimates  the s i z e s i n the  i n the o l d e r ages.  The than  anterior radius  the length  of the f i s h  of  the body-scale  of  the intercept.  keeping  grows  i n herring.  The r e g r e s s i o n  r e l a t i o n s h i p i s used The b a c k  Parker-Larkin  on h a l i b u t .  This  equation  i s because  are  a c t u a l l y calculated values  ion  of weight  is  i n which  range  of best  true  The  only  slower equation  the value  i s made b y  slopes  f o r the  as high  variability and/or  a s 3.6  from  a linear  plot  availability  the  available i s similar i s an inverse  values  of h a l i b u t  logarithmic  equivalent  of the  1«0 a n d 1*5,  approaches  t h e 45°  regress-  Parker-  i s the r e c i p r o c a l of  (l-x).  when t h e diagonal,  s i t u a t i o n s f o r other  f o r z i s obtained.  o f z depending  i s shown f o r r a i n b o w  There  an e x c e l l e n t f i t f o r the  and i n s i m i l a r  material food  gave  o f z between  f i t on a W a l f o r d  a value  population  of lengths  the observed  the slope  of values  f o r salmonids  species  to obtain  variable  on age - a n a l g e b r a i c  equation The  line  with  relatively  fish.  The  Larkin  only  calculation  the intercept constant  individual  data  of the scale  un t h e d e n s i t y  or non-availability trout.  This  of the  of food  dependence  o f z on  o f IJ or ¥ . oo oo r e l a t i o n s h i p b e t w e e n I* a n d z a s t h a t to that  oo of  1^  a n d K.  parameter  I t i s tentatively  suggested  of p h y s i o l o g i c a l importance  that  z might  be a  i n the Parker-Larkin  equation. Further  wwork  o f an experimental  nature  i s suggested  to  e s t a b l i s h the p h y s i o l o g i c a l s i g n i f i c a n c e of the parameters  Qf  the Parker-Larkin  equation.  ACKNOWLEDGMENT S T h i s i n v e s t i g a t i o n was c a r r i e d out a t the I n s t i t u t e of F i s h e r i e s of the U n i v e r s i t y  of B r i t i s h  Columbia,  The author s i n c e r e l y expresses h i s g r a t i t u d e t o Dr. P. A. L a r k i n f o r suggestions and c r i t i c i s m s d u r i n g the investigations• The author a l s o acknowledges Drs. K. S. Ketchen, F. H. C. T a y l o r ,  L. M. D i c k i e , Messrs. L. A. Sunde and C. E .  Stenton f o r k i n d l y p r o v i d i n g herring,  the author w i t h data on b r i l l ,  s c a l l o p s , sturgeon and c u t t h r o a t The a s s i s t a n c e  Centre, U n i v e r s i t y  trout  respectively.  given by the personnel of the Computing  of B r i t i s h Columbia,  i s a l s o acknowledged.  iv  TABLE  GP  CONTENTS Page  TITLE PAGE ABSTRACT  1  ACKNOWLEDGEMENTS  i i : L  TABLE OP CONTENTS.  *  LIST OF FIGURES  •!  LIST OF TABLES  ix  INTRODUCTION  1  BRILL - E o p s e t t a j o r d a n i  5  HALIBUT - Hippoglossus  stenolepis  13  LAKE STURGEON - Acipenser f u l v e s c e n s  29  WHITE STURGEON - Acipenser transmontanous  37  HERRING - Clupea p a l l a s i i  43  Body-scale r e l a t i o n s h i p Growth rate  43 ••  44  CUTTHROAT TROUT - Salmo c l a r k i i  51  RAINBOW TROUT - Salmo g a i r d n e r i i . . .  56  Paul Lake  56  Loon Lake  64  Beaver Lake  67  SCALLOPS - Placopecten magellanicus  72  DISCUSSION AND CONCLUSIONS  81  v  V  Table  of Contents - Cont'd.  Page  SUMMARY  84  LITERATURE CITED  86  vi  LIST Figure  OP  FIGURES  1.  P l o t of l ^ j _ on 1^ f o r female  2.  P l o t of  3.  1.3 1.3 P l o t of l ^ j on 1^ f o r female  4.  1.3 1.3 P l o t of 1^. on 1^ f o r male  5.  Growth curves showing age-weight r e l a t i o n s h i p f o r  +  on 1^ f o r male  Portlock-Albatross 6.  brill.  +  +1  P l o t of for  brill.  brill.  brill.  halibut.  on W^ f o r P o r t l o c k - A l b a t r o s s  1926.  7.  0,5 0.5 P l o t of WI on ¥ f o r Portlock-Albatross f o r 1926.  8  P l o t of ¥  C  t+1  for  t + 1  10.  0.45 on ¥^  f o r Portlock-Albatross  halibut  1956. halibut  P l o t of 1^. ^ on 1^ f o r P o r t l o c k - A l b a t r o s s  halibut  1926. +  1956.  1.5 1.5 P l o t of 1HJ. ^ on 1^ f o r P o r t l o c k - A l b a t r o s s +  for 13.  halibut  on 1^ f o r P o r t l o c k - A l b a t r o s s  for 12.  t + 1  P l o t of 1 + ^ for  11.  on ¥^ f o r P o r t l o c k - A l b a t r o s s  1956.  P l o t of ¥ for  halibut  t  0.45 9.  halibut  halibut  1926.  1.36 1.36 P l o t of l ^ j on 1^ f o r Portlock-Albatross +  for  1956.  14,  P l o t of l . j . i on 1^ f o r male lake sturgeon.  15.  P l o t of 1^.  +  +1  on 1^ f o r female lake sturgeon.  halibut  vii  P l o t of  2.64  2.64 on 1^ f o r male lake sturgeon.  17.  2.64 P l o t of 1^ ^  2.64 on 1^ f o r female lake sturgeon.  18.  P l o t of  19.  P l o t of  F i g u r e 16.  +  on 1^ f o r white sturgeon 1*89  0—6  1.89 on 1^ f o r white sturgeon from  years. 0.9  20.  P l o t of 6-30  21.  22.  o  n  1^  o  n  1^ ^  +  on 1^  25.  from B e l l a  3.1 on 1^  f o r male h e r r i n g from B e l l a  region.  P l o t of l ^ . j on 1^ f o r c u t t h r o a t +  Lake, B• C• 0.94 26.  f o r female h e r r i n g  region.  P l o t of Bella  from B e l l a  3.5  3.1 24.  female h e r r i n g  o r  region.  P l o t of Bella  white sturgeon from  on l ^ f o r male h e r r i n g from B e l l a  3.5 23.  r  region.  P l o t of 1^ ^ Bella  f°  years.  P l o t of Bella  0.9  P l o t of  t r o u t from Kiakho  0.94 °  n  1^  f o r cutthroat  t r o u t from Kiakho  Lake , B. C • 27.  P l o t of l ^ . j ^ on 1^ f o r rainbow t r o u t from Paul Lake,  28.  1.3 1.3 P l o t of l ^ + i on 1^. f o r rainbow t r o u t from Paul Lake,B.C.  29.  P l o t of 1^. TL °  30.  P l o t of  +  +  1.3  n  *°  r  B.C.  rainbow t r o u t from Loon Lake,B.C.  1.3 on 1^ f o r rainbow t r o u t from Loon Lake,B.C.  viii  F i g u r e 31«  P l o t of  on 1^ f o r 1952 year c l a s s rainbow t r o u t  from Beaver Lake, B. G. 32.  P l o t of  o  *t*  n  o r  ^ 5 3 year c l a s s rainbow t r o u t  from Beaver Lake, B. C. 1,14 33.  P l o t of  1.14 on 1^  f o r 1952 year c l a s s rainbow t r o u t  from Beaver Lake, B. C. 0.65 34.  0.65  P l o t of  on 1^  f o r 1953 year c l a s s rainbow  t r o u t from Beaver Lake, B. C. 35.  P l o t of l ^ ] _  36.  0.625 0.625 P l o f ©f * t + l *t * from the Hour ground.  +  °  f o r s c a l l o p s from the Hour ground.  n  0  n  3.6 37.  o r  s c a  H°P  s  °f  years  3.6  P l o t of l^. 2 on 1^ +  f o r s c a l l o p s of 6-9 years from  the Hour ground. 38.  P l o t of  39.  P l o t of  on 1^ f o r s c a l l o p s from the Buoy ground. 0.375  0.375 on 1^ f o r s c a l l o p s from the Buoy  ground. Oft-U s, oS-itr Hatch IAQ j  ix  LIST Table  OP  TABLES  1»  Growth increments between v a r i o u s ages of  2.  A n a l y s i s of variance  brill.  of growth increments at d i f f e r e n t  ages of both sexes of  brill.  3.  Observed and  calculated  4.  Weight i n pounds at ages 5 to 40 f o r P o r t l o c k Albatross halibut i n  5»  6.  1926.  Weight i n pounds at ages 5 to 40 f o r P o r t l o c k Albatross halibut i n  1956.  Length i n centimeters  a t each age  Portlock-Albatross 7.  lengths.  f o r 1926  and  1956  halibut.  A n a l y s i s of variance  on sturgeon f o r growth d i f f e -  rences between sexes. 8.  Observed and  c a l c u l a t e d lengths f o r male sturgeon.  9.  Observed and  c a l c u l a t e d lengths  f o r female sturgeon.  10.  Observed and  c a l c u l a t e d lengths  of white sturgeon  from C a l i f o r n i a waters. 11.  Back c a l c u l a t e d and  c a l c u l a t e d f o r k lengths  for  c a l c u l a t e d f o r k lengths  for  female h e r r i n g . 12.  Back c a l c u l a t e d and male h e r r i n g .  13.  A n a l y s i s of variance in  on o(_values of cutthroat, t r o u t  d i f f e r e n t age i n t e r v a l s .  14.  Observed and  c a l c u l a t e d lengths  15.  A n a l y s i s of variance  16.  Comparison of observed and  of c u t t h r o a t t r o u t .  of growth increments between ages. c a l c u l a t e d lengths  of  Paul Lake rainfepw t r o u t . 17.  Log  l e n g t h l o g weight r e l a t i o n of  various  s i z e s , sexes and  Paul Lake, B.  C.  rainbow  stages of maturity  t r o u t of from  X  Table 18.  Comparison of observed and c a l c u l a t e d lengths of Loon Lake rainbow t r o u t .  19.  Comparison of observed and c a l c u l a t e d s h e l l heights of s c a l l o p s i n m i l l i m e t e r s  20.  from Hour ground.  Comparison of observed and c a l c u l a t e d s h e l l heights of s c a l l o p s from Buoy ground.  21.  Growth parameters of Von B e r t a l a n f f y and ParkerLarkin.  1 INTRODUCTION In the study of the dynamics of f i s h p o p u l a t i o n s , are a number of parameters t h a t must be determined. to the estimations  there  In a d d i t i o n  of m o r t a l i t y r a t e s , age composition e t c . ,  growth r a t e s of f i s h are important since the growth of an organism i s one of the b a s i c determinants of y i e l d * I t i s a common p r a c t i c e to use age as a c r i t e r i o n of s i z e and  growth p o t e n t i a l , even though t h i s i s a r e l i a b l e index only  under s t a b l e environmental c o n d i t i o n s .  Under changing  mental c o n d i t i o n s age can no longer be considered of s i z e .  environ-  as a c r i t e r i o n  L a r k i n , Terpenning and Parker (1957) suggest a method  t h a ^ r e l a t e s growth to s i z e independent of age. contention opportunity  I t i s their  t h a t s i z e gives a b e t t e r i n d i c a t i o n of e c o l o g i c a l f o r growth than does age.  many f i s h may change the "ultimate by changing t h e i r e c o l o g i c a l n i c h e . g i c a l changes i n the l i f e  They a l s o mention that  s i z e " to which they are tending There may a l s o be p h y s i o l o -  of a f i s h t h a t are r e l a t e d to s i z e .  