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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 University, Waltair (India), 1954 M.Sc. Andhra University, Waltair (India), 1955 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 thesis 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 formulation for depicting the growth of organisms must y i e l d an empirical f i t that i s reasonably good. Its v a l i d i t y i s enhanced i f the equation yi e l d s information of bi o l o g i c a l i n t e r e s t . This investigation i s aimed at applying the Parker-Larkin (1959) growth equation to a number of aquatic organisms to describe the problems encountered i n making use of this technique. The data also analysed by the Von Bertalanffy growth equation to bring out the s i m i l a r i t i e s of the- constants of both the equations. The data pertaining to three species of marine f i s h , b r i l l , halibut and herring, four species of freshwater f i s h , rainbow trout, cutthroat trout and sturgeon and to a l a m e l l i -branch species scallops, have been analysed. I't i s pointed out that the exponent of the length-weight relationship should not be taken as 3. It i s shown that the length-weight relationship of rainbow trout varies depending on sex, maturity and siz e . In many species the Parker-SsLarkin growth equation predicted the lengths at various ages accurately. Von Bertalanffy 1s equation progressively overestimated the sizes. In white sturgeon the growth increments decrease at f i r s t and then become equal. In such a situation i t i s sugges-ted that the data be s p l i t into two stanzas for analysis since the analysis without s p l i t t i n g underestimates the sizes i n the early years and overestimates i n the older ages. The a n t e r i o r r a d i u s o f t h e s c a l e grows r e l a t i v e l y s l o w e r t h a n the l e n g t h o f t h e f i s h i n h e r r i n g . The r e g r e s s i o n e q u a t i o n o f t h e b o d y - s c a l e r e l a t i o n s h i p i s u s e d o n l y t o o b t a i n t h e v a l u e o f t h e i n t e r c e p t . The b a c k c a l c u l a t i o n o f l e n g t h s i s made by k e e p i n g t h e i n t e r c e p t c o n s t a n t w i t h v a r i a b l e s l o p e s f o r t h e i n d i v i d u a l f i s h . The P a r k e r - L a r k i n e q u a t i o n gave an e x c e l l e n t f i t f o r t h e d a t a on h a l i b u t . T h i s i s b e c a u s e t h e o b s e r v e d v a l u e s o f h a l i b u t a r e a c t u a l l y c a l c u l a t e d v a l u e s f r o m 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 o f w e i g h t on age - an a l g e b r a i c e q u i v a l e n t o f t h e P a r k e r -L a r k i n e q u a t i o n i n w h i c h t h e s l o p e i s t h e r e c i p r o c a l o f ( l - x ) . The r a n g e o f v a l u e s o f z between 1«0 and 1*5, when t h e l i n e o f b e s t f i t on a W a l f o r d p l o t a p p r o a c h e s t h e 45° d i a g o n a l , i s t r u e f o r s a l m o n i d s o n l y and i n s i m i l a r s i t u a t i o n s f o r o t h e r s p e c i e s a v a l u e as h i g h a s 3.6 f o r z i s o b t a i n e d . The v a r i a b i l i t y o f z d e p e n d i n g un t h e d e n s i t y o f t h e p o p u l a t i o n a n d / o r a v a i l a b i l i t y o r n o n - a v a i l a b i l i t y o f f o o d m a t e r i a l i s shown f o r r a i n b o w t r o u t . T h i s dependence o f z on t h e f o o d a v a i l a b l e i s s i m i l a r t o t h a t o f IJ o r ¥ . oo oo The r e i s an i n v e r s e r e l a t i o n s h i p between I* and z as t h a t oo o f 1^ and K. I t i s t e n t a t i v e l y s u g g e s t e d t h a t z m i g h t be a p a r a m e t e r o f p h y s i o l o g i c a l i m p o r t a n c e i n t h e P a r k e r - L a r k i n e q u a t i o n . F u r t h e r wwork o f an e x p e r i m e n t a l n a t u r e i s s u g g e s t e d t o e s t a b l i s h t h e 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 t h e p a r a m e t e r s Qf t h e P a r k e r - L a r k i n e q u a t i o n . ACKNOWLEDGMENT S This investigation was carried out at the Institute of Fisheries of the University of B r i t i s h Columbia, The author sincerely expresses his gratitude to Dr. P. A. Larkin for suggestions and cr i t i c i s m s during the investigations• The author also acknowledges Drs. K. S. Ketchen, F. H. C. Taylor, L. M. Dickie, Messrs. L. A. Sunde and C. E. Stenton for kindly providing the author with data on b r i l l , herring, scallops, sturgeon and cutthroat trout respectively. The assistance given by the personnel of the Computing Centre, University of B r i t i s h Columbia, i s also acknowledged. i v TABLE GP CONTENTS Page TITLE PAGE ABSTRACT 1 ACKNOWLEDGEMENTS i i : L TABLE OP CONTENTS. * v LIST OF FIGURES •! LIST OF TABLES i x INTRODUCTION 1 BRILL - Eopsetta jordani 5 HALIBUT - Hippoglossus stenolepis 13 LAKE STURGEON - Acipenser fulvescens 29 WHITE STURGEON - Acipenser transmontanous 37 HERRING - Clupea p a l l a s i i 43 Body-scale relationship 43 Growth rate •• 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 Table of Contents - Cont'd. Page SUMMARY 84 LITERATURE CITED 86 v i LIST OP FIGURES Figure 1. Plot of l ^ + j _ on 1^ for female b r i l l . 2. Plot of on 1^ for male b r i l l . 1.3 1.3 3. Plot of l ^ + j on 1^ for female b r i l l . 1.3 1.3 4. Plot of 1^.+1 on 1^ for male b r i l l . 5. Growth curves showing age-weight relationship for Portlock-Albatross halibut. 6. Plot of on W^  for Portlock-Albatross halibut for 1926. 7. Plot of W t + 1 on ¥ t for Portlock-Albatross halibut 0,5 0.5 I for 1926. 8 C Plot of ¥ t + 1 on ¥^ for Portlock-Albatross halibut for 1956. 0.45 0.45 9. Plot of ¥ t + 1 on ¥^ for Portlock-Albatross halibut for 1956. 10. Plot of 1+^ on 1^ for Portlock-Albatross halibut for 1926. 11. Plot of 1^.+^  on 1^ for Portlock-Albatross halibut for 1956. 1.5 1.5 for 1926. 12. Plot of 1HJ.+^  on 1^ for Portlock-Albatross halibut 13. Plot of l ^ + j on 1^ for Portlock-Albatross halibut 1.36 1.36 for 1956. 14, Plot of l . j . + i on 1^ for male lake sturgeon. 15. Plot of 1^.+1 on 1^ for female lake sturgeon. v i i 2.64 2.64 Figure 16. Plot of on 1^ for male lake sturgeon. 2.64 2.64 17. Plot of 1^+^ on 1^ for female lake sturgeon. 18. Plot of on 1^ for white sturgeon 1*89 1.89 19. Plot of on 1^ for white sturgeon from 0—6 years. 0.9 0.9 20. Plot of o n 1^ f ° r white sturgeon from 6-30 years. 21. Plot of o n 1^ ^ o r female herring from B e l l a B e l l a region. 22. Plot of 1^+^ on l ^ for male herring from Be l l a B e l l a region. 3.5 3.5 23. Plot of on 1^ for female herring from B e l l a B e l l a region. 3.1 3.1 24. Plot of on 1^ for male herring from B e l l a B e l l a region. 25. Plot of l ^ . + j on 1^ for cutthroat trout from Kiakho Lake, B• C• 0.94 0.94 26. Plot of ° n 1^ for cutthroat trout from Kiakho Lake , B. C • 27. Plot of l ^ . + j ^ on 1^ for rainbow trout from Paul Lake, B.C. 1.3 1.3 28. Plot of l^+i on 1^ . for rainbow trout from Paul Lake,B.C. 29. Plot of 1^.+TL ° n * ° r rainbow trout from Loon Lake,B.C. 1.3 1.3 30. Plot of on 1^ for rainbow trout from Loon Lake,B.C. v i i i Figure 31« Plot of on 1^ for 1952 year class rainbow trout from Beaver Lake, B. G. 32. Plot of o n * t * o r ^53 year class rainbow trout from Beaver Lake, B. C. 1,14 1.14 33. Plot of on 1^ for 1952 year class rainbow trout from Beaver Lake, B. C. 0.65 0.65 34. Plot of on 1^ for 1953 year class rainbow trout from Beaver Lake, B. C. 35. Plot of l ^ + ] _ ° n for scallops from the Hour ground. 0.625 0.625 36. Plof ©f *t+l 0 n * t * o r s c a H ° P s °f years from the Hour ground. 3.6 3.6 37. Plot of l^.+2 on 1^ for scallops of 6-9 years from the Hour ground. 38. Plot of on 1^ for scallops from the Buoy ground. 0.375 0.375 39. Plot of on 1^ for scallops from the Buoy ground. Oft-U s, oS-itr Hatch I A Q j i x LIST OP TABLES Table 1» Growth increments between various ages of b r i l l . 2. Analysis of variance of growth increments at d i f f e r e n t ages of both sexes of b r i l l . 3. Observed and calculated lengths. 4. Weight i n pounds at ages 5 to 40 for Portlock-Albatross halibut i n 1926. 