52 



THE STRUGGLE FOR EXISTENCE 



Change in the) f Natural increase) 

 number of prey > = < of prey per unit f - 



per unit of time. 



Change in the] 

 I number of pre- 

 jdators per unit 

 [of time 



of time 



J 



Increase in the 

 number of pre- 

 dators per unit 

 of time resulting 

 from the devour- 

 ing of the prey, 



Destruction of 

 the prey by the 

 predators per 

 unit of time 



Deaths of the 

 predators per 

 unit of time 



(20) 



We can introduce here the following notation: 

 dNt 



dt 



= rate of increase of the number of prey. 



61 = coefficient of natural increase of prey (birth rate 



minus death rate). 



biNi = natural increase of the number of prey at a given 



moment. 



/1 (Ni, N 2 ) = the function characterizing the consumption of prey 



by predators per unit of time. This is the greater 



the larger is the number of predators (N 2 ) and the 



larger is the number of the prey themselves (iVi). 



dN 2 



dt 



= rate of increase of the number of predators. 



F (Ni, N 2 ) = the function characterizing simultaneously the natal- 

 ity and the mortality of predators. 



We can now translate the equations (20) into mathematical lan- 

 guage by writing: 



dNi 



dt 



dNj 

 dt 



= biNi - MN lf #2) 



= F(N l ,NJ 



(21) 



In a particular case investigated by Volterra in detail, the functions 

 in these equations have been somewhat simplified. He put /1 (JYi, N 2 ) 

 = kiNzNi, e.g., the consumption of prey by predators is directly pro- 

 portional to the product of their concentrations. Also F(Ni, N 2 ) = 



