V GROWTHINTIMEOFTHETOTALORGANISM 1 69 



2. The mathematically simplest assumption to take into account the continuous 

 increase in size is to assume specific growth rate to be constant: 



;t- • - = ^ (5-2) 



dt y 



Integrated, this expression gives: 



y =-^oe" (5-3) 



from which specific growth rate can be calculated as: 



dy I logj^-logjo 



dt y log e {t-t^ 



(54) 



(e = base of natural logarithms). This formula is a better approximation than 

 (5.1) but also is a frequently inexact approximation. It presupposes that growth 

 is exponential according to equation (5.3). This, and the corresponding assump- 

 tion that specific growth rate is constant, is in general untrue; for as a rule growth 

 rates decrease during development. The error committed is considerable, so that 

 equations (5.1) and (5.4) have only limited applicability. 



In general, absolute and specific growth rates decrease during animal growth, 

 and eventually the organism attains a steady state or is "adult". There are, 

 however, also other types of animal growth (p. 180). Apart from these cases it is 

 a general rule that the curve of weight growth is a sigmoid. This implies that 

 absolute growth rate dw\dt first increases to a maximum and then decreases. 

 Specific growth rate dw\vodt or, what amounts to the same, logarithmic growth 

 rate d log w\dt is a monotonically decreasing function which implies that increase 

 per cent and per unit time decreases, or that the time needed for duplication of 

 weight continually increases. 



In the case that the growing body remains geometrically similar during its 

 increase in size, growth is called proportional. In this case, the surface s can be 

 obtained by multiplication of the square of any linear dimension / with a constant; 

 similarly the volume v (or the weight w, if specific weight is constant) is obtained 

 by multiplication of the third power with a constant: 



s = pl\ V = qi' (5-5) 



In the case of proportional growth, a simple relation between growth in length, in 

 surface, and in volume obtains : 



(5-6) 



That is, specific growth rate of a surface is twice, that of volume or weight three 

 times the specific growth rate of a linear dimension. 



Literature p. 253 