5» Weight i n pounds at ages 5 to 40 for Portlock-Albatross halibut i n 1956. 6. Length i n centimeters at each age for 1926 and 1956 Portlock-Albatross halibut. 7. Analysis of variance on sturgeon for growth d i f f e -rences between sexes. 8. Observed and calculated lengths for male sturgeon. 9. Observed and calculated lengths for female sturgeon. 10. Observed and calculated lengths of white sturgeon from C a l i f o r n i a waters. 11. Back calculated and calculated fork lengths for female herring. 12. Back calculated and calculated fork lengths for male herring. 13. Analysis of variance on o(_values of cutthroat, trout i n d i f f e r e n t age i n t e r v a l s . 14. Observed and calculated lengths of cutthroat trout. 15. Analysis of variance of growth increments between ages. 16. Comparison of observed and calculated lengths of Paul Lake rainfepw trout. 17. Log length log weight r e l a t i o n of rainbow trout of various sizes, sexes and stages of maturity from Paul Lake, B. C. X Table 18. Comparison of observed and calculated lengths of Loon Lake rainbow trout. 19. Comparison of observed and calculated s h e l l heights of scallops i n millimeters from Hour ground. 20. Comparison of observed and calculated s h e l l heights of scallops from Buoy ground. 21. Growth parameters of Von Bertalanffy and Parker-Larkin. 1 INTRODUCTION In the study of the dynamics of f i s h populations, there are a number of parameters that must be determined. In addition to the estimations of mortality rates, age composition etc., growth rates of f i s h are important since the growth of an organism i s one of the basic determinants of y i e l d * I t i s a common practice to use age as a c r i t e r i o n of size and growth potential, even though th i s i s a r e l i a b l e index only under stable environmental conditions. Under changing environ-mental conditions age can no longer be considered as a c r i t e r i o n of s i z e . Larkin, Terpenning and Parker (1957) suggest a method tha^ relates growth to size independent of age. It i s their contention that size gives a better indication of ecological opportunity for growth than does age. They also mention that many f i s h may change the "ultimate s i z e " to which they are tending by changing their ecological niche. There may also be physiolo-g i c a l changes i n the l i f e of a f i s h that are related to siz e . Thus f i s h growth may be considered as a series of cycles or growth stanzas each of which can be defined as a period during which the parameters used for describing growth processes can be considered constant, within reasonable l i m i t s . Parker and Larkin (1959) suggested the use of the d i f f e -r e n t i a l equation ^ = kw i n the description of growth of chinook salmon (Oncorhynchus tshawytscha) and steelhead trout (Salmo  gairdnerii). E s s e n t i a l l y the use of thi s equation implies that growth can be treated l i k e any other physiological function. 2 For instance,respiration rate i s commonly related to weight by the d i f f e r e n t i a l equation. /AO , x = kw A t where A 0 represents oxygen uptake w represents weight The re s p i r a t i o n rate equation i s usually expressed al g e b r a i c a l l y as:-log log k +xlog w y i e l d : Treating the growth equation i n the same way would log J = log k + x log w (1) The question arises whether growth rate i s related to i n i t i a l weight, average weight or f i n a l weight during the period t . The c l a s s i c solution to th i s kind of problem i s to deal i n instantaneous rates, integrating the expression dw/dt = kwx to y i e l d w<;Tx) = k t ( l - x ) + w^" x ) (2) Using the expression w = q l ^ to denote the relationship between weight and length i t can be demonstrated that growth i n length can be depicted as: where z = y(l-x) Setting l _ j . _ o = 0 The above equation can be written as since * t + l = ° ^ 2 hence 1. =ovn x=n Taking logarithms and d i v i d i n g by i z log>;.l t= | l o g < X + | l o g n (4) p l o t t i n g l o g 1 a g a i n s t l o g t ( i . e . n) y i e l d s a s t r a i g h t l i n e w i t h slope of i analogous to ( l ) above. This i s e s s e n t i a l l y a z r e g r e s s i o n of l e n g t h on age w i t h the slope b = — , The s o l u t i o n z i s again made d i f f i c u l t , t h i s time by the presence of z i n both terms on the r i g h t had s i d e , which would r e q u i r e i t e r a t i o n f o r an a r b i t r a r y estimate of z. To obviate these d i f f i c u l t i e s Parker and L a r k i n suggested z z a s o l u t i o n u s i n g a technique based on a p l o t of on 1^ , u s i n g an a p p r o p r i a t e valjie of z t h a t would minimize the r e l a t i v e v a r i a n c e of oi. • The s o l u t i o n i s most c o n v e n i e n t l y handled by a computer (An ALWAC I I I - E computer was used* T h i s program i s on f i l e a t the Computing Centre, U n i v e r s i t y of B r i t i s h Columbia)* An approximate s o l u t i o n can be obtained u s i n g the q u a d r a t i c f u n c t i o n = a + bz + c z ^ 4. Three values of r e l a t i v e standard deviation (S^) and t h e i r asso-ciated z values on simultaneous solution y i e l d the best z value (the value of z giving minimum standard deviation) as Approximate graphic methods for solution are also given by Parker and Larkin. Carlander and Whitney (1961) mention that there i s a d i f f e r e n t growth pattern for walleyes i n Clear Lake which exceed 25.0 inches i n length when the older f i s h beyond age VII are eliminated or 23.9 inches i n length when only the f i s h which com-pleted a given annual increment are considered. They made use of the Parker-Larkin growth equation to give a better f i t . The present work describes problems i n applying these techniques of growth representation to data for various species of aquatic organisms including ( l ) b r i l l (Eopsetta jordani), (2) halibut (Hippoglossus stenolepis), (3) lake sturgeon (Aci- penser fulvescens), (.4) white sturgeon (Acipenser transmontanous), (5) herring (Clupea p a l l a s i i ) , (6) cutthroat trout (Salmo c l a r k i i ) , (7) rainbow trout (Salmo g a i r d n e r i i ) , and (8) scallops (Placo-pecten magellanicus)• Since the usefulness of any empirical equation i s enhanced i f i t s constants y i e l d information of b i o l o g i c a l i n t e -r e s t , the present work has t r i e d to draw tentative conclusions on the significance of the constants included i n the Parker-Larkin equation. The raw data pertaining to the species studied and the input and output tapes of the computer work are stored i n the Institute of Fisheries of the University of B r i t i s h Columbia. BRILL (Eopsetta 3 ordani) Back calculated lengths for b r i l l were kindly provided by Dr. K. S. Ketchen of the P a c i f i c B i o l o g i c a l Station, Nanaimo. Walford plots of l - j . + ^ against 1^ separately for the two sexes are shown i n Figures 1 and 2. The data show a s l i g h t convergence towards the 45° diagonal and indicate that an appropriate z value for the equation l^. +^= <2\+ 1^ would be more than 1. For convenience i n computation, growth i n the f i r s t f i v e years only was considered. Using the quadratic method for estimating minimum r e l a t i v e variance the o r i g i n a l length data, and sets of values for l ^ * - * and i^«65 yielded an estimate of z of 1.3. Using the ALWAC III E Computer the same value of z was obtained. 1 3 1 3 The plots of 1^][ against 1^ are shown i n Figures 3 and 4. The corresponding growth formulae are:-Females l * ^ 3 = 21.2837 + l * * 3 Males =3 20.4991 + l ^ * 3 Combined l j * ^ «= 20.8964 + l * * 3 Mean growth increments between the various ages are given i n Table I. 8 * ' o o o° o • • • • •. • • • • • . • • • s • • • t • • • • • • • 50 Z* 1.3 t 100 1.3 1.3 Figure 3. P lo t of l t + 1 on l t for female b r i l l . 