50 THE STRUGGLE FOR EXISTENCE 



In other words the logistic curve possesses the property that with 

 an increase in the number of individuals the relative rate of growth 

 decreases linearly (this has been recently mentioned by Winsor ('32)). 

 Consequently the expression (13) according to which we must sub- 

 tract from the coefficient of increase b sl certain value proportional to 

 the accumulated number of individuals in order to obtain the rate of 

 growth, and the expression (9), according to which we must multiply 

 the geometric increase bN by a certain "degree of its realization," 

 coincide with one another. Both are based on a broad mathematical 

 assumption of a linear relation between the relative rate of growth and 

 the number of individuals. Volterra extended the equation (13) to 

 the competition of two species for common food, assuming that the 

 presence of a certain number of individuals of the first species (Ni) 

 decreases the quantity of food by hiNi, and the presence of N 2 individ- 

 uals of the second species decreases the quantity of food by h 2 N 2 . 

 Therefore, both species together decrease the quantity of food by 

 hiNi + h 2 N 2 , and the coefficient of multiplication of the first species 

 decreases in connection with the diminution of food : 



6i - Xi (hiNt + h 2 N 2 ) (16) 



But for the second species the degree of influence of the decrease of 

 food on the coefficient of multiplication b 2 will be different (X 2 ), and 

 we shall obtain: 



b 2 - X 2 (hNi + h 2 N 2 ) (17) 



Starting from these expressions Volterra ('26) wrote the follow- 

 ing simultaneous differential equations of the competition between 

 two species for common food: 



d ^ = fo-lufcNi + hNMNi 



dN 2 

 dt 



= [b 2 - XtQitNt + faN^Ns 



(18) 



These equations represent, therefore, a natural extension of the prin- 

 ciple of the logistic curve, and the equation (12) written by Gause 

 ('32b) coincides with them. Indeed the equation (12) can be trans- 

 formed in this manner: 



