STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 53 



k 2 N 2 Ni — ckN 2 . Here k 2 N 2 Ni is the increase in the number of preda- 

 tors resulting from the devouring of the prey per unit of time, and 

 d 2 N 2 — number of predators dying per unit of time {d 2 is the coef- 

 ficient of mortality). This translation of (20) gives 



*£ = b 1 N 1 -k 1 N 2 N 1 

 at 



= k 2 N 2 Ni — d 2 N 2 



dN, 

 dt 



(21a) 



These equations have a very interesting property, namely the 

 periodic solution, which has been discovered by both Lotka ('20) 

 and Volterra ('26). As the number of predators increases the prey 

 diminish in number, 5 but when the concentration of the latter be- 

 comes small, the predators owing to an insufficiency of food begin to 

 decrease. 6 This produces an opportunity for growth of the prey, 

 which again increases in number. 



(2) In our discussion up to this point we have noted how the proc- 

 ess of interaction between predators and prey can be expressed in a 

 general form covering a great many special cases (equation 21), and 

 how this general expression can be made more concrete by introduc- 

 ing certain simple assumptions (equation 21a). There is no doubt 

 that we shall not obtain any real insight into the nature of these 

 processes by further abstract calculations, and the reader will have 

 to wait for Chapter VI where the discussion is continued on the 

 sound basis of experimental data. 



Let us better devote the remainder of this chapter to two rather 

 special problems of the natural increase of both predators and prey 

 in a mixed culture simply in order to show how the biological reason- 

 ing can be translated into mathematical terms. 



In the general form the rate of increase in the number of individuals 



of the predatory species resulting from the devouring of the prey 



dN 2 

 dt 



B When the number of predators (N 2 ) is considerable the number of the prey 

 devoured per unit of time (k l NiN 2 ) is greater than the natural increase of the 

 prey during the same time (+&i2Vi), and (biNi — kiNiN 2 ) becomes a negative 

 value. 



6 With a small number of prey (Ni) the increase of the predators owing to 

 the consumption of the prey (+k 2 NiN 2 ) is smaller than the mortality of the 

 predators ( — dN 2 ), and (k 2 NiN 2 — dN 2 ) becomes a negative value. 