9 10 Table I . Growth increments between va r ious ages of b r i l l ^ 1 2 C ^ 2 3 ^ 4 ^ 4 5 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 var iance (Table I I ) i n d i c a t e s no s i g n i f i c a n t d i f f e -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 . there i s no s i g n i f i c a n t departure from the average growth ra te a t va r i ous ages or f o r e i t h e r sex o£ fo 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 age. Table I I . A n a l y s i s of va r i ance of growth increments a t d i f f e r e n t ages of both sexes of b r i l l . Source of Var iance d . f . Mean Square F r a t i o P r o b a b i l i t y T o t a l 159 Means 7 I n d i v i d u a l s 152 41.3912 Sexes 1 25.2571 0.610 > 0.25 Ages 3 43.1577 1.042 ^=-0.25 I n t e r a c t i o n 3 44.5880 1.077 >-0.25 z z The a n a l y s i s suggests t ha t the equat ion 1. , =0<C + 1, can u s e f u l l y be a p p l i e d to 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 convers ion from z to x should not be based on the assumption tha t y = 3 . The length-11 weigth relationship for male and female b r i l l , provided by Dr. K. S. Ketchen are: Male log ¥ ( Q i ) = 3.1349 log l ( m m ) - 6.6982 Female log ¥ ( 0 z ) = 3.3523 log l ( m m ) - 7.2478 Substituting these regression c o e f f i c i e n t s for y i n the r e l a t i o n z = y(l-x) the values of the exponent x for males and females are 0.58 and 0.61 respectively. The data were also analysed by using the von Bertalanffy equation according to the method described by Ricker (1958)*. Lengths at various ages could be calculated from the equations.* Females l t + 1 = 8l(.104j + 0.8958 l t Males l t + 1 = 85 (• 10,4) + 0.8958 l t Observed and calculated lengths by Parker-Larkin and Von Bertalanffy^equations^are given i n Table III. *The d i f f e r e n t i a l equation of von Bertalanffy, = HS - KW —Xt can be shown to y i e l d the expression 1+^ = l Q O ( l - e ~ ) + T K l , l^.e~ . By p l o t t i n g 1^+^ against l^Ofalford plot) the constants 1 and K can be estimated. Al t e r n a t i v e l y , p l o t t i n g OO l o g j ^ l ^ - l o g ^ l O Q - l ^ . ) against t gives a line of slope-K and y intercept (-Kto). T r i a l values of 1 can be chosen to provide a best f i t for the equation. 12 Table I I I . Observed and calculated lengths. Total length i n centimeters Observed Parke r-Larkin Von Bertalanffy Age i n years 1 2 3 4 5 Males 10.5 18.3 24.0 30.4 35.5 Females 10.8 18.3 25.0 30.6 36.2 Males 10.68 17.80 24.14 30.00 35.53 Females 10.73 18.17 24.63 30.69 36.40 Males 10,50 18.14 25.09 31.32 36.90 Females 10,80 18.54 25.04 30.87 36.09 Both equations do an adequate job of prediction although the Von Bertalanffy expression tends to progressively overestimate lengths. 13 HA Iii BUT. (Hippoglossus stenolepis). Halibut of Portlock-Albatross grounds are used for the present investigation. Growth rates estimated for 1926 and 1956 are obtained from Table5, page 15 of the 28th report of the International P a c i f i c Halibut Commission. Average weights at each age given by the Commission's report were stated to have been obtained by converting the lengths to weight by using a length-weight table. Average weight i n pounds at each age of Portlock-Albatross halibut for 1926 and 1956 i s shown i n Figure 5. The plot of W ^ + ^ against W^. for 1926 data of Figure 6 appear to diverge from the 45° diagonal l i n e . When data on weights are used the minimum r e l a t i v e variance w i l l y i e l d an optimum value of (1-x) as i s evident from the equation. (1-x) (1-x) ¥ t + l 3 8 k ( l " x ) + T t Here for the sake of convenience k(l-x) i s denoted as O^" and thus we have (1-x) , (1-x) ¥ t + l - C* + * t The 6}ptimum value of (1-x) obtained from 1926 data was 0.5 and the corresponding growth formula i s : -0.5 0,5 V l = G - 2 7 6 + V t The plot of w j j ' against W^ *5 i s shown i n Figure;. 7 • Making use of the above growth formula.weights at various ages computed and are shown i n Table 4 along with the observed weights. Agreement between observed and calculated weights i s good. This i s hardly surprising since the observed values are 14 450 400 350 300 250 150 \-100 50 0 o 1956 o 200 (- ° o o o o o o o o o ° 1926 „ ° o o o ° o o ° ° o O ° n o o ° O _ O O _ o o " o o ° ° „ o o o o o •0 15 20 25 30 35 40 AGE IN Y E A R S F i g u r e 5. Growth c u r v e s showing a g e - w e i g h t r e l a t i o n s h i p 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 . 15 16 Table 4 . Weight i n pounds at ages 5 to 40 for Portlock-Albatross halibut i n 1926 Weight Age i n years Observed Calculated 5 3 3.0 6 4 4.0 7 5 5.2 8 6 6.5 9 8 8.0 10 10 9.7 11 12 11.5 12 14 13.4 13 16 15.5 14 18 17.8 15 21 20.2 16 24 22.7 17 26 25.4 18 29 28.3 19 32 31.3 20 36 34.5 21 39 37.8 22 42 41.3 23 46 44.9 24 50 48.7 25 54 52.6 26 58 56.7 27 62 60.9 28 67 65.3 29 71 69.8 30 76 74.5 31 81 79.4 32 85 84.4 33 90 89.5 34 96 94.8 35 101 100.2 36 107 105.8 37 113 111.6 38 118 117.5 39 124 123.6 40 130 129.8 18 actually calculated values from a linear logarithmic regression of weight on age - an algebraic equivalent of the Parker-Larkin equation i n which the slope i s the reciprocal of (1-x). However the calculations confirm the usefulness of the Parker-Larkin, equation i n describing the growth of an average f i s h . A plot of ¥j.+^ against for 1956 data i s shown i n Figure 8. In t h i s data also the l i n e of best f i t diverges from the 45° diagonal. A value of 0.45 was obtained for (1-x). It would be noted that a change i n 0.05 for the value of (l-x) would r e s u l t i n an enormous change i n o(| . For example, for 0.45 as (1-x) the mean value of 0 6 was 0.385 whereas i t was 0.539 for 0.45 0.45 a value of 0.5 for (1-x). A plot of against i s given i n Figure 9. The formula for depicting weight i s : -0.45 0.45 V t + 1 = 0.385 + ¥ t Weights at various ages are tabulated i n Table 5. The average weights at each age for 1926 and 1956 were converted to lengths at age from a length-weight relationship expressed as: log W = 3.0417 log L « 4.70054 This relationship was obtained by f i t t i n g a regression for average length i n centimeters and average weight i n pounds for ages from 4 to 25. The data madeouse of here are given i n Table 4 of the 8th report of the International PacificeHalibut Commission 1934. Figure 8. P lo t of ¥ t + 1 on ¥ t for Por i lock-Albatros ha l ibut for 1956. 20 21 Table 5 . Weight in pounds at ages 5 to 40 for Portlock-Albatross halibut in 1956 Weight Age in years Observed Calculated 5 5 5.0 6 7 7.3 7 10 10.1 8 13 13.4 9 17 17.3 10 21 21.6 11 26 26.5 12 32 32.0 13 38 38.1 14 44 44.7 15 52 51.9 16 60 59.7 17 68 68.1 18 77 77.2 19 87 86.8 20 98 97.1 21 108 108.0 22 120 119.6 23 132 131.8 24 145 144.6 25 159 158.1 26 174 172.4 27 188 187.2 28 204 202.8 29 221 219.0 30 237 236.0 31 255 253.5 32 274 271.9 33 293 291.2 34 313 310.7 35 333 330.9 36 354 352.3 37 376 374.0 38 400 396.9 ,39 423 419.9 40 445 444.3 22 Walford plots of 1^+^ against 1^ are shown i n Figures 10 and 11 for 1926 and 1956 respectively. In both plots the Walford l i n e of best f i t converges to the 4 5 ° diagonal. The values of -z estimated are 1.5 and 1.36 respectively. These estimates of z are obtained from the expression y( l - x ) = z 1.5 1.5 Transformed plots of 1^ + 1 against 1^ for 1926 and 1.36 1.36 1 t + l a& a i n s' f c l j . for 1956 are given i n Figures 12 and 13. The formulae for depicting the growth rates are: 1.5 1.5 1t+l = 5 5 , 3 1 + H 1 9 2 6 1.36 1.36 l+^i = 47.76 + 1. 