STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 37 



curve, instead of a series of points restricted to the end of each repro- 

 ductive period. The smooth curve of Figure 4 is drawn according 

 to the equation of the logistic curve, and its close coincidence with 

 the results of the observations shows that the logistic curve represents 

 a good empirical description of the growth of the population. The 

 practical method of fitting such an empirical curve will also be con- 

 sidered further (Appendix II). The question that interests us just 

 now is this: what is, according to the logistic curve, the potential 

 rate of increase of Paramecium under our conditions, and how does it 

 become reduced in the process of growth as the environmental resist- 

 ance increases? 



According to Figure 4, the maximal possible number of Paramecia 

 in a microcosm of our type, or the saturating population, K = 375 

 individuals. As a result of the very simple operation of fitting the 

 logistic curve to the empirical observations, the coefficient of multi- 

 plication or the biotic potential of one Paramecium (b) was found. 

 It is equal to 2.309. This means that per unit of time (one day) 

 under our conditions of cultivation every Paramecium can potentially 

 give 2.309 new Paramecia. It is understood that the coefficient b is 

 taken from a differential equation and therefore its value automati- 

 cally obtained for a time interval equal to one day is extrapolated from 

 a consideration of infinitesimal sections of time. This value would 

 be realized if the conditions of an unoccupied microcosm, i.e., the 

 absence of environmental resistance existing only at the initial mo- 

 ment of time, existed during the entire 24 hours. It is automatically 

 taken into account here that if at the initial moment the population 

 increases by a certain infinitesimal quantity proportional to this 

 population, at the next moment the population plus the increment 

 will increase again by a certain infinitesimal quantity proportional 

 no longer to the initial population, but to that of the preceding mo- 

 ment. The coefficient b represents the rate of increase in the ab- 

 sence of environmental resistance under certain fixed conditions. At 

 another temperature and under other conditions of cultivation the 

 value b will be different. Table IV gives the constants of growth of 

 the population of Paramecia calculated on the basis of the logistic 

 curve. There is shown N or the number of Paramecia on the first, 

 second, third and fourth days of growth. These numbers represent 

 the ordinates of the logistic curve which passes near the empirical 

 observations and smoothes certain insignificant deviations. The 