Thus f i s h growth may be considered  as a s e r i e s of c y c l e s or  growth stanzas each of which can be d e f i n e d as a p e r i o d  during  which the parameters used f o r d e s c r i b i n g growth processes can be considered  constant,  w i t h i n reasonable  limits.  Parker and L a r k i n (1959) suggested the use of the d i f f e r e n t i a l equation ^  = kw  i n the d e s c r i p t i o n of growth of chinook  salmon (Oncorhynchus tshawytscha) and steelhead gairdnerii).  t r o u t (Salmo  E s s e n t i a l l y the use of t h i s equation i m p l i e s  that  growth can be t r e a t e d l i k e any other p h y s i o l o g i c a l f u n c t i o n .  2 For  i n s t a n c e , r e s p i r a t i o n r a t e i s commonly r e l a t e d to weight by  the d i f f e r e n t i a l  equation.  where  /AO A t  , x = kw  A  0  represents  oxygen uptake  w  represents  weight  The r e s p i r a t i o n r a t e equation i s u s u a l l y expressed a l g e b r a i c a l l y as:log  l o g k +xlog w  T r e a t i n g the growth equation i n the same way yield:  J=  log  The q u e s t i o n  log k + x log w  (1)  a r i s e s whether growth r a t e i s r e l a t e d to  i n i t i a l weight, average weight or f i n a l weight during t.  would  the p e r i o d  The c l a s s i c s o l u t i o n t o t h i s k i n d of problem i s to deal i n  instantaneous r a t e s , i n t e g r a t i n g the dw/dt =  kw  expression  x  to y i e l d w<;T  x)  = k t ( l - x ) + w^"  Using the expression  x)  (2)  w = q l ^ to denote the r e l a t i o n s h i p between  weight and l e n g t h i t can be demonstrated t h a t growth i n l e n g t h can be d e p i c t e d  where  as:  z = y(l-x)  Setting  l_j._  =  o  0  The above equation can be w r i t t e n as  since  *t+l  hence  1. x=n  2  Taking  °^  =  =ovn  logarithms  and  log>;.l = | l o g < X + |  slope  log 1 against  of i z  regression is  again  o f l e n g t h on  the  right  arbitrary To  a  obviate  ( i . e . n) y i e l d s  with  this  This  the  the  b = — , z presence  of  these  difficulties  b a s e d on  Parker  and  a plot  of  s o l u t i o n i s most c o n v e n i e n t l y  computer  (An ALWAC I I I - E computer was  quadratic  o f z t h a t would m i n i m i z e t h e  Computing C e n t r e , approximate  +  used*  cz^  be  1^, relative  h a n d l e d by  T h i s program i s  U n i v e r s i t y of B r i t i s h  s o l u t i o n can  function = a + bz  for  suggested z  on  o f oi.  An  of z i n b o t h  Larkin z  variance  a t the  solution  z.  a technique  The  The  with  a  s i d e , which would r e q u i r e i t e r a t i o n  valjie  •  line  is essentially  slope  t i m e by  a straight  appropriate  file  an  had  estimate  solution using  using  age  i z (4)  ( l ) above.  made d i f f i c u l t ,  t e r m s on an  log t  analogous to  by  log n  t  plotting  dividing  obtained  a on  Columbia)*  using  the  4. Three values of r e l a t i v e standard d e v i a t i o n (S^) and t h e i r c i a t e d z values on simultaneous  asso-  s o l u t i o n y i e l d the best z value  (the value of z g i v i n g minimum standard d e v i a t i o n ) as  Approximate graphic methods f o r s o l u t i o n are a l s o given by  Parker  and L a r k i n . Carlander and Whitney (1961) mention t h a t there i s a d i f f e r e n t growth p a t t e r n f o r walleyes i n C l e a r Lake which exceed 25.0  inches i n l e n g t h when the older f i s h beyond age VII are  e l i m i n a t e d or 23.9  inches i n l e n g t h when only the f i s h which com-  p l e t e d a given annual increment are c o n s i d e r e d .  They made use  of the P a r k e r - L a r k i n growth equation to give a b e t t e r f i t . The present work d e s c r i b e s problems i n a p p l y i n g these techniques  of growth r e p r e s e n t a t i o n to data f o r v a r i o u s species  of aquatic organisms i n c l u d i n g ( l ) b r i l l (2) h a l i b u t (Hippoglossus  (Eopsetta j o r d a n i ) ,  s t e n o l e p i s ) , (3) lake sturgeon  penser f u l v e s c e n s ) , (.4) white sturgeon (5) h e r r i n g (Clupea p a l l a s i i ) ,  (Aci-  (Acipenser transmontanous),  (6) c u t t h r o a t t r o u t (Salmo c l a r k i i ) ,  (7) rainbow t r o u t (Salmo g a i r d n e r i i ) , and  (8) s c a l l o p s ( P l a c o -  pecten m a g e l l a n i c u s ) • Since the u s e f u l n e s s of any e m p i r i c a l equation i s enhanced i f i t s constants y i e l d i n f o r m a t i o n of b i o l o g i c a l  inte-  r e s t , the present work has t r i e d to draw t e n t a t i v e conclusions on the s i g n i f i c a n c e of the constants i n c l u d e d i n the Larkin  Parker-  equation. The raw data p e r t a i n i n g to the species s t u d i e d and  the  input and output tapes of the computer work are s t o r e d i n the I n s t i t u t e of F i s h e r i e s of the U n i v e r s i t y of B r i t i s h Columbia.  BRILL (Eopsetta 3 ordani) Back c a l c u l a t e d lengths f o r b r i l l were k i n d l y p r o v i d e d by Dr. K. S. Ketchen of the P a c i f i c B i o l o g i c a l S t a t i o n , Nanaimo. Walford  p l o t s of l - j . ^ a g a i n s t 1^ s e p a r a t e l y f o r the +  sexes are shown i n F i g u r e s 1 and 2.  two  The data show a s l i g h t  convergence towards the 45° d i a g o n a l  and i n d i c a t e t h a t an  a p p r o p r i a t e z value f o r the equation l^. ^= <2\+ 1^ would be +  more than 1.  For convenience i n computation, growth i n the  f i v e years only was  considered.  first  Using the q u a d r a t i c method f o r  e s t i m a t i n g minimum r e l a t i v e v a r i a n c e the o r i g i n a l l e n g t h data, and of was  sets of values f o r l ^ * - * 1.3.  and i^«65 y i e l d e d an estimate  of z  Using the ALWAC I I I E Computer the same value of z  obtained. The  and 4.  13 p l o t s of 1^][  The  corresponding  Females  l*^  Males  Combined  lj*^  Mean growth increments Table  I.  a g a i n s t 1^  13  are shown i n F i g u r e s 3  growth formulae a r e : 21.2837 +  3 =  l**  3  =3 20.4991 +  l^*  3  «=  l**  3  20.8964 +  between the v a r i o u s ages are given i n  8  * ' o o •  •. • • •  . • • • ••  •  o° o • •  ••  s  •  t  •  •• • • • •  50  100  Z* 1.3  t  1.3 Figure 3.  P l o t of l  t  +  1  1.3 on  l  t  f o r female  brill.  9  10 Table I .  Growth increments between v a r i o u s ages of  brill  ^ 1 2  C^23  ^ 4  ^45  Females  22.3030  22.0045  19.9550  20.9125  Males  22.4320  18.9295  21.6280  19.0070  Combined  22.3675  20.4670  20.7915  19.9597  A n a l y s i s of v a r i a n c e (Table I I ) i n d i c a t e s no s i g n i f i c a n t  diffe-  rences between sexes or ages and no s i g n i f i c a n t i n t e r a c t i o n ,  i.e.  t h e r e i s no s i g n i f i c a n t d e p a r t u r e from the average growth r a t e a t v a r i o u s ages or f o r e i t h e r sex o£ f o r any p a r t i c u l a r sex a t any p a r t i c u l a r Table I I .  age.  A n a l y s i s of v a r i a n c e of growth increments d i f f e r e n t ages of b o t h sexes of  Source of Variance  d.f.  Total  159  Means  7  Mean Square  at  brill.  F ratio  Probability  152  41.3912  Sexes  1  25.2571  0.610  >  Ages  3  43.1577  1.042  ^=-0.25  Interaction  3  44.5880  1.077  >-0.25  Individuals  0.25  z  The a n a l y s i s suggests t h a t the e q u a t i o n 1. ,  z =0<C +  1,  can u s e f u l l y be a p p l i e d t o d e s c r i p t i o n of and comparison of growth r a t e s of b r i l l .  However, the c o n v e r s i o n from z t o x  s h o u l d not be based on the assumption t h a t y = 3 .  The length-  11 weigth r e l a t i o n s h i p f o r male and female b r i l l ,  provided  by  Dr. K. S. Ketchen a r e : Male  log ¥  ( Q i )  =  3.1349  log l  (  m  m  )  -  6.6982  Female  log ¥  ( 0 z )  =  3.3523  log l  (  m  m  )  -  7.2478  S u b s t i t u t i n g these r e g r e s s i o n c o e f f i c i e n t s f o r y i n the r e l a t i o n z = y ( l - x ) the values of the exponent x f o r males and females are 0.58  and 0.61  respectively.  The data were a l s o analysed by u s i n g the  von  B e r t a l a n f f y equation a c c o r d i n g to the method d e s c r i b e d by R i c k e r (1958)*. from the  Lengths at v a r i o u s ages could be  calculated  equations.* Females  l  t  +  1  =  8l(.104j +  0.8958 l  t  Males  l  t  +  1  =  85 (• 10,4) +  0.8958 l  t  Observed and c a l c u l a t e d lengths by P a r k e r - L a r k i n and  Von  B e r t a l a n f f y ^ e q u a t i o n s ^ a r e given i n Table I I I .  *The d i f f e r e n t i a l equation of von B e r t a l a n f f y , can be shown to y i e l d the e x p r e s s i o n TKl,  l^.e~  .  By p l o t t i n g 1^ ^ +  constants 1  1+^  =  l  = HS Q O  —Xt (l-e~ )  KW +  a g a i n s t l ^ O f a l f o r d p l o t ) the  and K can be estimated.  Alternatively, plotting  OO  l o g j ^ l ^ - l o g ^ l - l ^ . ) a g a i n s t t g i v e s a l i n e of slope-K O Q  y i n t e r c e p t (-Kto).  T r i a l values of 1  provide a best f i t f o r the  equation.  and  can be chosen to  12  Table I I I .  Observed and c a l c u l a t e d  lengths.  T o t a l l e n g t h i n centimeters  Age i n Observed  years Males  Females  Parke r - L a r k i n  Von  Bertalanffy  Males  Females  Males  Females  1  10.5  10.8  10.68  10.73  10,50  10,80  2  18.3  18.3  17.80  18.17  18.14  18.54  3  24.0  25.0  24.14  24.63  25.09  25.04  4  30.4  30.6  30.00  30.69  31.32  30.87  5  35.5  36.2  35.53  36.40  36.90  36.09  Both equations do an adequate job of p r e d i c t i o n although the Von  B e r t a l a n f f y expression tends to p r o g r e s s i v e l y  lengths.  