1956 Observed and calculated lengths of Portlock-Albatross halibut for the years 1926 and 1956 are presented i n Table 6. The agreement between the observed and calculated lengths i s excellent. The Von Bertalanffy equation could not be applied to the weight data as the l i n e of best f i t on the Walford plot diverges from the 4 5 ° diagonal. I t was used for the length data for both the years. Average sizes can be obtained by using the following 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 Figures 5-9 and the values obtained for (l-x) i t may be concluded that halibut grew faster i n weight 23 halibut for 1956, 27 Table 6 . Length i n centimeters at each age for 1926 and 1956 Portlock-Albatross halibut 1926 1956 Age i n years Observed Calculated Observed Calculated 5 50.4 50.4 59.6 59.6 6 55.4 55.5 66.6 67.4 7 59.6 60.3 74.8 75.0 8 63.3 64.9 81.6 82.3 9 69.5 69.5 89.1 89.3 10 74.8 73.8 95.5 96.3 11 79.5 78.0 102.5 102.9 12 83.6 82.2 109.7 109.5 13 87.4 86.2 116.1 116.0 14 90.8 90.1 121.8 122.2 15 95.5 94.0 128.7 128.4 16 99.8 97.7 134.9 134.5 17 102.5 101.4 140.5 140.4 18 106.2 105.1 146.4 146.3 19 109.7 108.6 152.4 152.1 20 114.0 112.1 158.5 157.8 21 117.1 115.6 163.6 163.5 22 120.1 119.0 169.4 169.1 23 123.6 122.4 174.8 174.5 24 127.0 125.6 180.3 180.0 25 130.3 129.0 185.8 185.4 26 133.4 132.1 191.4 190.7 27 136.3 135.3 196.3 196.0 28 139.9 138.5 201.7 201.6 29 142.6 141.6 207.1 206.4 30 145.8 144.7 211.9 211.6 31 148.9 147.7 217.0 216.6 32 151.3 150.7 222.2 221.6 33 154.1 153.8 227.2 226.7 34 157.4 156.7 232.2 231.7 35 160.1 159.7 237.0 236.5 36 163.1 162.5 241.8 241.4 37 166.1 165.4 246.6 246.3 38 168.5 168.3 251.7 251.2 39 171.2 171.1 256.3 256.0 40 173.9 173.9 260.7 260.7 28 f o r the p e r i o d of 1956 compared to t h a t of 1926. The same i s tr u e f o r growth i n l e n g t h . Smaller values of (1-x) and z are the i n d i c a t i o n s of f a s t growth r a t e . 29 LAKE STURGEON (Acipenser fulvescens) Back calculated lengths of Nelson River lake sturgeon were kindly provided by Mr. L. A. Sunde of the Department of Mines and Natural Resources, Manitoba. The oldest f i s h from which back calculations were made was 55 years. For computational con-venience only the f i r s t 21 years of back calculated growth were used. Sexes were treated separately. Walford plots of 1^.+^  on 1^ for males and females are shown i n Figures 14 and 15 respectively. In both cases the l i n e of best f i t approaches the 45° diagonal. The value of z for both males and females 2.64 2.64 was 2.64. The transformed plot of l^.+2 against 1^ for males and females are shown i n Figures 16 and 17 respectively. Equations to express the growth rates are:-2.64 2.64 Males l t + 1 = 794.11 + l t 2.64 2.64 Females l t + 1 « 820.11 +- l t Lengths at various ages were calculated using the above equations and are given i n Tables 8 and 9. There was good agreement between observed and calculated lengths. There were no s i g n i f i c a n t differences i n the growth rates of the sexes (Table 7). Table 7. Analysis of variance on sturgeon for growth differences between sexes. Source of Variance d.f. Sum of squares Mean square F r a t i o Probability Total Sexes Individual 39 1 Ls38 1702641.10 6733.20 1695907.90 6733.20 44629.15 0.15 ^..01 30 32 34 Growth i n length was apparently rapid during the f i r s t year and decreased steadily to the age of 9 i n females and 8 i n males. Classen (1944) observed marked changes i n growth of Acipenser sturio at the ages of 8 and 9 years and attributed the occurrence to a l t e r a t i o n of the general metabolism, due to development of gonads. Cuerrier and Rowssow (1951) reported that male lake sturgeon matured sexually at approximately 14 years of age and females i n about 25 years. Probst and Cooper (1954) observed i r r e g u l a r i t y i n growth between 14 and 19 years of age. 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 River lake sturgeon cannot be explained because of the lack of information either on the environment or on the gonad development. Since the li n e s of best f i t on a Walford plot tend to converge to the 45° diagonal, the data were analyzed by using the Von Bertalanffy equation. The formulae obtained were -Males l t + 1 = 84(0.020) + 0.980 l t Females l t + 1 = 76(0.023) + 0.977 l t The predicted lengths at various ages by Von Bertalanffy equation for both the sexes are given i n Tables 8 and 9. It can be seen from the tables that the lengths are overestimated progressively by the equation. 35 Table 8. Observed and calculated lengths for male sturgeon Fork length i n inches Age i n years Observed Parke r-Larkin 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 calculated lengths for female sturgeon Fork length i n inches. Calculated Age i n years Observed Parker-Larkin Von Bertalanffy 1 7.53 8.53 mm 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 37 WHITE STURGEON (Acipenser transmontanous) Data for this study was procured from Pycha's (1956) publication on white sturgeon. Total lengths i n inches at capture of the various age groups were used to express growth rather than back calculated lengths. A plot of 1^.+^  on 1^ . i s shown i n Figure "18. It can be seen from the Walford plot that white sturgeon puts on large annual increments f o r the f i r s t 6 years and thereafter there are r e l a t i v e l y constant increments, i . e . 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 analysis using lengths at ages from 0-30 years z was estimated as 1.45. But the estimated lengths were under-estimated for the early ages and overestimated for the older ages. Since the line of best f i t on Walford plot runs p a r a l l e l to the 45° l i n e from a length of 38.5 inches (corres-ponding to 6 years of age) the data was s p l i t at thi s size and analysed separately. The Parker-Larkin equations for depicting lengths up to the 6th year and from the 6th year onward were respectively. ^ 1.89 V i - 1 5 1- 1 5 + h and 0.9 0.9 V l =  l-4037 + h 1.89 1.89 0.9 Transformed plots of 1^.+^  against 1^ and 1^.+^  against 0.9 1^ . are shown i n Figures 19 and 20 respectively. Lengths at various ages estimated by the above equations are tabulated i n Table 10. This example has provided an, excellent agreement between observed and calculated lengths. The analysis underlines the fact that the data should be s p l i t at a length of 38.5 inches as the white sturgeon follows a di f f e r e n t growth pattern above Figure 18. Plot of lt+1 o n h f o r white sturgeon. 39 4 0 41 Table 10. Observed and calculated lengths of white sturgeon from C a l i f o r n i a waters n years Observed Calculated Parker-Larkin Von Bertalanffy 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 80.5 18 66.0 66.2 ;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 t h i s s i z e . Since the l i n e of best f i t ana Walford plot for 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 application of the Bertalanffy equation i s not possible. Beverton and Holt (1959) tabulated the values of R and I* as * oo 0»06 and 300 centimeters (120 inches) respectively. This could only be possible i f the data on the older f i s h was ignored. The Von Bertalanffy equation for estimating growth with these constants i s : -l t + 1 = 120(0.0582) + l t(0.9418) Lengths at various ages predicted by t h i s equation are given i n Table 10. It i s evident from the table that the lengths are grossly overestimated at a l l ages above 4 years. 43 HERRING (Glupea p a l l a s i i ) . Scales of herring from the Bella B e l l a area were kindly-provided 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 Station at Nanaimo. Scales were read and the back calculated lengths were used for the present investigation. Fish used i n this study were f i v e years of age caught i n 1955 from the B e l l a B e l l a region. Body-scale - relationship Measurements of 290 scales from herring ranging i n fork length from 112 to 244 millimeters were recorded. A regression l i n e to show the relationship of fork length to the anterior radius of the magnified scale image was constructed and the appropriate formula i s : log L = 0.88380 + 0.705 log S The correlation c o e f f i c i e n t for this data i s 0.93 which 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 d i f f e -rent from unity (t = 17.56)} hence the use of d i r e c t proportion i n back calculation would not be v a l i d . Apparently the anterior radius of the scale grows r e l a t i v e l y slower than length. These observations are partly explained by the work of Guyn (1939) who observed on P a c i f i c herring that the growth rate of the anterior f i e l d of the scale i s greater than that of body length up to a body length of about 40-50 mm. Thereafter i t decreases to become less than body length growth rate. After the sixth year the scale again grows faster than the body length. The most convenient means of calculating the annual growth of the body from the growth of the scales would appear to be to construct a nomograph which took cognizance of the changing 44 scale to body length relationship. However, i f one i s dealing only with the central period of growth as i n the present case back calculations to lengths above 5 cm. should be accurate. There are two possible ways of back calculating the lengths at previous ages. ( l ) By assuming the slope i s constant and the intercept i s variable and (2) by keeping the intercept constant on the assumption that individual 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 obviously ridiculous i n such variable data, for i n back calculation the ranges i n sizes at the end of the f i r s t year would be enormous. Accordingly, a l l the back calcu-lations are made by the second method, keeping the intercept constant and assuming variable slopes. Growth Rate. Back calculated fork lengths i n millimeters of the 5 year old herring are used. The sexes are treated separately. Walford plots of 1^.