STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 41 



The values of i for a population of Paramecia are given in Table 

 IV and we see that at the beginning of the population growth the 

 intensity of the struggle for existence is not great, but that afterwards 

 it increases considerably. Thus on the first day there are 0.058 "un- 

 realized" Paramecia for every one realized, but on the fourth day 61.7 

 "unrealized" ones are lost for one realized (Fig. 6). Figure 5 shows 

 graphically the changes of all the discussed characteristics in the 

 course of the population growth of Paramecia. 



(5) The intensity of the struggle for existence can evidently be ex- 

 pressed in this form only in case the population grows, i.e., if the num- 

 ber of individuals increases continually. If growth ceases the popu- 



dN 

 lation is in a state of equilibrium, and the rate of growth -^- = 0; 



X 



0.0S8 per 

 one 



First day 



Fourth day 



Fig. 6. Intensity of the struggle for existence in Paramecium caudatum. On 

 the first day of the growth of the population 0.058 "unrealized" Paramecia are 

 lost per one realized, but on the fourth day 61.7 per one. 



in this case the expression of intensity will take another form. Popu- 

 lation in a state of equilibrium represents a stream moving with a 

 certain rapidity: per unit of time a definite number of individuals 

 perishes, and new ones take their places. The number of these 

 liberated places is not large if compared with the number of organ- 

 isms that the population can produce in the same unit of time accord- 

 ing to the potential coefficients of multiplication. Therefore a con- 

 siderable part of the potentially possible increase of the population 

 will not be realized and liberated places will be occupied only by a 

 very small fraction of it. If the potential increase of a population in 

 the state of equilibrium per infinitesimal unit of time is &iiVi, and a 