overestimate  13 HA Iii BUT.  (Hippoglossus  stenolepis).  H a l i b u t of P o r t l o c k - A l b a t r o s s grounds are used f o r the present i n v e s t i g a t i o n . 1956  Growth r a t e s estimated f o r 1926  and  are obtained from Table5, page 15 of the 28th r e p o r t of  the 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.  Average weights  a t each age g i v e n by the Commission's r e p o r t were s t a t e d to have been obtained by c o n v e r t i n g the lengths to weight by u s i n g a length-weight  table.  Average weight i n pounds at each age of P o r t l o c k A l b a t r o s s h a l i b u t f o r 1926  and 1956  p l o t of W ^ ^ a g a i n s t W^. f o r 1926 +  i s shown i n F i g u r e 5.  The  data of Figure 6 appear to  diverge from the 45° d i a g o n a l l i n e .  When data on weights are  used the minimum r e l a t i v e v a r i a n c e w i l l y i e l d an optimum value of (1-x) as i s e v i d e n t from the (1-x) ¥  t+l  equation.  (1-x)  38  k  (  l  "  x  )  +  T  t  Here f o r the sake of convenience O^" and thus we  have  (1-x) ¥  t+l  k ( l - x ) i s denoted as  ,  (1-x)  - C*  +  *  t  6}ptimum value of (1-x) obtained from 1926  The  data was  0.5  and the corresponding growth formula i s : 0.5 Vl  0,5 = G  2  7  6  +  The p l o t of w j j '  V  t a g a i n s t W^*  5  i s shown i n Figure;. 7  Making use of the above growth formula.weights  a t v a r i o u s ages  computed and are shown i n Table 4 along w i t h the weights. good.  •  observed  Agreement between observed and c a l c u l a t e d weights i s  T h i s i s h a r d l y s u r p r i s i n g s i n c e the observed values are  14 450  400  350  300  o  250  1956  200 (-  o  ° o  o o  150 \-  o o o  100  °  °  50  o  o  1926  o  n  °  0  o  O  o  •0  o  _ O _ o o " o °  15  5.  Growth for  O  °  o °  20 AGE  Figure  o  o °  °  „ o o  o  o  o  o O  „ °  o  o  curves  25 IN  30  35  YEARS  showing  Portlock-Albatross  age-weight halibut.  relationship  40  15  16  Table  4 .  Weight i n pounds a t ages 5 to 40 f o r P o r t l o c k A l b a t r o s s h a l i b u t i n 1926 Weight  Age i n years 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  Observed 3 4 5 6 8 10 12 14 16 18 21 24 26 29 32 36 39 42 46 50 54 58 62 67 71 76 81 85 90 96 101 107 113 118 124 130  Calculated 3.0 4.0 5.2 6.5 8.0 9.7 11.5 13.4 15.5 17.8 20.2 22.7 25.4 28.3 31.3 34.5 37.8 41.3 44.9 48.7 52.6 56.7 60.9 65.3 69.8 74.5 79.4 84.4 89.5 94.8 100.2 105.8 111.6 117.5 123.6 129.8  18 a c t u a l l y c a l c u l a t e d values from a l i n e a r l o g a r i t h m i c r e g r e s s i o n of weight on age - an a l g e b r a i c e q u i v a l e n t of the P a r k e r - L a r k i n equation i n which the slope i s the r e c i p r o c a l of ( 1 - x ) . However the c a l c u l a t i o n s confirm the u s e f u l n e s s of the ParkerLarkin, equation i n d e s c r i b i n g the growth of an average A p l o t of ¥j. ^ a g a i n s t  f o r 1956  +  F i g u r e 8.  fish.  data i s shown i n  In t h i s data a l s o the l i n e of best f i t diverges from  the 45° d i a g o n a l .  A value of 0.45  was  obtained f o r ( 1 - x ) .  I t would be noted t h a t a change i n 0.05  f o r the value of ( l - x )  would r e s u l t i n an enormous change i n o(| . as (1-x) the mean value of 0 6 was a value of 0.5 i n F i g u r e 9.  f o r (1-x).  For example, f o r  0.385 whereas i t was  A p l o t of  0.45  against  0.539 f o r  0.45 i s given  The formula f o r d e p i c t i n g weight i s : V  0.45 t + 1  =  0.385 +  ¥  0.45 t  Weights a t v a r i o u s ages are t a b u l a t e d i n Table The average weights  a t each age f o r 1926  converted to lengths a t age from a length-weight expressed  0.45  5.  and 1956  were  relationship  as: l o g W = 3.0417 l o g L  T h i s r e l a t i o n s h i p was f o r average  « 4.70054 obtained by f i t t i n g a r e g r e s s i o n  l e n g t h i n centimeters and average weight i n pounds  f o r ages from 4 to 25.  The data madeouse of here are given i n  Table 4 of the 8th r e p o r t of the I n t e r n a t i o n a l P a c i f i c e H a l i b u t Commission  1934.  F i g u r e 8.  P l o t of ¥  t  +  1  on ¥  t  halibut for  for Porilock-Albatros 1956.  20  21  Table  5 .  Weight i n pounds at ages 5 to 40 for PortlockAlbatross halibut i n 1956 Weight  Age i n years 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ,39 40  Observed 5 7 10 13 17 21 26 32 38 44 52 60 68 77 87 98 108 120 132 145 159 174 188 204 221 237 255 274 293 313 333 354 376 400 423 445  Calculated 5.0 7.3 10.1 13.4 17.3 21.6 26.5 32.0 38.1 44.7 51.9 59.7 68.1 77.2 86.8 97.1 108.0 119.6 131.8 144.6 158.1 172.4 187.2 202.8 219.0 236.0 253.5 271.9 291.2 310.7 330.9 352.3 374.0 396.9 419.9 444.3  22 Walford p l o t s of 1^ ^ a g a i n s t 1^ are shown i n F i g u r e s 10 +  and 11 f o r 1926 and 1956 r e s p e c t i v e l y .  In both p l o t s the  l i n e o f best f i t converges to the 4 5 ° d i a g o n a l . -z estimated are 1.5 and 1.36 r e s p e c t i v e l y .  Walford  The values of  These estimates of  z are obtained from the expression y(l-x) = z 1.5 1.5 Transformed p l o t s o f 1 ^ a g a i n s t 1^ f o r 1926 and 1.36 ' lj. f o r 1956 are g i v e n i n F i g u r e s 12 and 13. +1  1  1.36 t+l & a  ains  fc  The formulae 1.5 1  t+l  = 5 5 , 3 1+  f o r d e p i c t i n g the growth r a t e s a r e : 1.5  H  1 9 2 6  1.36 1.36 l+^i = 47.76 + 1.  1956  Observed and c a l c u l a t e d lengths o f P o r t l o c k - A l b a t r o s s h a l i b u t f o r the years 1926 and 1956 are presented i n Table 6. The agreement between the observed  and c a l c u l a t e d lengths i s  excellent. The Von B e r t a l a n f f y equation could not be a p p l i e d t o the weight data as the l i n e of best f i t on the Walford d i v e r g e s from the 4 5 ° d i a g o n a l . data f o r both the y e a r s .  plot  I t was used f o r the l e n g t h  Average s i z e s can be obtained by  u s i n g the f o l l o w i n g equations. l  t  +  1  o 232(0.14) + l  t  0.8607  .1926  l  t  +  1  = 400(0.21) + l  t  0.7866  1956  These two equations give an overestimate  of lengths.  Judging from F i g u r e s 5-9 and the values obtained f o r (l-x)  i t may be concluded t h a t h a l i b u t grew f a s t e r i n weight  23  halibut for 1956,  27 Table  6 .  Length i n centimeters a t each age f o r 1926 and 1956 Portlock-Albatross halibut 1926  Age i n years 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  Observed 50.4 55.4 59.6 63.3 69.5 74.8 79.5 83.6 87.4 90.8 95.5 99.8 102.5 106.2 109.7 114.0 117.1 120.1 123.6 127.0 130.3 133.4 136.3 139.9 142.6 145.8 148.9 151.3 154.1 157.4 160.1 163.1 166.1 168.5 171.2 173.9  1956 Calculated 50.4 55.5 60.3 64.9 69.5 73.8 78.0 82.2 86.2 90.1 94.0 97.7 101.4 105.1 108.6 112.1 115.6 119.0 122.4 125.6 129.0 132.1 135.3 138.5 141.6 144.7 147.7 150.7 153.8 156.7 159.7 162.5 165.4 168.3 171.1 173.9  Observed 59.6 66.6 74.8 81.6 89.1 95.5 102.5 109.7 116.1 121.8 128.7 134.9 140.5 146.4 152.4 158.5 163.6 169.4 174.8 180.3 185.8 191.4 196.3 201.7 207.1 211.9 217.0 222.2 227.2 232.2 237.0 241.8 246.6 251.7 256.3 260.7  Calculated 59.6 67.4 75.0 82.3 89.3 96.3 102.9 109.5 116.0 122.2 128.4 134.5 140.4 146.3 152.1 157.8 163.5 169.1 174.5 180.0 185.4 190.7 196.0 201.6 206.4 211.6 216.6 221.6 226.7 231.7 236.5 241.4 246.3 251.2 256.0 260.7  28 for true the  the p e r i o d  o f 1956  compared  f o r growth i n l e n g t h . indications  to that  o f 1926.  The  same i s  S m a l l e r v a l u e s o f (1-x) a n d z a r e  of f a s t g r o w t h  rate.  29 LAKE STURGEON (Acipenser  fulvescens)  Back c a l c u l a t e d lengths of Nelson R i v e r lake sturgeon were k i n d l y provided by Mr. L. A. Sunde of the Department of Mines and N a t u r a l Resources, Manitoba.  The o l d e s t f i s h from which  back c a l c u l a t i o n s were made was 55 y e a r s . venience used.  F o r computational  con-  only the f i r s t 21 years of back c a l c u l a t e d growth were  Sexes were t r e a t e d s e p a r a t e l y .  Walford p l o t s of 1^. ^ +  on 1^ f o r males and females are shown i n F i g u r e s 14 and 15 respectively.  In both cases the l i n e of best f i t approaches  the 45° d i a g o n a l . was 2.64.  The value of z f o r both males and females 2.64 2.64 The transformed p l o t of l^. 2 a g a i n s t 1^ for +  males and females are shown i n F i g u r e s 16 and 17 r e s p e c t i v e l y . Equations t o express the growth r a t e s a r e : 2.64 2.64 Males l = 794.11 + l t  +  1  t  2.64 Females l  t  +  1  2.64 « 820.11 +- l t  Lengths a t v a r i o u s ages were c a l c u l a t e d u s i n g the above equations  and are given i n Tables 8 and 9.  agreement between observed  There was good  and c a l c u l a t e d l e n g t h s .  There were no s i g n i f i c a n t d i f f e r e n c e s i n the growth r a t e s of the sexes (Table 7 ) . Table 7.  A n a l y s i s of v a r i a n c e on sturgeon f o r growth d i f f e r e n c e s between sexes.  Source of Variance d.f. Total Sexes  39 1  Individual Ls38  Sum of squares Mean square F r a t i o 1702641.10 6733.20  6733.20  1695907.90  44629.15  0.15  Probability  ^..01  30  32  34 Growth i n l e n g t h was apparently r a p i d during the f i r s t year and decreased males.  