NOTES ]1 27

HART, S. D., A N D W. P. HOWMILLER. 1975. Studies On the decomposition of allochthonous detritus in two southern California streams. Verh. Tnt. Verein. Lininol. 19: 1665-1 674.

MERRIT.~, K. W., AND K. W. CUMMHNS. 1978. An introduction to the aquatic insects of North America. Kendall/Hunt, Dubuque, Iowa.

PETERSEN, W. C., AND K. W. CUMMINS. 1974. Leaf processing in a woodland stream. Freshwater Biol. 4: 343-368.

REICE, S. R. 1974. Environmental patchiness and the break- down of leaf litter in a woodland stream. Ecology 55: 1271-1282.

REICE, S. K. 1977. The role of anirnal associations and current velocity in sediment specific leaf litter decon~position. Oikos 29: 357-365.

SEDELL, J. K., F. J. TRISKA, AND N. S. TRISKA. 1975. The pro- cessing of conifer and hardwood leaves in two coniferous forest streams: 1. Weight loss and associated invertebrates. Verh. Int. Verein. Limnol. 19: 1617-1 627.

SIIORT, R . A., S. 19. CANTON, AND J. V. WARD. 1980. Detrital processiilg and associated macroinvertebrates in a Colorado mountain stream. Ecology. (In press)

TRISKA, F. J., AND J. R. SEDELL. 1976. Decomposition of four species of leaf litter in response to nitrate manipulation. Ecology 57: 783-792.

WARD, J. V. 1976a. Comparative litnnology of differentially regulated sections of a Colorado mountain river. Arch. Hydrobiol. 78 : 310-342.

1976b. Effects of thermal constancy and seasonal temperature displacement on community structure of stream macroinvertebrates, p. 302-307.111 G. W. Esch and R. W. McFarlane [ed.], Thermal ecology 11, ERDA Symposium Series (SONF-740425).

WARD, J. V., AND W. A. SHOIIT. 1978. Macroinvertebrate com- munity structure of four special lotic habitats in Colorado, U.S.A. Verh. Int. Verein. Limnol. 20: 1382-2387.

WARD, J. V., AND J. A. Stanford [ed.] 1970. The ecology of regulated streams. Plenum Publishing Corp., New York, N.Y. (In press)

A Motion for the Retirement of the Von Bertalanffy Function

Dcp.purfm~.t2f o j Fisheries and Ocearls, Fislzeries mzd A'Murine Service, P.O. Box 5667, Sf. Jolzn's, Nfld. A I C 5x2

ROI:F, D. A. 1980. A motion for the retirernent of the von Bertalanffy function. Can. 9. Fish. Acluat. Sci. 37: 127-129.

The reasons for the continued use of the von Bertalanffy equation are examined. It is suggested that these reasons are insutticient to overcome the drawbacks of this function and alternative procedures are recommended.

Key words: von Bertalanffy, growth, yield per recruit

RUFF, D. A. 1980. A motion for the retirement of the von BertalantTy function. Can. J . Fish. Acluat. Sci. 37: 127-129.

Nous analysons les raisons pour lesquelles on continue d'utiliser l'kquation de von BertalanfTy. Nous suggkrons que ces raisons sont insuffisantes pour l'emporter sur les dksavan- tages de cette formction, et rmous recornmandons d'autres methodeb.

Received June 18, 1979 Accepted October 4, 1979

ONE of the most ubiquitous equations in the fisheries l i terat~lre is the von Bcrtalanffy function. Historically it gained wide currency because it claimed to be developed f rom bioenergetic principles. However, as pointed out by several authors (Parker and Larkin 1959; Paloheirno and Dickie 1964; Ursin 1967) this equation, from a bioenergetic perspective, iq a t best a special case and at worst noilsense (Knight 1968). Given that the equation can no longer be justified o n the grounds upon which it was developed. why has it remained so much in use'?

Printed in Canada (55680) Imprim6 au Canada (55688)

R e ~ u le 18 juin 1979 Accept6 le 4 octobre 1979

I n this note I examine the reasons for this and suggest that they d o nut outweigh the disadvantages of the von Bertalanffy equation.

