304 



POPULATIONS 



illustrated them graphically. Figure 95, 

 adapted from his book, depicts on a relative 

 scale and in pictorial fashion the interopera- 

 tion of these factors for the Paramecium 

 caudatum illustration. The drawing adds 

 nothing to what has already been said, but 

 aflfords a succinct summary. 



It would not be a fair appraisal to leave 

 the logistic curve without mentioning that 

 it has been subject to criticism. Possibly the 



the empirical representation of growth phe- 

 nomena. It does not appear that either curve 

 has any substantial advantage over the other 

 in the range of phenomena which it will fit. 

 Each curve has three arbitrary constants, 

 which correspond essentially to the upper 

 asymptote, the time origin, and the time unit 

 or 'rate constant.' In each curve, the degree 

 of skewness, as measured by the relation of 

 the ordinate at the point of inflection to the 

 distance between the asymptote, is fixed. It 



DAYS 



Fig. 95. Schematic representation of Cause's "characteristics of competition" exhibited by a 

 population of paramecia growing logistically. In the text Cause's "degree of realization of the 

 potential increase" is referred to as "unutilized opportunity for growth," and his "environ- 

 mental resistance" as "utilized opportunity for growth."' ( From Cause. ) 



.Tiost general criticism is a simple one: 

 populations characteristically grow in a sig- 

 moid or S-shaped fashion; the logistic curve 

 is a sigmoid curve which describes their 

 growth; therefore, there is nothing unique 

 about the fact that the logistic equation 

 "works." In short (the criticism holds), it 

 is fallacious to designate the logistic as a 

 "law" of population growth. Other curves 

 can be apphed to population increase. For 

 example, Wright (1926) discusses the 

 "Gompertz" curve, named after Benjamin 

 Gompertz, who discovered it in 1825. Win- 

 sor (1932, p. 7) compared this function 

 with the logistic and came to the following 

 conclusion : 



"The Gompertz curve and the logistic possess 

 similar properties which make them useful for 



has been found in practice that the logistic 

 gives good fits on material showing an in- 

 flection about midway between the asymp- 

 totes. No such extended experience with the 

 Compertz curve is as yet available, but it 

 seems reasonable to expect that it will give 

 fits on material showing an inflection when 

 about 37 per cent of the total growth has 

 been completed. Ceneralizations of both 

 curves are possible, but here again there ap- 

 pears to be no reason to expect any marked 

 difference in the additional freedom pro- 

 vided." 



The reader may find for himself critical 

 comments about the logistic curve in the 

 papers by Hogben (1931) and by Wilson 

 and PuflFer (1933). The latter workers warn 

 particularly against using extrapolations of 

 the logistic curve in predicting the size of a 