s t e a d i l y to the age of 9 i n females and 8 i n  Classen (1944) observed marked changes i n growth of  Acipenser s t u r i o a t the ages of 8 and 9 y e a r s and a t t r i b u t e d the occurrence  to a l t e r a t i o n of the general metabolism, due to  development of gonads.  C u e r r i e r and Rowssow (1951) r e p o r t e d  t h a t male lake sturgeon matured s e x u a l l y a t approximately years of age and females (1954) observed of age.  i n about 25 y e a r s .  14  Probst and Cooper  i r r e g u l a r i t y i n growth between 14 and 19 years  The i r r e g u l a r i t y i n growth beyond the 9th year of age i n  Nelson R i v e r lake sturgeon cannot be explained because of the l a c k of i n f o r m a t i o n e i t h e r on the environment or on the gonad development. Since the l i n e s of best f i t on a Walford p l o t tend to converge to the 45° d i a g o n a l , the data were analyzed by using the Von B e r t a l a n f f y equation. -Males Females The  The formulae  obtained were  l  t  +  1  = 84(0.020) + 0.980 l  t  l  t  +  1  = 76(0.023) + 0.977 l  t  p r e d i c t e d lengths a t v a r i o u s ages by Von B e r t a l a n f f y  equation f o r both the sexes are given i n Tables 8 and 9. I t can be seen from the t a b l e s t h a t the lengths are overestimated p r o g r e s s i v e l y by the equation.  35 Table  8.  Observed and c a l c u l a t e d lengths f o r male sturgeon Fork l e n g t h i n inches  Age i n years  Observed  Parke r - L a r k i n  Von  Bertalanffy  -  1  6.70  6.85  2  11.16  13.45  3  15.29  16.92  4  18.49  19.49  5  21.21  21.64  24.2  6  23.43  23.43  25.4  7  25.19  25.04  26.6  8  26.70  26.50  27.7  9  28.29  27.84  28.8  10  29.89  29.08  29.9  11  31.21  30.24  31.0  12  32.41  31.33  32.1  13  33.94  32.36  33.1  14  34.29  33.34  34.1  15  35.13  34.27  35.1  16  35.89  35.17  36.1  17  36.61  36.03  37.0  18  37.32  36.85  37.9  19  37.88  37.66  38.9  20  38.39  38.41  39.8  21  38.85  39.17  40.6  Table  9.  Observed c a l c u l a t e d lengths f o r female  sturgeon  Fork l e n g t h i n inches. Calculated Age i n years  Observed  Parker-Larkin  Von B e r t a l a n f f y  1  7.53  8.53  2  13.40  13.90  3  17.21  17.35  4  20.01  19.91  5  22.34  22.02  24.5  6  24.29  23.85  25.7  7  26.23  25.46  26.9  8  28.01  26.93  27.9  9  29.02  28.28  29.1  10  30.20  29.52  30.1  11  31.26  30.69  31.2  12  32.20  31.79  32.2  13  33.06  32.83  33.2  14  33.98  33.82  34.1  15  34.86  34.77  35.1  16  35.72  35.67  36.0  17  36.53  36.53  36.9  18  37.33  37.37  37.8  19  38.01  38.17  38.6  20  .38.63  38.95  39.5  21  39.33  39.70  40.3  mm  -  -  37 WHITE STURGEON (Acipenser transmontanous) Data f o r t h i s study was procured from Pycha's (1956) p u b l i c a t i o n on white  sturgeon.  T o t a l lengths i n inches a t  capture of the v a r i o u s age groups were used to express growth r a t h e r than back c a l c u l a t e d l e n g t h s . shown i n F i g u r e "18. white  A p l o t of 1^. ^ on 1^. i s +  I t can be seen from the Walford p l o t t h a t  sturgeon puts on l a r g e annual increments f o r the f i r s t  6 years and t h e r e a f t e r there are r e l a t i v e l y constant i.e.  increments,  the l i n e of best f i t almost runs p a r a l l e l to the 45°  diagonal.  In an a n a l y s i s using lengths a t ages from 0-30  years z was estimated as 1.45. But the estimated lengths were under-estimated o l d e r ages.  f o r the e a r l y ages and overestimated f o r the  Since the l i n e of best f i t on Walford p l o t  p a r a l l e l t o the 45° l i n e from a l e n g t h of 38.5 inches ponding  runs  (corres-  to 6 years of age) the data was s p l i t a t t h i s s i z e and  analysed s e p a r a t e l y .  The P a r k e r - L a r k i n equations f o r d e p i c t i n g  lengths up to the 6th year and from the 6 t h year onward were respectively.  ^  Vi  - 151  15+  h  1.89  and 0.9  Vl Transformed  =-  l 4037  +  h  1.89 p l o t s of 1^. ^ +  0.9 1.89 0.9 a g a i n s t 1^ and 1^. ^ a g a i n s t +  0.9 1^.  are shown i n F i g u r e s 19 and 20 r e s p e c t i v e l y .  Lengths a t  v a r i o u s ages estimated by the above equations are t a b u l a t e d i n Table 10.  This example has provided an, e x c e l l e n t agreement  between observed and c a l c u l a t e d l e n g t h s .  The a n a l y s i s u n d e r l i n e s  the f a c t t h a t the data should be s p l i t a t a l e n g t h of 38.5 inches as the white  sturgeon f o l l o w s a d i f f e r e n t growth p a t t e r n above  Figure 18.  Plot of lt+1  o n  h  f o r  white sturgeon.  39  40  41 Table  10.  Observed and c a l c u l a t e d lengths of white sturgeon from C a l i f o r n i a waters  n years  Calculated  Observed  Parker-Larkin  Von B e r t a l a n f f y  0  10.5  10.5  10.5  1  18.0  18.0  16.9  2  23.0  23.4  22.9  3  28.0  27.9  28.5  4  32.0  31.8  33.8  5  35.3  35.3  38.9  6  38.5  38.5  43.6  7  41.0  40.6  48.0  8  43.6  43.0  52.2  9  45.8  45.3  56.1  10  47.9  47.6  59.9  11  50,0  49.9  63.4  12  52.2  52.2  66.7  13  54.5  54.5  69.8  14  56.8  56.8  72.7  15  59.0  59.1  75.4  16  61.2  61.5  78.0  17  63.6  63.8  18  66.0  66.2  80.5 ;82.8  19  68.3  68.6  84.9  20  70.7  70.9  86.9  21  73.1  73.3  88.9  22  75.5  75.7  90.7  23  78.0  78.1  92.4  24  80.4  80.5  93.9  25  82.8  82.9  95.5  26  85.2  85.4  96.9  27  87.7  87.8  98.3  28  90.2  90.2  99.5  29  92.8  92.7  100.7  30  95.3  95.1  101.8  42 this  size. Since the l i n e of best f i t ana Walford p l o t f o r sturgeon  above 6 years of age l i e s approximately p a r a l l e l to the 45° d i a gonal, the a p p l i c a t i o n of the B e r t a l a n f f y equation i s not p o s s i b l e . Beverton and H o l t (1959) t a b u l a t e d the values of R and *  I* as oo  0»06 and 300 centimeters (120 inches) r e s p e c t i v e l y . T h i s could only be p o s s i b l e i f the data on the o l d e r f i s h was  ignored.  The  Von B e r t a l a n f f y equation f o r e s t i m a t i n g growth w i t h these constants i s : l  t  +  1  = 120(0.0582) + l ( 0 . 9 4 1 8 ) t  Lengths a t v a r i o u s ages p r e d i c t e d by t h i s equation are g i v e n i n Table are g r o s s l y  10.  I t i s evident from the t a b l e t h a t the lengths  overestimated a t a l l ages above 4 y e a r s .  43 HERRING (Glupea p a l l a s i i ) . Scales of h e r r i n g from the B e l l a B e l l a area were kindlyprovided  by Dr. F. H. C. Taylor of the P a c i f i c B i o l o g i c a l S t a t i o n  a t Nanaimo.  Scales were read and the back c a l c u l a t e d  were used f o r the present were f i v e years of age  investigation.  caught i n 1955  lengths  F i s h used i n t h i s  study  from the B e l l a B e l l a r e g i o n .  Body-scale - r e l a t i o n s h i p Measurements of 290 l e n g t h from 112  to 244  s c a l e s from h e r r i n g ranging  m i l l i m e t e r s were recorded.  A  i n fork  regression  l i n e to show the r e l a t i o n s h i p of f o r k l e n g t h to the a n t e r i o r r a d i u s of the magnified s c a l e image was appropriate  constructed  and  the  formula i s : l o g L = 0.88380 + 0.705 l o g S  The  c o r r e l a t i o n c o e f f i c i e n t f o r t h i s data i s 0.93  i s highly s i g n i f i c a n t .  The  slope 0.705 i s s i g n i f i c a n t l y  r e n t from u n i t y ( t = 17.56)} hence the use i n back c a l c u l a t i o n would not be v a l i d .  who  are p a r t l y explained  diffe-  of d i r e c t p r o p o r t i o n  Apparently the a n t e r i o r  r a d i u s of the s c a l e grows r e l a t i v e l y slower than l e n g t h . observations  which  These  by the work of Guyn (1939)  observed on P a c i f i c h e r r i n g t h a t the growth r a t e of the  a n t e r i o r f i e l d of the s c a l e i s g r e a t e r than t h a t of body l e n g t h up to a body l e n g t h of about 40-50 mm.  Thereafter  to become l e s s than body length growth r a t e .  i t decreases  A f t e r the s i x t h  year the s c a l e again grows f a s t e r than the body l e n g t h . most convenient means of c a l c u l a t i n g the annual growth of body from the growth of the  s c a l e s would appear to be  c o n s t r u c t a nomograph which took cognizance of the  The the  to  changing  44 s c a l e to body l e n g t h r e l a t i o n s h i p . However, i f one  i s d e a l i n g only with the c e n t r a l p e r i o d  of growth as i n the present case back c a l c u l a t i o n s to above 5 cm.  should be accurate.  There are two  lengths  p o s s i b l e ways  of back c a l c u l a t i n g the lengths at previous ages.  (l)  By  assuming the slope i s constant and the i n t e r c e p t i s v a r i a b l e and  (2) by keeping the i n t e r c e p t constant  on the assumption t h a t  i n d i v i d u a l f i s h have a d i f f e r e n t slope from the other.  The  f i r s t method i s o b v i o u s l y r i d i c u l o u s i n such v a r i a b l e data, for  i n back c a l c u l a t i o n the ranges i n s i z e s at the end of the  f i r s t year would be enormous.  A c c o r d i n g l y , a l l the back c a l c u -  l a t i o n s are made by the second method, keeping the i n t e r c e p t constant and assuming v a r i a b l e s l o p e s . Growth Rate. Back c a l c u l a t e d f o r k lengths i n m i l l i m e t e r s of the 5 year  o l d h e r r i n g are used.  Walford  p l o t s of 1^.  +1  i n F i g u r e s 21 and 22.  The  sexes are t r e a t e d s e p a r a t e l y .  