STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 



35 



the relative number of the "still vacant places" for definite species in a 

 given microcosm at a definite moment of time. According to the 

 number of the still vacant places only a definite part of the potential 

 rate of increase can be realized. At the beginning of the population 

 growth when the relative number of unoccupied places is considerable 

 the potential increase is realized to a great extent, but when the al- 

 ready accumulated population approaches the maximally possible or 

 saturating one, only an insignificant part of the biotic potential will 

 be realized (Fig. 3). Multiplying the biotic potential of the popula- 

 tion (bN) by the relative number of still vacant places or its "degree 



K — N 



of realization" — = — , we shall have the increase of population per 



infinitesimal unit of time: 



Geometric increase 



Saturating population 



Holistic curve 



Unutilised opportu- 

 nity for growth 



Time 



Fig. 3. The curve of geometric increase and the logistic curve 

 Rate of growth 



Potential 

 increase of 



or increase perf = ^population 



unit of time 



Degree of realization 



of the potential in- 



" X i crease. Depends on 



per unit of the number of still 



time J vacant places. 



Expressing this mathematically we have : 



_ = W_- 



..(8) 



(9) 



This is the differential equation of the Verhulst-Pearl logistic 

 curve. 2 



(3) Before going further we shall examine the differential form 



2 It is to be noted that we have to do in all the cases with numbers of indi- 

 viduals per unit of volume or area, e.g., with population densities (N). 