+1 against 1^ for females and males are shown i n Figures 21 and 22. In both the figures the points could y i e l d a line of best f i t that would intersect the 45° diagonal. The values of z for the sexes were estimated and the correspond-ing equations are:-3.5 3.5 Females l t + 1 = 8856.22 + l t 3.1 3.1 Males l t + 1 = 2021.48 + l t z z Corresponding plots of 1^.+^  on 1^ are shown i n Figures 23 and 24. The data were analysed using the Von Bertalanffy equation 45 22.0 20.0 1 8 . 0 16.0 14.0 12.0 10.0 8.0 og o oo o o Q O U o °o o o o o o sP ° o m O g p g o o o 3 <fe o o o . o % o o o o o o o o ° o o o 80 10.0 12.0 14.0 16.0 18 0 20 0 22 F i g u r e 21. P l o t of l t + 1 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 r e g i o n . 46 22.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 F i g u r e 22. P l o t o f 1 ^ on l t f o r male h e r r i n g f r o m B e l l a B e l l a r e g i o n . 47 3.5 3.5 Figure 23. Plot of l t + 1 on l t for female herring from B e l l a B e l l a region. 48 49 and the growth rates can be depicted from the following formulae: Females l t + 1 = 25.4(.190) + 0.810 l t Males l t + 1 m 27.6(.122) + 0.878„l t Lengths at higher ages calculated by Parker-Larkin and Von Bertalanffy 1s equations are given i n Tables 11 and 12. Both the growth equations seem to predict the lengths equally well. By using the Parker-Larkin method lengths at e a r l i e r ages which do not l i e on a straight l i n e on the Walford plot could also be predicted, whereas application of the Von Bertalanffy* method i s confined to the l a s t three years of l i f e of the 5 year old herring. 50 Table 11. Back calculated and calculated fork lengths for female herring. Age i n Back Calculated years calculated Parker - Larkin Von Bertalanffy 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 calculated and calculated fork lengths for male herring. Age i n Back Calculated years calculated Parker - Larkin Von Bertalanffy 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 calculated lengths of 5 year old cutthroat trout caught i n 1958 from Kiakho Lake, B. C. were kindly provided by Mr. C. E. Stenton, Pish and Game Branch of B. C. A Walford plot of l.fc+1 <>n 1^ Is given i n Figure 25. The general trend of points for cutthroat trout i s p a r a l l e l to the 45° diagonal suggesting- that z = 1. The quadratic solution for z yi e l d s the value of 1.01j the computer solution was 0.94. The plot of 0.94 0.94 *t+l o n """t *"*s s ^ o w n * n figure 26. While the use of the Parker-Larkin equation would permit more accurate prediction of growth than a Von Bertalanffy l i n e f i t t e d on the Walford transformation, i t i s obvious that the Parker-Larkin equation does not eliminate the "hump" i n the scatter of points which occurs between the lengths of 10 to 20 centimeters. Table 13 gives the analysis of variance of values for various age interval 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 real existence of the "hump". Two explanations could be offered for t h i s hump: ( l ) there i s an i n f l e c t i o n i n growth rate at about 15 cm i n which case the data should be s p l i t at the i n f l e c t i o n and the two parts treated separately or (2) because d i f f e r e n t environmental conditions may have prevailed i n di f f e r e n t years, the year of growth which largely corresponds to the hump may have been p a r t i c u l a r l y favourable of the other years of growth unfavourable. 53 54 Table 13 • Analysis of variance on ^ values of cutthroat trout i n d i f f e r e n t age i n t e r v a l s . Source of variance d.f. Sum of squares Mean square Total 91 Mean 3 Individual 88 168.3548 61.4352 106.9196 20.4784 1.2149 F = 20.4784 = 16.8 P = <.01 1.2149 Both explanations f i n d support i n f i e l d data. In t h e i r f i r s t year. Kiakho Lake f i s h reside i n an outlet stream, migrating as yearlings between 10 and 15 centimeters into the lake. Hence there would be some j u s t i f i c a t i o n for s p l i t t i n g the data as representative of the two environments, just as Parker and Larkin (1959) did for steelhead trout. On the other hand, the stream environment i s s t r i k i n g l y variable from year to year i n i t s f a v o u r a b i l i t y for growth and survival of young cutthroat. Accordingly, d i f f e r e n t year classes enter the lake at d i f f e r e n t sizes and s t r i k i n g l y d i f f e r e n t densities. Each year class then would show a pattern of growth r e f l e c t i n g the p a r t i c u l a r conditions that prevailed i n the environment during i t s l i f e . This i s apparently true because 4 year old cutthroat caught i n 1958 show no hump at 10 to 20 centimeters. Moreover, they y i e l d an estimate of 0.7 for z (Stenton I960) which would r e f l e c t good growth conditions for larger f i s h combined with poor growth conditions for smaller f i s h . The analysis underlines that adequate estimation of z 55 hinges upon uniformity of environment. When the environment i s variable,, z could be calculated from observed increments i n growth i n the same year of f i s h of various sizes* In the Kiakho Lake situ a t i o n the added precaution 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 th i s wayj> C>C values for a par t i c u l a r year are indices of environmental conditions (as they should be according to Parker and Larkin). This procedure runs the r i s k of bias from selection f a s t growing f i s h by the fishery but i t seems a lesser e v i l than spurious estimation of z from fluctuating environmental conditions It i s also consistent with the contention that z i s a physiologi-cal constant and that differences i n observed growth rate are caused by changes i n environment. Lengths at various ages are calculated according to the equation 0.94 0.94 H+l " 4 ' 3 2 + h The observed and calculated lengths are shown i n Table 14 Table 14 • Observed and calculated lengthsof cutthroat trout 4 . „ Pork length i n centimeters Age i n years. 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 gairdnerii) S u i t a b i l i t y of an environment for f i s h i s re f l e c t e d i n the growth of the f i s h . For th i s purpose growth of rainbow trout from three lakes i n B r i t i s h Columbia was investigated. The lakes chosen for study were Paul Lake, Loon Lake and Beaver Lake. Paul Lake The growth of rainbow trout i n various years i n Paul Lake has been described i n several publications (Larkin et a l . 1950, Larkin and Smith 1954, Crossman and Larkin 1958). To avoid complications a r i s i n g from changes i n growth rate during the period of an explosive increase of redside shiners (Richardsonius  balteatus) the data selected for the present study apply to the 1946 year class, caught from 1946-49 as three year olds* A Walford plot of l ^ + j against 1^ for three year old rainbow trout i s shown i n Figure 27. These points could y i e l d a l i n e of best f i t that would intersect the 45° diagonal and hence the value of z could be expected to be more than one. By quadratic approximation z was estimated as 1.1, and by computer 1.3 1.3 1.3. Figure 28 shows 1^.