GROWTH IN TIME OF THE TOTAL ORGANISM 



175 



polation ; but is has no theoretical significance. The constants appearing in such 

 formulas have no physiological meaning, and are different in each particular 

 calculation. Fitting of data is obtained not by virtue of the particular equation 

 used, but of the number of arbitrary constants admitted. 



Instead of superimposing terms by applying series, complicated curves can 

 also be fitted by subdividing them into segments that can be approximated by a 

 simple equation; or the equation used can be transformed into a linear form, and 

 the curve then subdivided into a sequence of straight lines. As a rule, such 

 subdivision is gratuitous, and therefore fitting of data in this way only an expression 

 of the trivial fact that any curve can be approximated by cutting it into a sufficient 

 number of straight-line sections. 



TABLE 5 



FUNCTIONS USED FOR DESCRIBING GROWTH CURVES 



Growth rate 



( Weight) Weight 



Furtction Awldt w 



Growth rate 

 Inflexion (length) Length Inflexion 



(w;t) dljdt I (l,t) 



I. Exponential 



4. Parabola kwit w^tk 



5. Gompertz wkbe-^t w*e-l>e~" 



3 

 k //*3 



I lektii 



2. Decaying k(w*—w) w*(i—be — kt) None '(^2 — ') 

 exponential 3 \ ' / 



3. Logistic kw(w*—w) w*j(i + be — k't) o.sati*;-— — (1*^—1^) 



(k' = kw*) ^" 3 



None Brody (1945) 



0.63/*; — — — Robertson 



(k' = kw*) 



(1926) 



None klj'^t l^tkl'i None Schmalhausen 



('/l=/( = lj (1929) 



o.37i£'*;-— -kbe-kt ^ f 0.37I*;— -- Winsor(i932) 



k 3 ib' = bl3) k 



ib' = bl3) 



w = weight; / = length; Wq, w* = initial, final weight; l^, I* = initial, final length; e = basis of natural logarithms; 

 k, k' , b = constants; g = proportionality factor (wll^). 



Proportional growth (w = qP) is presupposed for the above relations between growth in weight and in length. 

 *References selected for detailed discussion of the functions concerned and application to observed growth data. 



Table 5 summarizes the more important equations applied to growth. It appears 

 unnecessary to enter into a discussion of the theoretical ideas brought forward in 

 connection with these formulas, as it can be shown that none of them is physio- 

 logically well-founded (Bertalanffy, 1951a), and it is sufficient to show that these 

 functions, for mathematical reasons, are ill-suited to represent growth curves as 

 empirically found. 



J. The exponential is most easily tested since it gives a straight line in semi- 

 logarithmic plot. This applies in certain cases (bacteria, insects, Helicidae, p. igGff.), 

 but is inapplicable in all cases where growth is limited and approaches a steady 

 state. 



2. The decaying exponential (sometimes called the "monomolecular" function) 

 applies in certain cases, namely, for growth of linear dimensions in certain animal 

 groups (Type I, p. igoff.). It is not suited for growth in weight which usually 

 presents a sigmoid curve. For dealing with the latter Brody (1945) subdivided 

 weight curves into a "self-accelerating phase", obeying the exponential or rather 

 consisting of a sequence of exponentials of different slopes; and a "self-inhibiting 



Literature p. 253 