a g a i n s t 1^ f o r females and males are shown In both the f i g u r e s the p o i n t s could  y i e l d a l i n e of best f i t t h a t would i n t e r s e c t the 45° The values of z f o r the sexes were estimated ing  equations  Females  l  Males  l  t  3.5 +  1  = 8856.22 + l  1  = 2021.48 + l  3.1 t  Corresponding  t  3.1 +  z  The  correspond-  are:3.5  and 24.  and the  diagonal.  t  z  p l o t s of 1^. ^ +  on 1^ are shown i n F i g u r e s  23  data were analysed u s i n g the Von B e r t a l a n f f y equation  45  22.0  20.0  o ooo  og o  QO  U  o 18.0  oo o ooo ° <ef o . o oo 3  % oo o oo  16.0  oo o° ooo  °oo o  O  m  gp  sP  o g  14.0  12.0  10.0  8.0 80  Figure  10.0  21.  12.0  P l o t of l  14.0  t  +  1  on  16.0  l  t  18 0  20 0  22  f o r female h e r r i n g from  Bella Bella  region.  46  22.0  10.0  Figure  22.  12.0  Plot  of  14.0  1^  on Bella  16.0  l  t  18.0  f o r male  Bella  20.0  herring  region.  from  22.0  47  3.5 F i g u r e 23.  P l o t of l  t  +  1  3.5 on  l  t  f o r female h e r r i n g from  B e l l a B e l l a region.  48  49 and the growth r a t e s can be d e p i c t e d from the f o l l o w i n g Females  l  t  +  1  = 25.4(.190) + 0.810 l  t  Males  l  t  +  1  m 27.6(.122) + 0.878„l  t  formulae:  Lengths a t higher ages c a l c u l a t e d by P a r k e r - L a r k i n and Von B e r t a l a n f f y s equations 1  Both the growth equations well.  are given i n Tables 11 and 12.  seem to p r e d i c t the lengths e q u a l l y  By u s i n g the P a r k e r - L a r k i n method lengths a t e a r l i e r  ages which do not l i e on a s t r a i g h t l i n e  on the Walford  plot  could a l s o be p r e d i c t e d , whereas a p p l i c a t i o n of the Von B e r t a l a n f f y * method i s confined to the l a s t three years of life  of the 5 year o l d h e r r i n g .  50 Table 11.  Back c a l c u l a t e d and c a l c u l a t e d f o r k lengths f o r female  Age i n  Back calculated  years  herring.  Calculated Parker - L a r k i n  Von B e r t a l a n f f y  1  10.71  10.80  2  15.29  14.98  3  17.77  17.37  17.77  4  19.41  19.15  19.26  5  20.56  20.58  20.41  Table  12.  Back c a l c u l a t e d and c a l c u l a t e d f o r k lengths f o r male h e r r i n g .  Age i n years  Back calculated  Calculated Parker - L a r k i n  Von B e r t a l a n f f y  1  10.32  10.31  2  14.24  15.09  3  16.48  16.06  16.06  4  18.04  17.77  17.61  5  19.05  19.19  18.99  51 CUTTHROAT TROUT (Salmo c l a r k i i ) Back c a l c u l a t e d lengths of 5 year o l d c u t t h r o a t t r o u t caught i n 1958 from Kiakho Lake, B. C. were k i n d l y provided by Mr.  C. E . Stenton, P i s h and Game Branch of B. C.  p l o t of  l.fc+1 <>  n  1^ I s given i n F i g u r e 25.  A Walford  The general t r e n d  of p o i n t s f o r c u t t h r o a t t r o u t i s p a r a l l e l t o the 45° d i a g o n a l suggesting- t h a t z = 1. value of 1.01j 0.94 *t+l  o  n  The q u a d r a t i c s o l u t i o n f o r z y i e l d s the  the computer s o l u t i o n was 0.94.  0.94 """t *"* ^ ss  o w n  *  n  The p l o t of  f i g u r e 26.  While the use of the P a r k e r - L a r k i n equation would permit more accurate p r e d i c t i o n of growth than a Von B e r t a l a n f f y l i n e f i t t e d on the Walford  t r a n s f o r m a t i o n , i t i s obvious  t h a t the  P a r k e r - L a r k i n equation does not e l i m i n a t e the "hump" i n the s c a t t e r of p o i n t s which occurs between the lengths of 10 t o 20 centimeters.  Table 13 gives the a n a l y s i s of v a r i a n c e of  values f o r v a r i o u s age i n t e r v a l s , the s i g n i f i c a n t F value r e f l e c t i n g the r e a l e x i s t e n c e of the "hump". could be o f f e r e d f o r t h i s hump:  Two explanations  ( l ) there i s an i n f l e c t i o n i n  growth r a t e a t about 15 cm i n which case the data should be s p l i t a t the i n f l e c t i o n and the two p a r t s t r e a t e d s e p a r a t e l y or (2) because d i f f e r e n t environmental  c o n d i t i o n s may have  p r e v a i l e d i n d i f f e r e n t y e a r s , the year of growth which l a r g e l y corresponds  t o the hump may have been p a r t i c u l a r l y  of the other years of growth  unfavourable.  favourable  53  54 Table  13 •  A n a l y s i s of v a r i a n c e  on ^  t r o u t i n d i f f e r e n t age  Source of variance  d.f.  Sum  values  of c u t t h r o a t  intervals.  Mean square  of squares  Total  91  168.3548  Mean  3  61.4352  20.4784  88  106.9196  1.2149  Individual  F = 20.4784 = 1.2149 Both explanations  f i n d support  16.8  P  =  <.01  i n f i e l d data.  In t h e i r  first  y e a r . Kiakho Lake f i s h r e s i d e i n an o u t l e t stream, migrating y e a r l i n g s between 10 and  15 centimeters  i n t o the l a k e .  Hence  there would be some j u s t i f i c a t i o n f o r s p l i t t i n g the data r e p r e s e n t a t i v e of the two  environments, j u s t as Parker  L a r k i n (1959) d i d f o r steelhead t r o u t .  as  as  and  On the other hand, the  stream environment i s s t r i k i n g l y v a r i a b l e from year to year i n i t s f a v o u r a b i l i t y f o r growth and  s u r v i v a l of young c u t t h r o a t .  A c c o r d i n g l y , d i f f e r e n t year c l a s s e s enter the lake a t d i f f e r e n t s i z e s and  strikingly different densities.  Each year c l a s s then  would show a p a t t e r n of growth r e f l e c t i n g the p a r t i c u l a r c o n d i t i o n s t h a t p r e v a i l e d i n the environment during i t s l i f e . T h i s i s apparently 1958  true because 4 year o l d c u t t h r o a t caught i n  show no hump at 10 to 20 centimeters.  y i e l d an estimate  of 0.7  f o r z (Stenton  Moreover, they  I960) which would  r e f l e c t good growth c o n d i t i o n s f o r l a r g e r f i s h combined w i t h poor growth c o n d i t i o n s f o r smaller The  fish.  a n a l y s i s u n d e r l i n e s t h a t adequate e s t i m a t i o n of z  55 hinges upon u n i f o r m i t y of environment.  When the environment i s  variable,, z could be c a l c u l a t e d from observed increments i n growth i n the same year of f i s h of v a r i o u s s i z e s *  In the Kiakho Lake  s i t u a t i o n the added p r e c a u t i o n might be taken of s p l i t t i n g the growth i n stream and lake environments.  Having estimated z i n  t h i s wayj> C>C values f o r a p a r t i c u l a r year are i n d i c e s of environmental c o n d i t i o n s (as they should be a c c o r d i n g t o Parker and L a r k i n ) .  T h i s procedure runs the r i s k of b i a s from  selection  f a s t growing f i s h by the f i s h e r y but i t seems a l e s s e r e v i l than spurious e s t i m a t i o n of z from f l u c t u a t i n g environmental c o n d i t i o n s I t i s a l s o c o n s i s t e n t w i t h the c o n t e n t i o n that z i s a p h y s i o l o g i c a l constant and t h a t d i f f e r e n c e s i n observed growth r a t e are caused by changes i n environment. Lengths a t v a r i o u s ages are c a l c u l a t e d a c c o r d i n g to the equation  0.94  H+l  0.94  " ' 4  3  2  +  h  The observed and c a l c u l a t e d lengths are shown i n Table 14 Table 14 •  Observed  . „ Age i n years. 4  and c a l c u l a t e d l e n g t h s o f c u t t h r o a t t r o u t Pork l e n g t h i n centimeters Observed  Calculated  I  7.01  7.07  II  11.54  12.34  III  18.71  17.74  IV  23.98  23.25  V  28.99  28.83  56 RAINBOW TROUT (Salmo g a i r d n e r i i ) S u i t a b i l i t y of an environment f o r f i s h i s r e f l e c t e d i n the growth of the f i s h .  For t h i s purpose growth of rainbow t r o u t  from three lakes i n B r i t i s h Columbia was i n v e s t i g a t e d .  The lakes  chosen f o r study were Paul Lake, Loon Lake and Beaver Lake. Paul Lake The  growth of rainbow t r o u t i n v a r i o u s years  i n Paul Lake  has been d e s c r i b e d i n s e v e r a l p u b l i c a t i o n s ( L a r k i n e t a l . 1950, L a r k i n and Smith 1954, Crossman and L a r k i n 1958).  To avoid  complications a r i s i n g from changes i n growth r a t e d u r i n g the p e r i o d of an e x p l o s i v e i n c r e a s e of r e d s i d e shiners b a l t e a t u s ) the data s e l e c t e d f o r the present  study apply to the  1946 year c l a s s , caught from 1946-49 as three year A Walford  (Richardsonius  olds*  p l o t of l ^ j a g a i n s t 1^ f o r three year o l d +  rainbow t r o u t i s shown i n Figure 27.  These p o i n t s could y i e l d  a l i n e of best f i t t h a t would i n t e r s e c t the 45° diagonal and hence the value of z could be expected to be more than one.  By  q u a d r a t i c approximation z was estimated as 1.1, and by computer 1.3 1.3 1.3. F i g u r e 28 shows 1^. ^ p l o t t e d a g a i n s t 1^ • The general +  equation f o r Paul Lake t r o u t i n terms of l e n g t h i s l  1.3 t  +  1  = 39.5609 + l  Mean growth increments between ages are C<  1  2  CX._  « 43.7436 = 35.3792  1.3 t  o  o  0  10  Figure  15  27  0  Plot  20  of l  t  +  1  25  on  l  30  t  o  35  f o r rainbow t r o u t  40  f r o m P a u l L a k e , B.  C.  58  59 A n a l y s i s of v a r i a n c e (Table 15)on QC v a l u e s show s i g n i f i c a n t  diffe-  rences between ages. Table 15. Source  A n a l y s i s of Variance of Growth Increments Between Ages.  of  Variance  d.f.  Mean square  Total Means Individuals  133 1 132  2343.2247 216.3224  Prom t h i s i t may  P ratio  Probability  10.8  <^.01  be i n f e r r e d t h a t 3 year Paul Lake t r o u t grow  f a s t e r i n t h e i r second year of l i f e than i s p r e d i c t e d (see Table 16 below).  