+^  plotted against 1^ • The general equation for Paul Lake trout i n terms of length i s 1.3 1.3 l t + 1 = 39.5609 + l t Mean growth increments between ages are C< 1 2 « 43.7436 C X . _ = 35.3792 o o o 0 10 15 20 25 3 0 35 40 Figure 27 0 P l o t of l t + 1 on l t f o r rainbow t r o u t from Paul Lake, B. C. 58 59 Analysis of variance (Table 15)on QC values show s i g n i f i c a n t d i f f e -rences between ages. Table 15. Analysis of Variance of Growth Increments Between Ages. Source of Variance d.f. Mean square P r a t i o Probability Total 133 Means 1 2343.2247 Individuals 132 216.3224 10.8 <^.01 Prom t h i s i t may be inferred that 3 year Paul Lake trout grow faster i n their second year of l i f e than i s predicted (see Table 16 below). Thus the Parker-Larkin equation i s not a good f i t to the data - i . e . the rate of change of increments i s not describable by only two parameters. The data i s analyzed by the Von Bertalanffy equation as H+l ~  55(«323)  + °«677 1t and the calculated lengths are shown i n Table 16. Table 16. Comparison of observed and calculated lengths of Paul Lake rainbow trout. Age i n Observed Fork Calculated Fork Length i n years Length i n centimeters centimeters Parker-Larkin Von Bertalanffy 1 8.17 8.17 2 22.91 21.78 22.93 3 33.04 33.07 33.04 Analysis of rainbow trout growth data after the establish-ment of shiners i n Paul Lake indicates another possible source of error i n estimating z values. Back calculated growth data for three year old trout caught i n 1955 and 1956 y i e l d a z of 0.27, suggesting rapidly increasing increments which on extrapolation 60 to the fourth and f i f t h year would produce enormous trout of 51.0 cm and 91.17 cm respectively. The spurious z value can be explained from the work of Larkin and Smith (1954) on the growth of rainbow trout i n Paul Lake. Small trout eat plankton and bottom organisms for which there i s intensive competition by shiners. At lengths ranging from 15 to 25 centimeters trout switch to a diet of shiners during the summer months, their growth rate responding accordingly. Parker and Larkin (1959) denote this type of change as an "ecological growth stanza" and the data should obviously be s p l i t into two groups - f i s h below 15 cm and f i s h above 25 cm. Por rainbow trout from Paul Lake t h i s i s an impractical procedure because many f i s h mature at age 3. In consequence there are only two growth increments ( l to 2 and 2 to 3) available for z estimations. S p l i t t i n g the f i s h into two size groups results i n size hierarchy effects within each group. - which can cause underestimation of z values* The best procedure would seem to be calcu l a t i o n of z from pre-shiner data and using t h i s value, to estimate for small and large f i s h separately, any changes inOCoccasioned by the shiner introduction. The assumption would be made that z i s a ^physio-l o g i c a l constant," an assumption consistent with the contentions of Parker and Larkin. Paul Lake rainbow trout offer s t i l l another complication i n growth analysis, because of va 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 length and growth i n weight was calculated for data collected before and after the introduction of shiners into Paul Lake. F i s h were separated 61 according to size, stage of maturity and sex. The measurements of lengths and weights were converted to logarithms and regressions were calculated by the method of leas t squares. The length-weight relationship for the periods 1947 and 1959 were log ¥ = - 1.47528 + 2.75216 log L.. . . - r (1947 ) log ¥ = - 1.81648 + 2.91714 log L.. . . - r(l959) where ¥ = weight i n grams L = fork length i n centimetres. Analysis of covariance was applied to test differences i n the length-weight relationship among the years 1947 and 1959. The relationship was found to d i f f e r s i g n i f i c a n t l y at P ^.01 with respect to the regression c o e f f i c i e n t and the adjusted means. For each period separately, the relationships for the f i s h below and above 25 cm. i n length are 1.57847 + 2.82398 log L 1.04865 + 2.47248 log L 1.86505 + 2.95497 log L 2.06575 + 3.07974 log L A comparison of slopes for f i s h below 25 cm. i n size for 25 cm. log =25 cm. log ¥ = 1959 -<25 cm. log ¥ -25 cm. log ¥ = -62 the periods 1947 and 1959 was not s i g n i f i c a n t but for f i s h above 25 cm i n size the slopes were s i g n i f i c a n t l y different at the 1% l e v e l . For 1947 the slopes for f i s h below and above 25 cm i n size 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 for the period 1959. In 1947 trout below 25 cm size were r e l a t i v e l y heavier than the larger f i s h , whereas the reverse was true i n 1959. The explanation for t h i s phenomenon would appear to be available from the history of the lake. For the period 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 di e t of trout of various sizes (Larkin and Smith 1953). In contrast to 39.8$ amphipods i n the diet during 1931, i n 1947-49 Daphnia formed the major food item for a l l sizes (Larkin et a l . 1950). Presumably, the sc a r c i t y of Gammarus did not aff e c t growth rates of trout of less than 25 cm size because of the abundance of Daphnia. But for trout above 25 cm i n size Daphnia were perhaps an inadequate source of food, and with competition •^or Gammarus, growth rates were low. Moreover, i t would be expected that during the 1946-47 period, trout of smaller size would be i n r e l a t i v e l y better condition than large trout. This was evident i n the slopes of 2,47 and 2.82 for large and small trout respectively. From 1952 onward trout over 25 cm started preying on shiners, while f i s h of small size were adversely affected by competition with shiners for plankton and bottom organisms. As a r e s u l t , trout regression c o e f f i c i e n t s for 1959 indicate r e l a t i v e l y 63 better condition of the larger f i s h . Moreover for trout above 25 cm. the regression 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 different for the years 1947 and 1959 i . e . before and after the introduct-ion of shiners and large trout of 1959 were heavier than those of 1947. The competition for food between shiners and small trout was not re f l e c t e d i n a lower condition of small trout as compared to the pre-shiner priod. There were also changes i n the length-weight r e l a t i o n -ships with regard to sexes, and maturity (see Table 17), Table 17. Log length log weight r e l a t i o n of ;raihbow trout of various sizes, sexes and stages of maturity from Paul Lake, B. C. Slo pe Intercept 1946-47 1957-59 1946-47 1957-59 <^25 cm. 2.82398 2.95497 -1.57847 -1.86505 ^ 25 cm 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 64 Length measurements are thus inadequate indications of the weight increments and b a s i c a l l y growth comprises weight increments* Considering a l l of the above observations, the Parker-Larkin growth equation would appear to be inadequate for description of the growth of rainbow trout i n Paul Lake. The short l i f e cycle, change i n food habits and changes i n length-weight r e l a t i o n seem to m i l i t a t e against the use of any theore-t i c a l system of orderly related increments. Loon Lake Back calculated lengths of 3 year old rainbow trout caught i n 1952 were used. A Walford plot of l ^ . + j against 1^ i s shown i n Figure 29* These points tend to converge to the 45° diagonal* Analysis of the data yielded a z value of 1.3. The 1.3 1.3 plot.' of 1^ + 1 against 1^ is- shown i n Figure 30. The Parker-Larkin growth equation for rainbow trout of Loon Lake i s 1.3 1.3 V l = 2 4 ' 4 6 + h The agreement between the observed and calculated lengths was very satisfactory. (Table 18). Table 18. Comparison of observed and calculated lengths of Loon Lake rainbow trout. Age i n Fork length i n centimeters Observed Calcu: ated years Parker-Larkin Von Bertalanffy 1 7.08 7.06 2 15.97 15.81 15.99 3 23.17 23.18 23.60 Loon Lake, B. C. 6 6 Loon Lake, B. C. 67 In Loon Lake where there are only rainbow trout , the decline i n growth rate follows a def in i te trend with increasing length (Larkin et a l . 