54 THE STRUGGLE FOR EXISTENCE 



can be represented by means of a certain geometrical increase which 

 is realized in proportion to the unutilized opportunity of growth. 

 This unutilized opportunity is a function of the number of prey at a 

 given moment: f(Ni). Therefore, 



dt 



= b 2 N 2 f(Nd (22) 



The simplest assumption would be that the geometric increase in 

 the number of predators is realized in direct proportion to the num- 

 ber of prey (XiVi) . Were our system a simple one we could say with 

 Lotka and Volterra that the rate of growth might be directly con- 

 nected with the number of encounters of the second species with the 

 first. The number of these encounters is proportional to the number 

 of individuals of the second species multiplied by the number of in- 

 dividuals of the first (aNiN 2 ), where a is the coefficient of proportion- 

 ality. If it were so, the increase of the number of predators would 

 be in direct proportion to the number of the prey. Indeed, if the 

 number of the prey iV\ has doubled and is 2Ni, the number of their 

 encounters with the predators has also doubled, and instead of being 

 aNiNz is equal to aN 2 2Ni. Consequently the increase in the number 

 of predators instead of the former b 2 N 2 \Ni would become equal to 



dN 2 

 dt 



= b 2 N 2 \ 2Ni, and the relative increase (per predator) would be 



therefore: -= — r— = b 2 \ 2Ni. In other words, the relative increase 



N 2 dt 



— -j^- would be a rectilinear function of the number of prey N h i.e., 



JS 2 dt 



with a rise of the concentration of the prey the corresponding values 

 of the relative increase of the predators could be placed on a straight 

 line (ab in Fig. 7) . But experience shows the following : If we study 

 the influence of the increase in the number of the prey per unit of 

 volume upon the increase from one predator per unit of time, we will 

 find that this increase rises at first rapidly and then slowly, approach- 

 ing a certain fixed value. A further change in density of the prey 

 does not call forth any rise in the increase per predator. In the limits 

 which interest us we can express this relationship with the aid of a 

 curve rapidly increasing at first and then approaching a certain 

 asymptote. Such a curve is represented in Figure 7 (ac). The con- 

 centration of prey (Ni) is marked on the abscissae, and the relative 



