ig2 GROWTH PRINCIPLES AND THEORY 2 



characteristically different. The curve of weight growth is sigmoid, with a point 

 of inflexion at about one-third (exactly: 8/27) of the final weight. The curve of 

 linear growth is a decaying exponential without a turning point (Figs. 8, 9). 



This characteristic course of growth is easily understood. If a body, with not 

 too much change of shape, increases in size, its surfaces increase approximately 

 with the second power of the length, but its volume and mass with the third 

 power. Hence, the ratio between surface and weight is continually shifted in 

 disfavor of the surface. Consequently, so long as the animal is small, surface- 

 proportional anabolism prevails over weight-proportional catabolism, and the 

 animal grows. The larger it grows, the more the surplus remaining for growth is 

 reduced, and eventually a steady state will be reached where anabolism and 

 catabohsm balance each other, and growth comes to an end. 



This is the most common form of growth curves, found in fish, in a number of 

 invertebrate classes and also, with certain restrictions, in mammals. The validity 

 of these growth equations has been shown in many examples {e.g. Piitter, 1920; 

 Bertalanffy, 1934, 1951a) of the animal classes indicated in Table 6 (Figs. 8-14). 

 In well analyzed cases, the numerical approximation of data is so exact that it 

 would be satisfactory even for a law of physics. 



Methods to calculate growth data according to equations (5.28) and (5.29) include: 



1. Graphical (Fig. 10). The linear form of equation (5.29) is: 



log (/* — /) = log (/* — /o) — kt log e (5.30) 



Hence, if the final value (/*) is properly chosen, semilogarithmic plot of (/* — /) 

 gives a straight line. 



If /* is not correctly chosen, a concave curve is obtained if the assumed /* 

 is too large, and a convex curve if it is too small. Examination of the plotted 

 growth data and some trial, if necessary, usually gives the desired result in calcu- 

 lations of linear growth. Calculated (/* — /) can be directly read from the 

 semilogarithmic plot (Fig. 10); k can be calculated from: 



k ^ log (/* — l^) — log (/* — U,)jt log e; (5.31) 



E - l*k. (5.32) 



2. Least-square method. The coefficients in equation (5.26) can be calculated 

 from the normal equations: 



(5-33) 



/_, Zw Z. cU 



{n = number of observations). For weight, a;"^ instead of/ is taken. This method 

 is recommended particularly for calculation of weight-growth curves as the taking 

 of cubic roots increases the error committed, and hence the graphical method (j) 

 often gives unsatisfactory results. 



Further methods and details for calculations according to equations (5.25, 5.27) can be 

 found in Buch Andersen and Fischer (1929), Bertalanffy (1934), Brody (1945)? and 

 Beverton and Holt (1957). 