Thus the P a r k e r - L a r k i n equation i s not a good f i t to  the data - i . e . the r a t e of change of increments by only two parameters.  i s not d e s c r i b a b l e  The data i s analyzed by the Von  Bertalanffy  equation as H+l  ~  (« )  55  323  °«  +  677  t  1  and the c a l c u l a t e d lengths are shown i n Table Table 16.  16.  Comparison of observed and c a l c u l a t e d lengths of Paul Lake rainbow t r o u t .  Age i n years  Observed Fork Length i n centimeters  1 2 3  8.17 22.91 33.04  C a l c u l a t e d Fork Length i n centimeters Parker-Larkin Von B e r t a l a n f f y 8.17 21.78 33.07  22.93 33.04  A n a l y s i s of rainbow t r o u t growth data a f t e r the  establish-  ment of s h i n e r s i n Paul Lake i n d i c a t e s another p o s s i b l e source of error i n estimating z values. three year o l d t r o u t caught  Back c a l c u l a t e d growth data f o r  i n 1955  and 1956 y i e l d a z of  0.27,  suggesting r a p i d l y i n c r e a s i n g increments which on e x t r a p o l a t i o n  60 to the f o u r t h and f i f t h year would produce enormous t r o u t of 51.0  cm and 91.17 cm r e s p e c t i v e l y .  The spurious z value can  be e x p l a i n e d from the work of L a r k i n and Smith (1954) on the growth of rainbow t r o u t i n Paul Lake.  Small t r o u t eat plankton  and bottom organisms f o r which there i s i n t e n s i v e competition by s h i n e r s .  A t lengths ranging from 15 to 25 centimeters  trout  s w i t c h t o a d i e t of s h i n e r s during the summer months, t h e i r growth r a t e responding  accordingly.  Parker and L a r k i n (1959)  denote t h i s type of change as an " e c o l o g i c a l growth stanza" and the data should o b v i o u s l y be s p l i t i n t o two groups - f i s h below 15 cm and f i s h above 25 cm.  Por rainbow t r o u t from Paul Lake  t h i s i s an i m p r a c t i c a l procedure age  3.  because many f i s h mature a t  In consequence there are only two growth increments  2 and 2 t o 3) a v a i l a b l e f o r z e s t i m a t i o n s .  ( l to  S p l i t t i n g the f i s h  i n t o two s i z e groups r e s u l t s i n s i z e h i e r a r c h y e f f e c t s w i t h i n each group. - which can cause underestimation of z v a l u e s * The  best procedure  would seem t o be c a l c u l a t i o n of z from p r e -  s h i n e r data and u s i n g t h i s v a l u e , to estimate f o r small and l a r g e f i s h s e p a r a t e l y , any changes i n O C o c c a s i o n e d by the shiner introduction.  The assumption would be made t h a t z i s a ^ p h y s i o -  l o g i c a l constant," an assumption c o n s i s t e n t w i t h the contentions of Parker and L a r k i n . Paul Lake rainbow t r o u t o f f e r s t i l l another  complication  i n growth a n a l y s i s , because of v a r i a t i o n i n the length-weight relationship.  The r e l a t i o n between growth i n l e n g t h and growth  i n weight was c a l c u l a t e d f o r data c o l l e c t e d before and a f t e r the i n t r o d u c t i o n of s h i n e r s i n t o Paul Lake.  F i s h were separated  61  a c c o r d i n g to s i z e , stage of m a t u r i t y and sex. The measurements of lengths and weights were converted to logarithms and r e g r e s s i o n s were c a l c u l a t e d by the method of l e a s t squares. 1947  The length-weight  r e l a t i o n s h i p f o r the p e r i o d s  and 1959 were log ¥  = - 1.47528 + 2.75216 l o g L . . . . - r (1947 )  log ¥  = - 1.81648 + 2.91714 l o g L . . . . - r ( l 9 5 9 )  where ¥  =  weight i n grams  L  =  fork length i n centimetres.  A n a l y s i s of covariance was a p p l i e d to t e s t d i f f e r e n c e s i n the length-weight The  r e l a t i o n s h i p among the years 1947 and 1959.  r e l a t i o n s h i p was found to d i f f e r s i g n i f i c a n t l y a t P ^ . 0 1  w i t h r e s p e c t to the r e g r e s s i o n c o e f f i c i e n t and the adjusted means. For each p e r i o d s e p a r a t e l y , the r e l a t i o n s h i p s f o r the f i s h below and above 25 cm. i n l e n g t h are  25 cm. log  1.57847 + 2.82398 l o g L  =25 cm. log ¥  =  1.04865 + 2.47248 l o g L  1959 -<25  cm. log ¥  1.86505 + 2.95497 l o g L  -25 cm. log ¥  = - 2.06575 + 3.07974 l o g L  A comparison of slopes f o r f i s h below 25 cm. i n s i z e f o r  62 the p e r i o d s 1947 and 1959 was not s i g n i f i c a n t but f o r f i s h  above  25 cm i n s i z e the slopes were s i g n i f i c a n t l y d i f f e r e n t a t the 1% l e v e l .  For 1947 the slopes f o r f i s h below and above 25 cm  i n s i z e were s i g n i f i c a n t l y d i f f e r e n t at the Vfi> l e v e l whereas they were not d i f f e r e n t f o r the p e r i o d  1959.  In 1947 t r o u t below 25 cm s i z e were r e l a t i v e l y heavier than the l a r g e r f i s h , whereas the reverse was true  i n 1959.  The e x p l a n a t i o n f o r t h i s phenomenon would appear to be from the h i s t o r y of the For the p e r i o d  available  lake. 1946-49 there were no s i g n i f i c a n t d i f f e -  rences i n the d i e t of t r o u t of v a r i o u s s i z e s ( L a r k i n and Smith 1953).  In c o n t r a s t  to 39.8$ amphipods i n the d i e t d u r i n g  1931,  i n 1947-49 Daphnia formed the major food item f o r a l l s i z e s ( L a r k i n e t a l . 1950). Presumably, the s c a r c i t y of Gammarus d i d not a f f e c t growth r a t e s of t r o u t of l e s s than 25 cm s i z e because of the abundance  of Daphnia.  But f o r t r o u t above 25 cm i n s i z e Daphnia  were perhaps an inadequate source of food, and w i t h competition •^  or  Gammarus, growth r a t e s were low.  Moreover, i t would be  expected that during the 1946-47 p e r i o d , would be i n r e l a t i v e l y b e t t e r  condition  evident i n the slopes of 2,47 and 2.82  t r o u t of smaller s i z e than large t r o u t .  This  was  f o r l a r g e and small t r o u t  respectively. From 1952 onward t r o u t over 25 cm s t a r t e d p r e y i n g on s h i n e r s , while f i s h of small s i z e were a d v e r s e l y a f f e c t e d by competition w i t h shiners  f o r plankton and bottom organisms.  a r e s u l t , trout regression  As  c o e f f i c i e n t s f o r 1959 i n d i c a t e r e l a t i v e l y  63 b e t t e r c o n d i t i o n of the l a r g e r f i s h .  Moreover f o r t r o u t above  25 cm. the r e g r e s s i o n c o e f f i c i e n t s were s i g n i f i c a n t l y d i f f e r e n t f o r the years 1947 and 1959 i . e . before and a f t e r the i n t r o d u c t i o n of s h i n e r s and l a r g e t r o u t of 1959 were heavier than of 1947.  those  The competition f o r food between s h i n e r s and small  t r o u t was not r e f l e c t e d i n a lower  c o n d i t i o n of small t r o u t as  compared to the p r e - s h i n e r p r i o d . There were a l s o changes i n the length-weight  relation-  ships w i t h regard t o sexes, and m a t u r i t y (see Table 17), Table 17.  Log l e n g t h l o g weight r e l a t i o n of ;raihbow  t r o u t of  v a r i o u s s i z e s , sexes and stages of m a t u r i t y from Paul Lake, B. C. Slo pe  Intercept  1946-47  1957-59  1946-47  2.82398  2.95497  -1.57847  -1.86505  2.47248  3.07974  -1.04865  -2.06575  Females"  2.71613 •  2.91285  -1.42076  -1.80537  Immature  2.70832  2.90824  -1.42002  -1.79895  Maturing  2.27125  3.02107  -0.73438  -1.97916  Males  2.78928  2.92746  -1.53017  -1.84281  Immature  2.96087  2.79529  -1.76698  -1.67601  Maturing  2.46134  3.49777  -1.03361  -2.72078  Total  2.75216  2.91714  -1.47528  -1.81648  <^25 ^  cm.  25 cm  1957-59  64  Length measurements are thus inadequate the weight increments  i n d i c a t i o n s of  and b a s i c a l l y growth comprises weight  increments* C o n s i d e r i n g a l l of the above observations, the L a r k i n growth equation would appear to be inadequate  Parkerfor  d e s c r i p t i o n of the growth of rainbow t r o u t i n Paul Lake. short l i f e  The  c y c l e , change i n food h a b i t s and changes i n l e n g t h -  weight r e l a t i o n seem to m i l i t a t e a g a i n s t the use of any t i c a l system of o r d e r l y r e l a t e d  theore-  increments.  Loon Lake Back c a l c u l a t e d lengths of 3 year o l d rainbow t r o u t caught i n 1952  were used.  shown i n F i g u r e 29* diagonal*  A Walford p l o t of l ^ . j a g a i n s t 1^ i s +  These p o i n t s tend to converge to the  45°  A n a l y s i s of the data y i e l d e d a z value of 1.3. 1.3  plot.' of 1 ^  +1  The  1.3 a g a i n s t 1^  is- shown i n F i g u r e 30.  The  Parker-  L a r k i n growth equation f o r rainbow t r o u t of Loon Lake i s 1.3 V l  The  1.3 =  2  4  '  4  6  +  h  agreement between the observed  very s a t i s f a c t o r y . Table 18.  (Table  and c a l c u l a t e d lengths  was  18).  Comparison of observed  and c a l c u l a t e d lengths of  Loon Lake rainbow t r o u t . Age  in  years 1 2 3  Observed 7.08 15.97 23.17  Fork l e n g t h i n centimeters Calcu: ated Von B e r t a l a n f f y Parker-Larkin 7.06 15.81 23.18  15.99 23.60  Loon Lake, B. C.  66  Loon Lake, B. C.  67 In Loon Lake where there are only rainbow t r o u t , the  decline  i n growth rate follows  a definite  t r e n d with i n c r e a s i n g l e n g t h  ( L a r k i n et a l . 1950),  As a r e s u l t the estimate of exponent z i n the  P a r k e r - L a r k i n equation i s a r e l i a b l e measure to express growth rate of rainbow t r o u t i n Loon Lake.  