STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 



55 



' 1 dN 2 \ 



increase in the number of predators ( ^ ^^ J, or the rate of growth 



VV2 at / 



per predator at different densities of prey, is marked on the ordinates. 



The curve connecting the relative increase of the predators with 



the concentration of the prey can be expressed by the equation: 



y = a (1 — e~ iz ), 7 which in our case takes the following form: 



N 2 dt 



= b 2 (l - e~™i) 



(23) 



The properties of this curve are such that as the prey becomes 



Concentration of the prey N, -*■ 



Fig. 7. The connection between the relative increase of the predators and 

 concentration of the prey. 



more concentrated the relative increase of the predators rises also, 

 approaching gradually the greatest possible or potential increase 62 

 (see Fig. 7). The meaning of the equation (23) is that instead of 

 the assumption of a linear alteration of the relative increase which 



7 This is the simplest expression of the curve of such a type, which is widely 

 used in modern biophysics. It is deduced from the assumption that the rate 

 of increase is proportional to still unutilized opportunity for increase taken in 



an absolute form: — = fc(o — y). For experimental verification see Chapter 

 dx 



VI. 



