NOTES 



ON THE COMPATIBILITY OF A NEW 



EXPRESSION FOR GROSS CONVERSION 



EFFICIENCY WITH THE VON BERTALANFFY 



GROWTH EQUATION' 



Gross food conversion efficiency {K{) is defined by 



K^ = growth increment/food ingested (1) 



dW 



form 



dt 



II 



where / is the ingestion rate (Ivlev 1939; Ricker 

 1966); data from feeding experiments are usually 

 fit to an allometric model of the form 



K^ = cW 



(2) 



where W is the body weight, and c and a are em- 

 pirical constants which, however, have the disadvan- 

 tage of always predicting values oi K^ > 0, al- 

 though the fish and other aquatic animals to which 

 the model is meant to apply usually experience size 

 constraints and hence must reach a value of W 

 where /fj = 0. It is therefore preferable to choose 

 a functional form for K^ which falls to zero as W 

 approaches W^. Furthermore, recent analysis of 

 feeding studies of a number of fish species indicates 

 that Ki can approach arbitrarily close to unity for 

 the smallest fishes, which suggests the alternate 

 equation 



K, = i- {w/wy 



(3) 



where W^ is the weight at which K-^ = 0, and ft is 

 an empirical constant estimated from the slope of 



log (1 - K,) = (ilogW - p log W^ 



(4) 



(Pauly 1986). 



In this note we show that Equation (3) is com- 

 patible with the von Bertalanffy growth function 

 (VBGF), both in its standard (von Bertalanffy 1938) 

 and generalized forms (Richards 1959; Pauly 1981), 

 which is not true of Equation (2). 



We assume that the ingestion rate (/) can be ex- 

 pressed as an allometric expression of weight of the 



iICLARM Contribution No. 316. 



FISHERY BULLETIN: VOL. 85, NO. 1, 1987. 



/ = HW^, 



(5) 



where H and d are empirical constants. From Equa- 

 tion (1) we then obtain for the growth rate 



dW/dt = K^ HW^ (6) 



which combined with Equation (3) gives 



dWIdt = (1 - (WIWJP) HW (7) 



and hence 



dW/dt = HW^ - kW"^ (8) 



where m = d + fi and k = HIW^. Equation (8) is 

 the differential form of the VBGF, and can be in- 

 tegrated for various values of the constants m and 

 d. Setting d = 2/3 and m = 1 (i.e., ft = 1/3) yields 

 the "normal" VBGF for weight, 



Wt = W^il - e-^(«-«o))3 



(9) 



where K = kIS, while if m = 1 and < d < 1 we 

 get the generalized VBGF sensu Pauly (1981), 



Wt = W^{1 - e-^('-«o)) 



ZID 



(10) 



where Z) = 3(1 - d). This second form is probably 

 more useful as it allows for the exponent of the 

 allometric relationship linking ingestion and weight 

 (Equation (5)) to take wider range of values, as 

 needed to fit various data sets and/or to mimic 

 various models in the literature (see, e.g., Paloheimo 

 and Dickie 1966 or Ursin et al. 1985). 



The compatibility shown here between the recent- 

 ly proposed Equation (3) expressing K^ as a func- 

 tion of fish weight and the VBGF is encouraging, 

 as it supports the method suggested by Pauly (1986) 

 for combining these two equations when estimating 

 the food consumption of fish populations and leads 

 to a mathematically consistent approach for the 

 analysis of feeding and growth data. 



LITERATURE CITED 



Bertalanffy, L. von. 



1938. A quantitative theory of organic growth (Inquiries on 



139 