The data were analysed by the Von  B e r t a l a n f f y equation as the l i n e of best f i t coverges  to the 4 5 ° d i a g o n a l . l  The estimated equation  = 60(.173) + l  t  of the Walford p l o t  t  is  0.8270  The agreement between observed and c a l c u l a t e d lengths as t h a t of P a r k e r - L a r k i n equation (Table  i s as good  18).  Beaver Lake Back c a l c u l a t e d lengths  of three year o l d rainbow t r o u t of  the year c l a s s e s 1952 and 1953 caught i n 1955 and 1956  respective-  l y are p l o t t e d to give the Walford l i n e represented i n F i g u r e s 31 and 32.  For the 1952 year c l a s s the l i n e of best f i t would i n t e r -  s e c t the 4 5 ° d i a g o n a l and the value o f . z m s  estimated as  1.14.  F i g u r e 33 shows the transformed data r a i s e d to the power  1.14.  A n a l y s i s of data f o r the 1953 year c l a s s gave a z of 0.65  which  indicates  that the growth increments get bigger as the f i s h grow .65  older.  The p l o t of  .65 a g a i n s t 1^  two d i f f e r e n t values  is  shown i n F i g u r e 34.  These  of z might be due to the v a r y i n g growth r a t e s  of the year c l a s s e s responding a c c o r d i n g l y to the s t r e n g t h of the year c l a s s e s . A s i m i l a r s i t u a t i o n can be demonstrated i n the data from Paul Lake, where z values from 0.8 to 1.4 were obtained f o r r e n t i n d i v i d u a l year c l a s s e s from 1946 to 1949. r i z e d on the b a s i s  diffe-  I t may be summa-  of these observations that the a p p l i c a t i o n of  P a r k e r - L a r k i n equation i s made d i f f i c u l t f o r rainbow t r o u t due to short l i f e  span and v a r i a t i o n s i n year c l a s s  strength.  the  68  69  71  72 SCALLOPS (Placopecten  magellanicus)  Br. L. D i c k i e of the A t l a n t i c B i o l o g i c a l S t a t i o n , St. Andrews, New  Brunswick, k i n d l y provided back c a l c u l a t e d s h e l l  heights of s c a l l o p s (Placopecten magellanicus) which were used i n his  study of t h i s species on v a r i o u s A t l a n t i c seaboard grounds  ( D i c k i e 1954,  1955),  S c a l l o p s from Hour ground and Buoy ground  are used f o r the present  study.  The Walford p l o t of 1^.  +1  for  on 1^ i s shown i n F i g u r e  s c a l l o p s from the Hour ground.  Growth i s sigmoid,  35  so t h a t  the p o i n t s f i r s t diverge from the 45° diagonal up to a s h e l l height of 70 to 80 mm, six  years.  corresponding  to an age  of  approximately  Beyond t h i s s h e l l height the l i n e of best f i t  approaches the 45° d i a g o n a l , thus showing an a c c e l e r a t i n g growth and then a d e c e l e r a t i n g growth. B e r t a l a n f f y formula  For t h i s reason the  Von  can only be a p p l i e d to the o l d e r specimens.  For the P a r k e r - L a r k i n method the data must be s p l i t at the p o i n t of  i n f l e x i o n i . e . approximately  at the age  of s i x y e a r s .  For  the f i r s t  s i x years of growth, a n a l y s i s of the data y i e l d e d a 0.625 z value of 0.625. The P a r k e r - L a r k i n t r a n s f o r m a t i o n of 1.,, on 0.625 1^ i s shown i n F i g u r e 36. The formula f o r expressing, the t  +  1  growth during the a c c e l e r a t i n g growth phase i s : l  0.625 t  +  = 2.798 + l  1  C a l c u l a t e d and observed  0.625 t  s h e l l heights are shown i n Table  19.  The d e c e l e r a t i n g phase of growth from s i x to nine years y i e l d e d a z of 3.6.  The  equation f o r the d e c e l e r a t i n g phase of growth i s : 3.6 1  T+1.  3.6 =  6 7 4  «  4 6 9  +  H  75  Table, 19.  Comparison of observed and c a l c u l a t e d h e i g h t s of s c a l l o p s i n m i l l i m e t e r s  shell  from  Hour ground.  Age i n years  Observed  Calculated Parker - L a r k i n  Von  Bertalanffy  1  6.7  6.7  2  16.2  17.9  3  31.3  32.9  4  53.0  51.0  5  72.0  71.9  6  83.4  83.8  7  90.9  90.3  91.6  8  96.6  95.6  95.7  9  99.9  100.3  99.3  10  103.6  102.4  11  105.0  105.2  76 For the sake of convenience are expressed  of computations the heights  i n centimeters and thus the above equation d e p i c t s  the growth i n centimeters.  The  c a l c u l a t e d values from s i x to 3.6 nine years are given i n Table 19, and the p l o t of l + . i a g a i n s t 3.6 " 1^ i s shown i n F i g u r e 37. 1H  1  Since the l i n e of best f i t f o r the d e c e l e r a t i n g growth p e r i o d approaches the 45° diagonal on Walford graph, the were a l s o analysed by the Von B e r t a l a n f f y method. v a r i o u s ages could be obtained from the l The  t  +  1  Lengths a t  equation:-  = 126(.122) + 0.878 l  t  c a l c u l a t e d lengths are shown i n Table  data  19.  -1  J  78 S c a l l o p s from Buoy ground from one were a l s o analysed. F i g u r e 38.  The p l o t of 1^ ^ +  to s i x years of  a g a i n s t 1^ i s shown i n  I t i s c l e a r from the f i g u r e that there i s a change  i n the growth p a t t e r n beyond the 6th year of l i f e 60-80 mm  of s h e l l h e i g h t .  The  The  i n F i g u r e 39.  A value of 0.375 f o r z has been 0.375 0.375 p l o t of l ^ ] _ on 1^ i s shown  transformed The  or between  l i n e of best f i t has a d i v e r g i n g  t r e n d from the 45° d i a g o n a l . estimated.  age  +  P a r k e r - L a r k i n equation to p r e d i c t the  shell  h e i g h t s d u r i n g the a c c e l e r a t i n g growth phase i s : 0.375 V i  0.375 " °-  6  5  6  2  h  +  Observed and p r e d i c t e d values are given i n Table Table  20.  Comparison of observed  and  20.  calculated shell  heights  of s c a l l o p s from Buoy ground.  Age  The  i n years  Observed  Calculated  1  5.55  5.55  2  12.06  12.24  3  21.50  22.50  4  36.25  36.92  5  56.56  56.05  6  80.44  80.44  P a r k e r - L a r k i n equation i s s a t i s f a c t o r i l y a p p l i e d to the  s c a l l o p s of Hour ground and Buoy ground which i s evident from the agreement between observed  and c a l c u l a t e d v a l u e s .  79  80  the Buoy ground.  81 DISCUSSION AND CONCLUSIONS The u s e f u l n e s s  of an e m p i r i c a l equation i s enhanced i f  i t s constants y i e l d e a s i l y information  of b i o l o g i c a l i n t e r e s t .  I t i s s o l e l y on t h i s b a s i s that the Von B e r t a l a n f f y equation has had a wide and v a r i e d use i n f i s h e r i e s b i o l o g y .  The parameter K  of Von B e r t a l a n f f y ' s equation i s supposed to be p r o p o r t i o n a l to the c o e f f i c i e n t of catabolism  i . e . i t i s the rate a t which the  animal a t t a i n s the e&symptotic s i z e .  I n t r a and i n t e r  species  growth comparisons n e a r l y always show t h a t K i s high when low and v i c e versa  (Holt I960).  Taylor  L^is  (1959) showed that  changes i n the value of K are temperature dependent.  He a l s o  showed (1959 and I960) the inverse r e l a t i o n s h i p e x i s t i n g between K and Lc© f or cod and r a z o r clam.  The values of K and L ^  Von B e r t a l a n f f y and z of P a r k e r - L a r k i n present i n v e s t i g a t i o n s are given Table 21.  equation obtained i n the  i n Table 21.  Growth parameter of Von B e r t a l a n f f y and equations.  Species Eopsetta jordani  Clupea p a l l a s i i  Salmo g a i r d n e r i i  Sex  of  L  oo  K  Parker-Larkin z  M  85  cm.  0.11  1.3  P  81  cm.  0.11  1.3  M  27.6  cm.  0.13  3.1  P  25.4  cm.  0.21  3.5  Paul Lake  55  em.  0.39  1.3  Loon Lake  60  cm.  0.19  1.3  Hippoglossus stenolepis  1926  232  cm.  0.16  1.5  1956  400  cm.  0.24  1.36  Acipenser fulvescens  M  210  cm.  0.02  2.64  P  180  cm.  0.023  2.64  82  I t can be seen from the t a b l e t h a t there i s an inverse r e l a t i o n s h i p between L ^ a n d and K.  z i n the same way as between L e o  From t h i s i t may t e n t a t i v e l y be concluded t h a t the  parameter z of the P a r k e r - L a r k i n growth equation i s an index of the p h y s i o l o g i c a l a c t i v i t y .  Parker and L a r k i n ( 1 9 5 9 )  suggested  t h a t x or z of t h e i r growth equation may be d e r i v e d from a comparative  study of m e t a b o l i c : r a t e over a range of s i z e .  The  values of z can a l s o be explained i n terms of the f a c t o r s t h a t affect LG©.  Due t o i t s p l a s t i c i t y , growth i s a f f e c t e d by the  a v a i l a b i l i t y of food m a t e r i a l .  The a v a i l a b i l i t y of food i s  dependent n o t only on the physico-chemical f a c t o r s of the environment but a l s o on the d e n s i t y of the p o p u l a t i o n .  In terms  of the Von B e r t a l a n f f y equation i t i s the parameter Looor Woa t h a t i s a f f e c t e d by v a r i a t i o n s i n the food consumption and H o l t 1 9 5 7 ) .  (Beverton  From the present i n v e s t i g a t i o n on t r o u t from  Paul Lake and Beaver Lake i t i s e v i d e n t t h a t the values o f the parameter z were v a r i a b l e which was e x p l a i n a b l e by v a r y i n g year c l a s s s t r e n g t h and ensuing competition f o r food.  When there i s  no s u f f i c i e n t food there i s a l e s s e n i n g o f L e o and higher L o o where there i s s u f f i c i e n t food.  In order t o make the e m p i r i c a l  growth data l i n e a r i t r e q u i r e s a h i g h z value i n the former case and a low z i n the l a t t e r i n s t a n c e .  I t i s too e a r l y to a t t r i b u t e  any p h y s i o l o g i c a l i n t e r p r e t a t i o n t o the parameter z. Assuming that f i s h growth i s i s o m e t r i c the exponent x o f ^  = kw ,  appears to serve as a measure of the complex of p h y s i o l o g i c a l processes. Parker and L a r k i n ( 1 9 5 9 p. 7 2 6 , F i g . l ) mentioned t h a t —the—value  of z i s l i k e l y t o l i e between 1 . 