Probably the most general reason for the continued use of this equation is that it is a convenient statistical description of the data. But is it so convenient'? There a re principally two statistical reasons for selecting a particular equation: firstly, the equation m a y be very flexible and fit a wide range of data and secondly the equation may be easily fitted to the data. T h e first is certainly true of the von Bertalanffy function but the second is decidely not true, as testified by the recur- rence of papers providing methods of fitting it and other

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128 CAN. J . FISH. AQUAT. SCI., VOL. 37, I980

logistic type curves (Walford 1946; Yoshihara 1956); Tornlinso~m and Abramsor~ 1961 : Fabens 1965; Grosen- ba~agh 1965; Al1e1-i 1966: Bhattacharya 1946; Ricklefs 1963; Rafail 19'73; R. K. Misra unpublished data). Indeed, if cwe was given the task of designing an equa- tion that was extremely dif'f'ac~mlt to fit in a statistically satisfactory manner the vora Bertalanffy function is pre- cisely the sort one might come up with! Its particular form defies transformation to nonle statistically tractable model and all methods call for iterative solutions which make statistical testing dificult. Altho~agh, as previously stated, the K on Bertalarsff'y function is flexible with respect to its shape it always assumes that an asymptotic length exists. which, when there is no evidence of curvature in the data leads to some ridiculous fits (Knight 1968 b . From a statistical point of view, what we require is a function that is flexibIe in shape, reduces to a linear futsction, and can be fitted to data lasing some standard technique such as least-sqarares estima- tion. Knight 6 1969) suggested that in some circeran- stanses a parabc~la might be a reasonable model. Rafail (1972) suggested a modified parabola as an alternative model and showed that in several fish species, plaice, haddock, North Sea cod, Nc~rth Sea sole, and Serra~zris crlandrifrnus, this parabola gives as good a fit or better than the voni Bertalandfy equation. Rafail'c n~oclification nukes the fitting of this eya~atiorr rather tedious. How- ever, the normal form of the parabola, L, = ca + bt + ctZ7 where I - , is 1cngtl-a at time t ant1 a , k , c are coalstants (whish can he fitted by thc standard least-squares method) is quite suficient. This may be demonstrated by cor~sidering how well a parabola can be fitted to data generated under the most restrictiale assumption that fish are actuallv growing according to the von Wertalanffy ftrnction.

The nunmber of ages ob\erved in a pc~pulation varies with k and Lr (Beverton and Holt 1959). This variation must be taken into consideration when comparing the parabolic fit to the von BertalanEy eq~aation. Using the data OF Beverton rend Holt ( 1059) to define the cet of most frequently observed combinations of L,, k , and number of ages H compared the difference in predicted sizes between the von Bertalanffy function and a parabola fitted to this eq~aation by thc method sf

Beast squares. This measure is defined as

x 100%, &here V, is the predicted size given by the von BeH-talanffgr function, P, that from the parabola and PI

the number of age-groups. The largest mean percent- age differer-acc between the two functions is only ap- proximately 5 % and for over 90% of the combinations the diff'erence is less than 3 % . This difference would have been even smaller if the first 1 to 6 age-groups (the number depending on the particuar combination of E x and k ) had been ignored as is done in most studies because these age-groups are not caught. Thus, eve?? if the fi,sh qr.t.ttl accordistg to the ttorz B c r r ~ l w ~ l f f y flrrrction the fit obtained with the parabola would be, in most cases, indistingui~hable from that obtained with

the von Bertalanffy equation. It would seen) that the parabola will fit a wider range of data than even Knight (1968) gives it credit for! The great advantage of the parabola is that it may be fitted by standard st:ntistical techniques which are available in "canned" statistical coniputer proyramcs. F~artl-aer, multiple growth curves can be ctsrnpared statistically and thc e=qmatiorr reduce? to a linear function and hence there is no attempt to force asymptotic behavior when none is observed.

It shoamld not be thought that I am advocating the replacement of the von Bertalanffy function with the parabola. The choice o i a partic~ilar equation should be dictated by circumstances: for cxarnple. if a , is very small the growth curve may be lineari~ed. in sorne cases. by taking logarithlsas of the variable5 (this proccd~ire may also stabilize the variances which is desirable for linear regression anslycis). The analysis of the growth of witch (G/.vptscc~phalras c.yr~oglosslas) by Bowers (1960) clearly den~onstrates the usefulness of this approach.

Another re:tson for renlaining faithFul to the von Bertalal-affy function is that it is lased in the Beverton- Holt yield per recruit model. In the days of abaci and slide rules an easily obtained solution of this model for an arbitary growth function was not available and hence tl-ae solutiorrs given by Beverton 6 1954) nnci Jones ( I'd57 1, presuming a von Bertalanff y growth equation, were ol considerable value. But thew are the days of the digital computer and there are a host of "'canned9' roaltines available to integrate the yield furlction nun~erically. There is, therefore. absolutely no reason to be tied to any particular growth function.

