STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 45 



number of individuals of this species when grown separately under 

 given conditions, Ni is the already accumulated number of individuals 

 of the first species at a given moment in the mixed population, and m 

 is the number of the places of the first species in terms of the number 

 of individuals of this species, which are taken up by the second species 

 at a given moment. The unutilized opportunity for growth of the 

 first species in the mixed population can be better understood if we 

 compare it with the value of the unutilized opportunity for the sep- 

 arate growth of the same species. In the latter case the unutilized 

 opportunity for growth is expressed by the difference (expressed in a 

 relative form) between the maximal number of places and the number 

 of places already occupied by the given species. Instead of this for 

 the mixed population we write the difference between the maximal 

 number of places and that of the places already taken up by our 

 species together with the second species growing simultaneously. 



(2) An attempt may be made to express the value m directly by 

 the number of individuals of the second species at a given moment, 

 which can be measured in the experiment. But it is of course un- 

 likely that in nature two species would utilize their environment in 

 an absolutely identical way, or in other words that equal numbers of 

 individuals would consume (on an average) equal quantities of food 

 and excrete equal quantities of metabolic products of the same chemi- 

 cal composition. Even if such cases do exist, as a rule different spe- 

 cies do not utilize the environment in the same way. Therefore the 

 number of individuals of the second species accumulated at a given 

 moment of time in a mixed population in respect to the place it 

 occupies, which might be suitable for the first species, is by no means 

 equivalent to the same number of individuals of the first species. 

 The individuals of the second species have taken up a certain larger 

 or smaller place. If N2 expresses the number of individuals of the 

 second species in a mixed population at a given moment, than the 

 places of the first species which they occupy in terms of the number of 

 individuals of the first species, will be m = aN 2 . Thus, the coefficient 

 a is the coefficient reducing the number of the individuals of the 

 second species to the number of places of the first species which they 

 occupy. This coefficient a shows the degree of influence of one 

 species upon the unutilized opportunity for growth of another. In 

 fact, if the interests of the different species do not clash and if in the 

 microcosm they occupy places of a different type or different "niches" 

 then the degree of influence of one species on the opportunity for 



