STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 33 



tial equations of the struggle for existence, which enable us to draw 

 definite conclusions about the course and the results of competition. 

 Therefore all the value of the ulterior deductions depends on the 

 question whether certain fundamental premises have been correctly 

 formulated. Consequently before proceeding any further to consider 

 more complicated mathematical equations of the struggle for exist- 

 ence we must with the greatest attention, relying upon the experi- 

 mental data already accumulated, decide the following question: 

 what are the premises we have a right to introduce into our differen- 

 tial equations? As the problem of the struggle for existence is a 

 question of the growth of mixed populations and of the replacement 

 of some components by others, we ought at once to examine this 

 problem : what is exactly known about the multiplication of animals 

 and the growth of their homogeneous populations? 



Of late years among ecologists the idea has become very wide 

 spread that the growth of homogeneous populations is a result of the 

 interaction of two groups of factors : the biotic potential of the species 

 and the environmental resistance [Chapman '28, '31]. The biotic 

 potential 1 represents the potential rate of increase of the species 

 under given conditions. It is realized if there are no restrictions of 

 food, no toxic waste products, etc. Environmental resistance can be 

 measured by the difference between the potential number of organ- 

 isms which can appear during a fixed time in consequence of the po- 

 tential rate of increase, and the actual number of organisms observed 

 in a given microcosm at a determined time. Environmental resist- 

 ance is thus expressed in terms of reduction of some potential rate of 

 increase, characteristic for the given organisms under given condi- 

 tions. This idea is a correct one and it clearly indicates the essential 

 factors which are operating in the growth of a homogeneous popula- 

 tion of organisms. 



However, as yet among ecologists the ideas of biotic potential and 

 of environmental resistance are not connected with any quantitative 

 conceptions. Nevertheless Chapman in his interesting book Animal 

 Ecology arrives at the conclusion that any further progress here can 

 only be achieved on a quantitative basis, and that in future "this 

 direction will probably be one of the most important fields of biologi- 

 cal science, which will be highly theoretical, highly quantitative, and 

 highly practical." 



1 What Chapman calls a "partial potential." 



