An analysis of the length-weight 

 relationship of larval fish: limitations 

 of the general allometric model 



Pierre Pepin 



Department of Fisheries and Oceans 



PO. Box 5667. St. John's. Newfoundland, Canada A1C 5X1 



Substantial morphological and 

 physiological changes occur during 

 the early ontogeny of fish. After 

 hatching, both shape and size un- 

 dergo significant alterations in as- 

 sociation with yolk absorption and 

 a subsequent increase in muscula- 

 ture. In fish, the larval stage gen- 

 erally consists of a period of rapid 

 growth, which can vary substan- 

 tially in duration within and among 

 species (Sinclair and Tremblay, 

 1985, Ware and Lambert, 1985; 

 Houde, 1989; Pepin, 1991). Length 

 and weight may increase by factors 

 of approximately 10 and 1,000, re- 

 spectively, over a time interval that 

 often spans less than 10% of a spe- 

 cies' life time. Such high develop- 

 ment rates are associated with high 

 metabolic rates (Giguere et al., 

 1989) which can lead to substantial 

 variation in condition as a result of 

 fluctuations in food availability 

 (Houde and Schekter, 1980; Werner 

 and Blaxter, 1980). Studies of varia- 

 tions in growth characteristics and 

 condition have been an important 

 keystone in understanding early 

 life history survival (Houde, 1987). 

 Changes in length or weight 

 through time have been used to as- 

 sess the general growth rates of 

 populations. Most often, the model 

 used to describe length-weight re- 

 lationships of fish larvae is a gen- 

 eral allometric function 



W = aL b , 



(la) 



logW = loga + blogL (lb) 



that can be estimated with minimal 

 computing power, by using least 

 squares, and for which the fit is gen- 

 erally strong (e.g. Laurence, 1978). 

 The latter point may seem reason 

 enough to assume that a general 

 allometric model is an adequate 

 description of the data. However, 

 some inferences derived from this 

 type of information concern varia- 

 tion in condition. For example, de- 

 viations from a general length- 

 weight relationship (e.g. Fulton's 

 ( 1911) index of condition [K=W/L 3 , 

 where W and L are the weight and 

 length of individuals]) have been 

 used to describe the relative state 

 of health of individuals (e.g. West- 

 ernhagen and Rosenthal, 1981; 

 Checkley, 1984; Ciechomski et al., 

 1986; Harris et al., 1986; Frank and 

 McRuer, 1989; Drolet et al., 1991). 

 It is therefore necessary to ensure 

 that the model used to describe the 

 length-weight relationship not only 

 provides a strong fit to the data but 

 also that it accurately describes the 

 functional form of that relationship. 

 Zweifel and Lasker (1976) suggest 

 that changes in length or weight of 

 fish larvae through time can be de- 

 scribed by a Gompertz model 



L = L e 



W = W e K ' (1 ~ e ' 



(2a) 

 (2b) 



where W is weight, L is length, and 

 a and b are constants. A logarith- 

 mic transformation of Equation 1 

 leads to a linear relationship 



where L Q and W are the length and 

 weight at time t=0, K and K' are 

 the specific growth rates at time £=0, 

 and a and a' are the rates of decay 



in growth rates. Only when a=a' 

 does the length-weight relationship 

 reduce to the form shown in Equa- 

 tion lb. Otherwise, the logarithmic 

 length-weight relationship will ex- 

 hibit a degree of nonlinearity (Laird 

 et al., 1968; Zweifel and Lasker, 

 1976). Barton and Laird (1969) 

 noted that fitting the general allo- 

 metric model is relatively insensi- 

 tive to slight departures from the 

 true time relations for growth in 

 length and weight (i.e. oc^ct'). Con- 

 sequently, a general allometric re- 

 lationship (Eq. lb), with only two 

 parameters, may be considered to 

 provide an adequate description to 

 the data, despite the fact that a 

 more complex model (e.g. a Gom- 

 pertz length-weight relationship) 

 better describes the patterns of 

 growth in length and weight. Al- 

 though the importance of such 

 subtle differences may appear to be 

 minor, consistent departures from 

 a general allometric model can lead 

 to significant bias in predicting or 

 interpreting weight at length. This 

 can be particularly important when 

 trying to model growth during the 

 early life history (e.g. Rose and 

 Cowan, 1993 ) or in estimating size- 

 dependent metabolic processes (e.g. 

 Checkley, 1984; Ki0rboe, 1989; 

 Giguere et al., 1989). Furthermore, 

 there is potential for inaccurate 

 inferences in instances where con- 

 dition is being studied (e.g. W/L 3 ). 

 Westernhagen and Rosenthal ( 1981 ) 

 and Ciechomski et al. (1986) noted 

 a decrease in Fulton's condition in- 

 dex after hatch, followed by an in- 

 crease some time after first feed- 

 ing. Although this pattern may be 

 due to food deprivation, it can also 

 arise because of a developmentally 

 determined nonlinear allometric 

 length-weight relationship (i.e 

 a*ct'). 



In this study, I present evidence 

 that, despite a strong fit to a gen- 

 eral allometric length-weight rela- 



Manuscript accepted 9 November 1994. 

 Fishery Bulletin 93:419-426 < 1995). 



419 