1950), As a re su l t the estimate of exponent z i n the Parker-Larkin equation i s a r e l i a b l e measure to express growth rate of rainbow trout i n Loon Lake. The data were analysed by the Von Bertalanffy equation as the l ine of best f i t of the Walford p lo t coverges to the 4 5 ° diagonal . The estimated equation i s l t = 60(.173) + l t 0.8270 The agreement between observed and calculated lengths i s as good as that of Parker-Larkin equation (Table 18). Beaver Lake Back calculated lengths of three year old rainbow trout of the year classes 1952 and 1953 caught i n 1955 and 1956 respect ive-l y are p lot ted to give the Walford l ine represented i n Figures 31 and 32. For the 1952 year class the l i n e of best f i t would i n t e r -sect the 4 5 ° diagonal and the value o f . z m s estimated as 1.14. Figure 33 shows the transformed data ra i sed to the power 1.14. Analys is of data for the 1953 year c lass gave a z of 0.65 which indicates that the growth increments get bigger as the f i s h grow .65 .65 o lder . The p lo t of against 1^  i s shown i n Figure 34. These two d i f f erent values of z might be due to the varying growth rates of the year classes responding accordingly to the strength of the year c lasses . A s imi lar s i tua t ion can be demonstrated i n the data from Paul Lake, where z values from 0.8 to 1.4 were obtained for d i f f e -rent i n d i v i d u a l year classes from 1946 to 1949. I t may be summa-r i z e d on the basis of these observations that the appl i ca t ion of the Parker-Larkin equation i s made d i f f i c u l t for rainbow trout due to short l i f e span and var ia t ions i n year class strength. 68 69 71 72 SCALLOPS (Placopecten magellanicus) Br. L. Dickie of the A t l a n t i c B i o l o g i c a l Station, St. Andrews, New Brunswick, kindly provided back calculated s h e l l heights of scallops (Placopecten magellanicus) which were used i n his study of t h i s species on various A t l a n t i c seaboard grounds (Dickie 1954, 1955), Scallops from Hour ground and Buoy ground are used for the present study. The Walford plot of 1^.+1 on 1^ i s shown i n Figure 35 for scallops from the Hour ground. Growth i s sigmoid, so that the points f i r s t diverge from the 45° diagonal up to a s h e l l height of 70 to 80 mm, corresponding to an age of approximately six years. Beyond t h i s s h e l l height the l i n e of best f i t approaches the 45° diagonal, thus showing an accelerating growth and then a decelerating growth. For this reason the Von Bertalanffy formula can only be applied to the older specimens. For the Parker-Larkin method the data must be s p l i t at the point of i n f l e x i o n i . e . approximately at the age of six years. For the f i r s t six years of growth, analysis of the data yielded a 0.625 z value of 0.625. The Parker-Larkin transformation of 1.,, on 0.625 t + 1 1^ i s shown i n Figure 36. The formula for expressing, the growth during the accelerating growth phase i s : -0.625 0.625 l t + 1 = 2.798 + l t Calculated and observed s h e l l heights are shown i n Table 19. The decelerating phase of growth from six to nine years yielded a z of 3.6. The equation for the decelerating phase of growth i s : -3.6 3.6 1T+1. = 6 7 4 « 4 6 9 + H 75 Table, 19. Comparison of observed and calculated s h e l l heights of scallops i n millimeters from Hour ground. Age i n Calculated years Observed Parker - Larkin 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 of computations the heights are expressed i n centimeters and thus the above equation depicts the growth i n centimeters. The calculated values from six to 3.6 nine years are given i n Table 19, and the plot of l + . i against 3.6 1 H" 1 1^ i s shown i n Figure 37. Since the li n e of best f i t for the decelerating growth period approaches the 45° diagonal on Walford graph, the data were also analysed by the Von Bertalanffy method. Lengths at various ages could be obtained from the equation:-l t + 1 = 126(.122) + 0.878 l t The calculated lengths are shown i n Table 19. -1 J 78 Scallops from Buoy ground from one to six years of age were also analysed. The plot of 1^+^ against 1^ i s shown i n Figure 38. It i s clear from the figure that there i s a change i n the growth pattern beyond the 6th year of l i f e or between 60-80 mm of s h e l l height. The li n e of best f i t has a diverging trend from the 45° diagonal. A value of 0.375 for z has been 0.375 0.375 estimated. The transformed plot of l ^ + ] _ on 1^ i s shown i n Figure 39. The Parker-Larkin equation to predict the s h e l l heights during the accelerating growth phase i s : -0.375 0.375 V i " ° - 6 5 6 2 + h Observed and predicted values are given i n Table 20. Table 20. Comparison of observed and calculated s h e l l heights of scallops from Buoy ground. Age 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 The Parker-Larkin equation i s s a t i s f a c t o r i l y applied to the scallops of Hour ground and Buoy ground which i s evident from the agreement between observed and calculated values. 79 8 0 the Buoy ground. 81 DISCUSSION AND CONCLUSIONS The usefulness of an empirical 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 . It i s solely on this basis that the Von Bertalanffy equation has had a wide and varied use i n f i s h e r i e s biology. The parameter K of Von Bertalanffy's equation i s supposed to be proportional to the c o e f f i c i e n t of catabolism i . e . i t i s the rate at which the animal attains the e&symptotic size. Intra and inter species growth comparisons nearly always show that K i s high when L^is low and vice versa (Holt I960). Taylor (1959) showed that changes i n the value of K are temperature dependent. He also showed (1959 and I960) the inverse relationship existing between K and Lc© f or cod and razor clam. The values of K and L ^ of Von Bertalanffy and z of Parker-Larkin equation obtained i n the present investigations are given i n Table 21. Table 21. Growth parameter of Von Bertalanffy and Parker-Larkin equations. Species Sex Loo K z Eopsetta jordani M 85 cm. 0.11 1.3 P 81 cm. 0.11 1.3 Clupea p a l l a s i i M 27.6 cm. 0.13 3.1 P 25.4 cm. 0.21 3.5 Salmo g a i r d n e r i i Paul Lake 55 em. 0.39 1.3 Loon Lake 60 cm. 0.19 1.3 Hippoglossus 1926 232 cm. 0.16 1.5 stenolepis 1956 400 cm. 0.24 1.36 Acipenser M 210 cm. 0.02 2.64 fulvescens P 180 cm. 0.023 2.64 82 It can be seen from the table that there i s an inverse relationship between L ^ a n d z i n the same way as between Leo and K. From th i s i t may tentatively be concluded that the parameter z of the Parker-Larkin growth equation i s an index of the physiological a c t i v i t y . Parker and Larkin ( 1 9 5 9 ) suggested that x or z of their growth equation may be derived from a comparative study of metabolic:rate over a range of siz e . The values of z can also be explained i n terms of the factors that a f f e c t L G©. Due to i t s p l a s t i c i t y , growth i s affected by the a v a i l a b i l i t y of food material. The a v a i l a b i l i t y of food i s dependent not only on the physico-chemical factors of the environment but also on the density of the population. In terms of the Von Bertalanffy equation i t i s the parameter Looor Woa that i s affected by variations i n the food consumption (Beverton and Holt 1 9 5 7 ) . From the present investigation on trout from Paul Lake and Beaver Lake i t i s evident that the values of the parameter z were variable which was explainable by varying year class strength and ensuing competition for food. When there i s no s u f f i c i e n t food there i s a lessening of Leo and higher Loo where there i s s u f f i c i e n t food. In order to make the empirical growth data linear i t requires a high z value i n the former case and a low z i n the l a t t e r instance. I t i s too early to attribute any physiological interpretation to the parameter z. Assuming that f i s h growth i s isometric the exponent x of ^  = kw , appears to serve as a measure of the complex of physiological processes. Parker and Larkin ( 1 9 5 9 p. 7 2 6 , F i g . l ) mentioned that —the—value of z i s l i k e l y to l i e between 1 . 0 and 1 . 