0 and 1 . 5 , i f the  83  d a t a appear t o approach the 45° d i a g o n a l . chinook salmon they worked on.  T h i s i s true f o r the  In the present s e r i e s of obser-  v a t i o n s the value of z i s more than 1.0 and a value as h i g h as 3.6 was obtained. 1.5.  For t r o u t the value of z was between 1.0 and  From t h i s i t may be concluded t h a t t h i s range of values  i s true f o r salmonids  only.  I t i s evident from t h i s study t h a t the P a r k e r - L a r k i n growth equation can be a p p l i e d to many a q u a t i c organisms and i n many i n s t a n c e s the agreement between the observed and c a l c u l a t e d values i s good.  However, t o evaluate the u s e f u l n e s s  of the v a r i o u s constants as t o o l s of p h y s i o l o g i c a l and/or e c o l o g i c a l events of the growth p a t t e r n , f u r t h e r work i s suggested, probably i n the experimental  field.  84  SUMMARY The P a r k e r - L a r k i n equation dw/dt = kw to observed scallops. (1)  x  has been f i t t e d  data on l e n g t h s , weights of f i s h e s and heights of The  f o l l o w i n g i s the summary of the f i n d i n g s .  Conversion of z to values of x should not assume t h a t the exponent r e l a t i n g to l e n g t h to weight i s n e c e s s a r i l y 3.  (2)  Back c a l c u l a t e d lengths may years and may  r e f l e c t bad and good growth  give a spurious e s t i m a t i o n of a z value  a p p r o p r i a t e f o r comparisons.  In a v a r i a b l e environment z  should be c a l c u l a t e d from increments  i n growth f o r f i s h of  v a r i o u s s i z e s i n the same year even though t h i s may  procedure  be b i a s e d by s e l e c t i o n of f a s t growing f i s h by the  fishery. (3)  In short l i v e d species w i t h h i g h l y v a r i a b l e growth r a t e s combinations  of complications make the e s t i m a t i o n of z  from f i e l d data h i g h l y u n r e l i a b l e .  In rainbow t r o u t  from Paul Lake i t i s necessary to recognize growth stanzas. inadequate  ecological  However the component "stanzas" are  then  f o r z e s t i m a t i o n because of the great v a r i a b i l i t y  i n growth r a t e and s e l e c t i o n of f a s t growing f i s h by the fishery.  E a r l y m a t u r i t y and d i f f e r i n g  r e l a t i o n s h i p s f o r both sexes and  length-weight  stages of m a t u r i t y e t c .  f u r t h e r confound the a n a l y s i s . (4)  Separation o f ' e c o l o g i c a l growth stanzas should be based on a s i z e r a t h e r than on age  c r i t e r i o n to a v o i d b i a s from  extremely f a s t or slow growing i n d i v i d u a l s . (5)  The Von B e r t a l a n f f y equation was  found to overestimate  s i z e i n the o l d e r ages i n many s p e c i e s .  the  85 (6)  F i s h from f r e s h water as w e l l as from the marine environment are d e s c r i b e d adequately by the P a r k e r - L a r k i n equation.  (7)  When the growth increments decrease a t f i r s t and then become equal as i n white sturgeon i t i s suggested that the data be s p l i t i n t o two stanzas f o r a n a l y s i s .  (8)  When the l i n e of best f i t on Walford p l o t s tends t o approach the 45° d i a g o n a l the value of z l i e s between 1.0 and 1.5 i n the case of salmonids.  In other s p e c i e s a value as  h i g h as 3.6 i s obtained. (9)  T e n t a t i v e b i o l o g i c a l i n t e r p r e t a t i o n i s attempted  to e x p l a i n  the parameter z of the P a r k e r - L a r k i n equation by drawing a comparison w i t h the parameters of the Von B e r t a l a n f f y equation. (IG)  The r e g r e s s i o n equation of the body-scale r e l a t i o n s h i p i s used only t o o b t a i n the value of the i n t e r c e p t .  A l l the  back c a l c u l a t i o n s are made by keeping the i n t e r c e p t constant w i t h v a r i a b l e slopes f o r the i n d i v i d u a l  fish.  86 LITERATURE CITED Anonymous. 1960. U t i l i z a t i o n of P a c i f i c H a l i b u t stocks: y i e l d per recruitment. Rept. I n t . P a c i f i c H a l i b u t Comm., No. 28, 52 pp. Von  B e r t a l a n f f y , L. 1938. A q u a n t i t a t i v e theory o f organic growth. Human B i o l o g y 10(2): 181-213.  Von  B e r t a l a n f f y , L. 1957. Q u a n t i t a t i v e laws i n metabolism and growth. Quart. Rev. B i o l . , 32(3): 217-231.  Beverton, R. J . H. and S. J . H o l t . 1957. On the dynamics of exploited f i s h populations. U.K. Min. Agr. and P i s h . , P i s h . Invest., Ser. 2, 19: 533 pp. Beverton, R. J . H. and S. J . H o l t . 1959. A review of the l i f e span and m o r t a l i t y r a t e s of f i s h i n nature and t h e i r r e l a t i o n to growth and other p h y s i o l o g i c a l c h a r a c t e r i s t i c s . Ciba Foundation C o l l o q u i a on Aging. V o l . 5. Boone, J . A. M.S. An assessment of the length-weight r e l a t i o n ship of rainbow t r o u t , Salmo g a i r d n e r i i . Univ. B r i t i s h Columbia. Carlander, K. D. and R. R. Whitney. 1961. Age and growth of walleyes i n C l e a r Lake, Iowa. 1935-57. Trans. Am. F i s h . S o c , V o l . 90, No. 2: 130-138. Classen, J . E . A. 1944. E s t u d i o b i o - e s t a d i s t i c o d e l e s t u r i o n o s a l l o d e l G u a d a l q u i v i r (Acipenser s t u r i o L . ) . I n s t . Esp. de Oceanogr., Madrid, No. 19: 112 pp. Clemens, W. A. and G. V. Wilby. 1949. F i s h e s of the P a c i f i c coast of Canada. B u l l . F i s h . Res. Bd. Canada, No. 48: 368 pp. C u e r r i e r , Jean-Paul and G. Rowssow. 1951. Age and growth of lake sturgeon from Lake S t . F r a n c i s , S t . Laurence R i v e r . Canadian F i s h C u l t . , No. 10: 17-29. D i c k i e , L. M. 1955. F l u c t u a t i o n s i n abundance of the g i a n t s c a l l o p , Placopecten magellanicus (Gmelin), i n the Digby area of the Bay of Fundy. J . F i s h . Res. Bd. Canada, 12(6): 797-857. Gwyn, A. G. 1939. The development and r e l a t i v e growth of the s c a l e s of the P a c i f i c Herring (Clupea p a l l a s i i ) . M.A. t h e s i s , Univ. of B r i t i s h Columbia. H o l t , S. J . 1960. " L e t t e r s to the e d i t o r . " V o l . 25, No. 2.  J . du C o n s e i l ,  L a r k i n , P. A., G. C. Anderson, W. A. Clemens and D. C. G. Mackay, 1950. The production of Kamloops t r o u t , (Salmo g a i r d n e r i i kamloops, Jordan) i n Paul Lake, B r i t i s h Columbia. S c i . Pubis. B. C. Game Dept., No. 1: 37 pp.  87 L a r k i n , P. A . a n d S. B . S m i t h . 1 9 5 4 . Some e f f e c t s o f i n t r o d u c t i o n o f t h e r e d s i d e s h i n e r on t h e kamloops t r o u t i n P a u l Lake, B r i t i s h Columbia. T r a n s . Am. P i s h . S o c , 8 3 ( 1 9 5 3 ) : 161-175. L a r k i n , P. A . , J . G. T e r p e n n i n g a n d R. R. P a r k e r . 1957. S i z e a d e t e r m i n a n t o f g r o w t h r a t e i n r a i n b o w t r o u t , Salmo gairdnerii. T r a n s . Am. F i s h . S o c , 8 6 : 8 4 - 9 6 . L a r k i n , P. A . a n d K. V . A y y a n g a r . 1961, A p p l i c a t i o n Parker equation t o growth o f aquatic organisms. ( : S S 11th A l a s k a S c i . Congress.(V^"f ) Ostle,  as  of the Proc.  B. 1956. S t a t i s t i c s i n r e s e a r c h . (Basic concepts and techniques f o rresearch workers). Iowa S t a t e C o l l e g e Press. 487 pp.  P a r k e r , R. R. a n d P. A . L a r k i n . 1959. A concept o f growth i n fishes. J . F i s h . R e s . Bd. Canada, 1 6 ( 5 ) : 721-745. P r o b s t , R. tion Lake 84: Pycha,  T. a n d E . L . C o o p e r . 1954. Age, growth and p r o d u c of the lake sturgeon (Acipenser fulvescens) i n the Winnibago r e g i o n , Wisconsin. T r a n s . Am. F i s h . S o c , 207-227.  R. L . 1956. P r o g r e s s r e p o r t on w h i t e C a l i f . F i s h a n d Game, 4 2 ( 1 ) : 2 3 - 3 5 .  sturgeon  studies.  R i c k e r , W. E . 1958. Handbook o f c o m p u t a t i o n s for biological statistics of fish populations. B u l l . F i s h . Res. Bd. C a n a d a , No. 1 1 9 : 300 p p . S n e d e c o r , G. W. 1956. S t a t i s t i c a l methods. State College Press. 534 p p .  5th Edition.  Iowa  S t e n t o n , C. E . 1960. E c o l o g y o f t h e y e l l o w s t o n e c u t t h r o a t (Salmo c l a r k i i l e w i s i G i r a r d ) i n Kiakho Lake, B r i t i s h Columbia. M.S. t h e s i s , U n i v . o f B r i t i s h Columbia.  trout  S t e v e n s o n , J . A . a n d L . M. D i c k i e . 1954. Annual growth r i n g s and r a t e of growth of the g i a n t s c a l l o p Placopectus magellanicus (Gmelin) i n t h e Digby a r e a o f t h e Bay o f Fundy. J . Fish. Res. Bd. Canada, 1 1 ( 5 ) : 660-671. T a y l o r , C. C. 1958. Cod growth and temperature. E x p l o r . Mer, 2 3 ( 3 ) : 366-370. T a y l o r , C. C. J . Cons.  J. Cons. I n t .  1959. Temperature and growth o f t h e r a z e r I n t . E x p l o r . Mer, 2 5 ( 5 ) : 93-101.  clam.  T h o m p s o n , W. F . a n d F . H. B e l l . 1934. B i o l o g i c a l s t a t i s t i c s o f the P a c i f i c H a l i b u t f i s h e r y . (2) E f f e c t o f changes i n i n t e n s i t y upon t o t a l y i e l d and y i e l d p e r u n i t o f gear. R e p t . I n t . P a c i f i c H a l i b u t Comm., N o . 8: 4 9 p p . ~ W a l f o r d , L . A. 1 9 4 6 . A new g r a p h i c m e t h o d o f d e s c r i b i n g t h e growth of animals. B i o l . B u l l . , 9 0 ( 2 ) : 141-147.  

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