104 Growth and Form 



and never reach a limit, i.e., growth would never stop. To show how an 

 automatic brake can be added to this equation, we derive it in another 

 and more general way. Consider that growing organisms display two 

 properties : 



a. To get any progeny at all, you must start with at least one in- 

 dividual or, in the case of higher animals, with at least two of the right 

 sexes (i.e., there is no spontaneous generation of life). 



b. The more parents you have, the more progeny you will get. In 

 other words, the rate of increase in the number of organisms is propor- 

 tional to the number already present. 



Now rate of increase is simply the change in the number of organ- 

 isms per unit time, or: 



total change in number of organisms 

 total change in time 



We symbolize this as AJV/Af. Since the rate of increase is proportional to 

 the number, N, of organisms already present, we can write: 



This equation can be solved by the integral calculus. That is, the relation 

 between IV and t can be deduced from the way in which N changes with 

 time. By these means (the mechanical manipulations are unimportant 

 here), we obtain the same equation as already has been derived, i.e., 

 equation 4. 



Equation 5 can be elaborated to account for the stoppage of growth. 

 What we want is to have the rate of growth start out high and give us 

 the first part of the growth curve and then to decline steadily to zero, at 

 which point the number of organisms would remain stationary. For some 

 organisms that do not remain stationary after cessation of growth but 

 actually die out, we would want the rate of growth to become negative 

 after a time ( a negative rate of increase is itself a decrease ) . These qual- 

 ifications are accomplished by making k not a constant but a variable. 

 For example, one way of doing this is to set k equal to (a — bN), where 

 a and b are constants. Equation 5 would then become: 



^=(a-hN)N 6 



At the start, when the number of organisms was very small, bN would be 

 negligible, and the rate of growth would be substantially equal to aN 