The final reason for continuing to use the yon Bea-talanffy function is that it permits a comparison of growth curves already found in the literature. How- ever, the variety of techniques used to fit this equation and the clifferences in their errors make comparisons, albeit non-statistical, highly suspect. However, if con-a- parisons are desired there is no reasor-e why the von BertalanfTy equation should not be fitted to the data for this purpose but another model used for other pur- poses, although for thc reasons given I do not favor this practise with respect to equations that do not per- m it statistical comparison.

Whether the von Bertalanffy growth eq~aatic~n was an intellectual breakthrough or a statistical blunder is a subject for historians of science: hut it has most as- suredly outworn its usefulness and I paat forward the motion for its retirement.

Acknowledgmerlds - I thank Drs D. J. Fairbairn and W. K. Misra, Departnaent of Fisheries and Oceans, Re- search and Resources Service, St. John's. Nfld., for their helpful criticisms in the preparation of this paper. The comments of the rcferces were sf considerab1e value.

ALLEN, K. W. 19GG. A method of fitting growth curves of the von Bertalanffy type to observed data. J. Fish. Mes. Board Can. 23 : 163-1 79.

BEVERTCZN, W. J. H. 1954. Notes on the use of theoretical

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NOTES 129

models in the ~ t u d y of the dynamics of exploited fish populatior~s. U.S. Fish. Lab., Beaufort, N.C. Misc. Contrib. 2: 159 p.

REVLRTON, R. J. H., AND S. J . WOLT. 1959. A review of thc lifespan and mortality rate5 in nature and their relation to growth and other physiological characteristics. Ciba Found. Colloq. Ageing 5 : 142-147. (The lifespan of animals.)

W~TATTACIIARY~, C. G. 1066. Fitting a class of growth curves. Sankhya ZPB( I !2): 1-10.

Rowkns, A. R. 1960. Growth of the witch (Giyprocepiral~a~ q~rogloss~rs L.) in the Irish Sea. J. Cons. Int. Explor. Mer. 25: 168-176.

FABLNS, A. T. 1965. Properties and fitting of the von Berta- Banfry growth curve?. Growth 29: 265-289.

GR~SLNBALGII, L. W. 1965. Generalization and regaratneteriaa- tion of some sigmoid and other non-linear f~inctio~ms. B~ornetrics 21 : 708-714.

Joxss, R. 19-57. A n ~ u e l ~ simplified ver4on of the fish yield ecluation. Doc. No. P. 21, prewsted at the Libbola joint meeting of Bnt. Cornrn. Northw. Atl. Fish., Tnt. Counc. Explor. Sea, and Food Agric. Organ., United Nations, 8 p.

KNIGHT, W. 1968. Asymptotic growth: an example of non- sense disguised as mathematrcs. J. Fish. Res. Board Can. 25 : 1 303-1307.

1969. A formulation sf the \Ion Bertalanffy growth curve when the growth rate is roughly constant. J. Fish. Res. Board Can. 26 : 3069-3072.

P.I\LOHEHMO, 9. E., AND L. M. BHCKIE. 1964. Food and growth of fi$hes. 1 . A growth curve derived from experimental data. J. F i ~ h . Res. Board Can. 22: 521-542.

PARKER, R. R., A N D P. A. LARKIN. 1959. A cencept of growth in fiches. 9. Fish. Rcs. Board Can. 16: 721-745.

RAFAIL. S. Z. 1972. Fitting a parabola to growth data of fishes and some alsplicatioas to fisheries. Mar. Biol. 15: 255-264.

1973. A simple and precise method for fitting a von UertalanfTy growth curve. Mar. Riol. 19 : 353-358.

RICKI~K, W. E. 8975. Computation and interpretation of biological $tatistics of fish populations. Hull. Fish. Res. Roard Can. 191 : 382 p.

I%H~.KI.E~S, R. E. 1967. A graphical method of fitting equations to growth curves. Ecology 48: 978-983.

~OR~LINSON, P. K., AND N. J. ADRAMSON. 1961. Fitting a von Bertalanffy growth curve by least squares: including tables of polynomials. Calif. Dep. Fish. Game Fish. Bull. 116: 69 g.

URSIN, E. 1967. A rnaathematical model of some aspects of fish growth respiration and mortality. J. Fish. Res. Board Can. 24: 2355-2453.

W~LFORD, L. A. 1946. A new graphic method of describi~ag the growth of animals. BioI. BulL 90: 141-147.

YOSHIHARA, T. 1950. On a graphical method of cieter~niraation of parameters contained in the logistic type cuves. Bull. Jpn. Soc. Sci. Fish. l6(7): 323-328.

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