34 THE STRUGGLE FOR EXISTENCE 



(2) There is no need to search for a quantitative expression of the 

 potential rate of increase and of the environmental resistance, as 

 this problem had already been solved by Verhulst in 1838 and quite 

 independently by Raymond Pearl and Reed in 1920. However, 

 ecologists did not connect their idea of biotic potential with these 

 classical works. The logistic curve, discovered by Verhulst and 

 Pearl, expresses quantitatively the idea that the growth of a popula- 

 tion of organisms is at every moment of time determined by the rela- 

 tion between the potential rate of increase and "environmental re- 

 sistance." The rate of multiplication or the increase of the number 



of organisms (N) per unit of time (t) can be expressed as — =-. The 



rate of multiplication depends first on the potential rate of multipli- 

 cation of each organism (6), i.e., on the potential number of offspring 

 which the organism can produce per unit of time. The total potential 

 number of offspring that can be produced by all the organisms per 

 unit of time can be expressed as the product of the number of organ- 

 isms (N) and the potential increase (6) from each one of them, i.e., 

 bN. Therefore the potential increase of the population in a certain 

 infinitesimal unit of time will be expressed thus: 



dN 



This expression represents a differential equation of the population 

 growth which would exist if all the offspring potentially possible were 

 produced and actually living. It is an equation of geometric increase, 

 as at every given moment the rate of growth is equal to the number 

 of organisms (N) multiplied by a certain constant (b). 



As has been already stated, the potential geometrical rate of popu- 

 lation growth is not realized, and its reduction is due to the environ- 

 mental resistance. This idea was quantitatively expressed by Pearl 

 in such a form that the potential geometric increase at every moment 

 of time is only partially realized, depending on how near the already 

 accumulated size of the population (N) approaches the maximal popu- 

 tion (K) that can exist in the given microcosm with the given level 

 of food resources, etc. The difference between the maximally pos- 

 sible and the already accumulated population (K — N), taken in a rel- 



ative form, i.e., divided by the maximal population ( — — — J, shows 



m 



