44 THE STRUGGLE FOR EXISTENCE 



of the environmental resistance which we have determined on the basis of 

 a purely 'physiological investigation coincides completely with the value 

 of the environmental resistance calculated according to the logistic equa- 

 tion, using the latter as an empirical expression of growth. In this 

 way we have proved that the logistic equation actually expresses the 

 mechanism of the growth of the number of unicellular organisms 

 within a limited microcosm. All this will be described in detail in 

 the next chapter. 



in 



(1) We are now sufficiently prepared for the acquaintance with the 

 mathematical equations of the struggle for existence, and for a critical 

 consideration of the premises implied in them. Let us consider first 

 of all the case of competition between two species for the possession 

 of a common place in the microcosm. This case was considered 

 theoretically for the first time by Vito Volterra in 1926. An experi- 

 mental investigation of this case was made by Gause ('32b), and at 

 the same time Lotka ('32b) submitted it to a further analysis along 

 theoretical lines. 



If there is competition between two species for a common place in 

 a limited microcosm, we can quite naturally extend the premises im- 

 plied in the logistic equation. The rate of growth of each of the 

 competing species in a mixed population will depend on (1) the po- 

 tential rate of population increase of a given species (hNi or b 2 N 2 ) 

 and (2) on the unutilized opportunity for growth of this species, just 

 as in the case of a population of the first and second species growing 

 separately. But unutilized opportunity for growth of a given species 

 in a mixed population is a complex variable. It measures the num- 

 ber of places which are still vacant for the given species in spite of 

 the presence of another species, which is consuming the common food, 

 excreting waste products and thereby depriving the first one of some 

 of the places. Let us denote as before by Ni the number of individ- 

 uals of the first species, though, as we shall see further on, we shall 

 have to deal in many cases not with the numbers of individuals but 

 with masses of species ( = the weight of the organisms present or its 

 equivalent) and we shall introduce corresponding alterations. The 

 unutilized opportunity for growth, or the degree of realization of the 

 potential increase for the first species in a mixed population, may be 



expressed thus: — * — -, where Ky is the maximal possible 



