STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 49 



(5) We have just discussed a very important set of equations of 

 the competition of two species for a common place in the microcosm, 

 and it remains to make in this connection a few historical remarks. 

 Analogous equations dealing with a more special case of competition 

 between two species for a common food were for the first time given 

 in 1926 by the Italian mathematician Vito Volterra who was not 

 acquainted with the investigations of Ross and of Pearl. 



Volterra assumed that the increase in the number of individuals 



dN 

 obeys the law of geometric increase : —r- = bN, but as the number of 



individuals (N) accumulates, the coefficient of increase (6) diminishes 

 to a first approximation proportionally to this accumulation (b — X2V), 

 where X is the coefficient of proportionality. Thus we obtain : 



d -£ =(b-\N)N (13) 



at 



It can be easily shown, as Lotka ('32) remarks, that the equation 

 of Volterra (13) coincides with the equation of the logistic curve of 

 Verhulst-Pearl (9). In fact, if we call the rate of growth per indi- 

 vidual a relative rate of growth and denote it as : -^ -r-, then the 

 equation (13) will have the following form: 



i.^ = 6_XiV (14) 



N dt 



This enables us to formulate the equation (13) in this manner: the 



relative rate of growth represents a linear function of the number 



of individuals N, as b — X N is the equation of a straight line. If 



we now take the equation of the Verhulst-Pearl logistic curve (9): 



dN K — N 



— = bN — = — , and make the following transformations: 

 dt K. 



dN .» 



( l -*"}■ rf = ^(i-^4 weshaIlhave: 



1 dN , & , 7 , 1e x 



N-Tt= b ~K N (15) 





