V GROWTHINTIMEOFTHETOTALORGANISM 20I 



by some mathematical function. Experimental evidence shows that, as a rule and 

 in first approximation, the rate of physiological processes can be expressed as a 

 function of body mass by the simplest possible expression, namely, as a power 

 function of the body mass present. Hence the equation: 



dwjdt = -fiw'" — y.w" (5- 1 6) 



hardly contains a hypothesis beside the assumption that anabolic and catabolic 

 processes can be summated in two overall terms. 



For the two exponents applies: 



o < m < I J 



^ / ^ (5-39) 



o < n < I ) 



for it would be absurd to assume that metabolic processes are inversely propor- 

 tional to body mass, and physiologically un-understandable that they are pro- 

 portional to a power > i of body mass. 



If the above assumptions are accepted, the family of growth equations follow by 

 mathematical considerations. Growth may be i. unlimited, or 2. limited. 



1. Unlimited growth. In this case, growth in time gives a straight line in the 

 semilogarithmic plot, as found in many empirical cases. 



If exponential growth is found, there must he m = n = i and 



dw 



^ = ^^ (5-37) 



because no other value of tn and n yields the exponential. 



2. Limited growth. In this case, there must be (a) a steady-state value w* which 

 is approached in the process; (b) growth rates dw/dt must be positive between 

 w = o and w = w*. Regarding (a), the steady-state value is obtained by equating 

 (5.16, p. 179) to zero: 



G 



n — m 



If* = 



According to (b), w* must be a maximum; otherwise, there would be negative 

 growth, that is, not increase but decrease o[ w between Wq and w*. For w* being 

 a maximum, the second derivative must be negative: 



(m— 1) „„,., («— 1) 



w = r^mw ' ' " — xTiw '" " < o 

 It follows : 



r,?nw* ^"' ^^ < xnw* ^" '^ 



H 

 „* (" — '") ^ 7^* (n — m) 



W' 



m 



m < n (540) 



If, therefore, equation (5.16) is to interpret growth approaching a final maxi- 

 mum value, it follows for mathematical reasons that m < n. 



Literature p. 253 