5 , i f the 83 data appear to approach the 45° diagonal. This i s true for the chinook salmon they worked on. In the present series of obser-vations the value of z i s more than 1.0 and a value as high as 3.6 was obtained. For trout the value of z was between 1.0 and 1.5. From th i s i t may be concluded that this range of values i s true for salmonids only. It i s evident from t h i s study that the Parker-Larkin growth equation can be applied to many aquatic organisms and i n many instances the agreement between the observed and calculated values i s good. However, to evaluate the usefulness of the various constants as tools of physiological and/or ecological events of the growth pattern, further work i s suggested, probably i n the experimental f i e l d . 84 SUMMARY The Parker-Larkin equation dw/dt = kwx has been f i t t e d to observed data on lengths, weights of fishes and heights of scallops. The following i s the summary of the findings. (1) Conversion of z to values of x should not assume that the exponent r e l a t i n g to length to weight i s necessarily 3. (2) Back calculated lengths may r e f l e c t bad and good growth years and may give a spurious estimation of a z value appropriate for comparisons. In a variable environment z should be calculated from increments i n growth for f i s h of various sizes i n the same year even though t h i s procedure may be biased by selection of f a s t growing f i s h by the fi s h e r y . (3) In short l i v e d species with highly variable growth rates combinations of complications make the estimation of z from f i e l d data highly unreliable. In rainbow trout from Paul Lake i t i s necessary to recognize ecological growth stanzas. However the component "stanzas" are then inadequate for z estimation because of the great v a r i a b i l i t y i n growth rate and selection of f a s t growing f i s h by the fi s h e r y . Early maturity and d i f f e r i n g length-weight relationships for both sexes and stages of maturity etc. further confound the analysis. (4) Separation of'ecological growth stanzas should be based on a size rather than on age c r i t e r i o n to avoid bias from extremely f a s t or slow growing individuals. (5) The Von Bertalanffy equation was found to overestimate the size i n the older ages i n many species. 85 (6) F i s h from fresh water as well as from the marine environment are described adequately by the Parker-Larkin equation. (7) When the growth increments decrease at 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 into two stanzas for analysis. (8) When the line of best f i t on Walford plots tends to approach the 45° diagonal the value of z l i e s between 1.0 and 1.5 i n the case of salmonids. In other species a value as high as 3.6 i s obtained. (9) Tentative b i o l o g i c a l interpretation i s attempted to explain the parameter z of the Parker-Larkin equation by drawing a comparison with the parameters of the Von Bertalanffy equation. (IG) The regression equation of the body-scale relationship i s used only to obtain the value of the intercept. A l l the back calculations are made by keeping the intercept constant with variable slopes for the individual f i s h . 86 LITERATURE CITED Anonymous. 1960. U t i l i z a t i o n of P a c i f i c Halibut stocks: y i e l d per recruitment. Rept. Int. P a c i f i c Halibut Comm., No. 28, 52 pp. Von Bertalanffy, L. 1938. A quantitative theory of organic growth. Human Biology 10(2): 181-213. Von Bertalanffy, L. 1957. Quantitative laws i n metabolism and growth. Quart. Rev. B i o l . , 32(3): 217-231. Beverton, R. J . H. and S. J . Holt. 1957. On the dynamics of exploited f i s h populations. U.K. Min. Agr. and Pish., Pish. Invest., Ser. 2, 19: 533 pp. Beverton, R. J . H. and S. J. Holt. 1959. A review of the l i f e -span and mortality rates of f i s h i n nature and their r e -l a t i o n to growth and other physiological c h a r a c t e r i s t i c s . Ciba Foundation Colloquia on Aging. Vol. 5. Boone, J . A. M.S. An assessment of the length-weight r e l a t i o n -ship of rainbow trout, 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 Clear Lake, Iowa. 1935-57. Trans. Am. Fish . S o c , Vol. 90, No. 2: 130-138. Classen, J . E. A. 1944. Estudio bio-estadistico del esturion o sa l l o del Guadalquivir (Acipenser sturio L.). Inst. Esp. de Oceanogr., Madrid, No. 19: 112 pp. Clemens, W. A. and G. V. Wilby. 1949. Fishes of the P a c i f i c coast of Canada. B u l l . Fish. Res. Bd. Canada, No. 48: 368 pp. Cuerrier, Jean-Paul and G. Rowssow. 1951. Age and growth of lake sturgeon from Lake St. Francis, St. Laurence River. Canadian Fish Cult., No. 10: 17-29. Dickie, L. M. 1955. Fluctuations i n abundance of the giant scallop, Placopecten magellanicus (Gmelin), i n the Digby area of the Bay of Fundy. J . Fish. 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 scales of the P a c i f i c Herring (Clupea p a l l a s i i ) . M.A. thesis, Univ. of B r i t i s h Columbia. Holt, S. J . 1960. "Letters to the editor." J . du Conseil, Vol. 25, No. 2. Larkin, P. A., G. C. Anderson, W. A. Clemens and D. C. G. Mackay, 1950. The production of Kamloops trout, (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. and S. B. S m i t h . 1954. 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 L a k e , B r i t i s h C o l u m b i a . 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 and R. R. P a r k e r . 1957. S i z e as 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  g a i r d n e r i i . T r a n s . Am. F i s h . S o c , 86: 84-96. L a r k i n , P. A. and K. V. A y y a n g a r . 1961, A p p l i c a t i o n o f t h e P a r k e r e q u a t i o n t o g r o w t h o f a q u a t i c o r g a n i s m s . P r o c . 1 1 t h A l a s k a S c i . Congress. ( V ^ " f ( : S S ) O s t l e , B. 1956. S t a t i s t i c s i n r e s e a r c h . ( B a s i c c o n c e p t s and t e c h n i q u e s f o r r e s e a r c h w o r k e r s ) . Iowa S t a t e C o l l e g e P r e s s . 487 pp. P a r k e r , R. R. and P. A. L a r k i n . 1959. A c o n c e p t o f g r o w t h i n f i s h e s . J . F i s h . R e s . Bd. Canada, 1 6 ( 5 ) : 721-745. P r o b s t , R. T. and E . L. C o o p e r . 1954. Age, g r o w t h and p r o d u c -t i o n o f t h e l a k e s t u r g e o n ( A c i p e n s e r f u l v e s c e n s ) i n t h e Lake W i n n i b a g o r e g i o n , W i s c o n s i n . T r a n s . Am. F i s h . S o c , 84: 207-227. P y c h a , R. L. 1956. P r o g r e s s r e p o r t on w h i t e s t u r g e o n s t u d i e s . C a l i f . F i s h and Game, 4 2 ( 1 ) : 23-35. R i c k e r , W. E . 1958. Handbook o f c o m p u t a t i o n s f o r b i o l o g i c a l s t a t i s t i c s o f f i s h p o p u l a t i o n s . B u l l . F i s h . R es. Bd. Canada, No. 119: 300 pp. S n e d e c o r , G. W. 1956. S t a t i s t i c a l methods. 5 t h E d i t i o n . Iowa S t a t e C o l l e g e P r e s s . 534 pp. 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 t r o u t (Salmo c l a r k i i l e w i s i G i r a r d ) i n K i a k h o L a k e , B r i t i s h C o l u m b i a . M.S. t h e s i s , U n i v . o f B r i t i s h C o l u m b i a . S t e v e n s o n , J . A. and L. M. D i c k i e . 1954. A n n u a l g r o w t h r i n g s and r a t e o f g r o w t h o f t h e g i a n t s c a l l o p P l a c o p e c t u s m a g e l l a n i c u s ( G m e l i n ) i n t h e D i g b y a r e a o f t h e Bay o f Fundy. J . F i s h . R e s . Bd. Canada, 1 1 ( 5 ) : 660-671. T a y l o r , C. C. 1958. Cod g r o w t h and t e m p e r a t u r e . J. Cons. I n t . E x p l o r . Mer, 2 3 ( 3 ) : 366-370. T a y l o r , C. C. 1959. Tempe r a t u r e and g r o w t h o f t h e r a z e r c l a m . J . Cons. I n t . E x p l o r . Mer, 2 5 ( 5 ) : 93-101. Thompson, W. F. and 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 t h e 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 g e a r . R e p t . I n t . P a c i f i c H a l i b u t Comm., No. 8: 49 pp. ~ W a l f o r d , L . A. 1946. A new g r a p h i c method o f d e s c r i b i n g t h e g r o w t h o f a n i m a l s . B i o l . B u l l . , 9 0 ( 2 ) : 141-147. 

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