A. J. Lotka — Mode of Growth of Material Aggregates. 199 



Art. XXII. — Studies on the Mode of Growth of Material 

 Aggregates y by Alfred J. Lotka. * 



I. 



Iisr a material system in which the physical conditions vary 

 with the time, certain individual constituent elements may 

 have a transitory existence as such, each lasting just so long as 

 its conditions and those of its neighborhood continue within 

 certain limits. 



Although the "life period" of each individual element may 

 be thus limited, an aggregate of a number of such individuals 

 may nevertheless have a prolonged existence, and may even 

 grow, provided that the variations in the conditions of the sys- 

 tem do not exceed certain limits, and that by some process or 

 other new individuals are continually formed as the old are 

 eliminated. 



Aggregates of this kind play an important role in nature, 

 and a study of their mode of growth or decay would be of 

 considerable interest. 



The general problem is evidently one of extreme complexity, 

 but some special cases which lend themselves to simple mathe- 

 matical treatment would seem to be rather instructive. 



Let us then consider an aggregate which at time t consists 

 of X individuals of a specified kind. If ~N t is the number of 

 individuals in the aggregate at time t, let us examine the rela- 

 tion between jST and ~N t . 



In the first place we note, that if 



B t == number of individuals added to the aggregate per unit 

 time at the instant t, 



and if 



D t = number of individuals eliminated therefrom per unit 

 time at the instant t, 

 then, 



d^ „ ^ 



1ST 



= N + / 1 (B t -D t )<ft. (2) 



D t is in general a function of ~N t , its value depending : 



1. On the character of the limitation of the " life "period " 

 of the individuals. 



2. On the " distribution of ages " among the individuals. 



B t may also in some cases be more or less directly dependent 

 upon ~N t and upon the "distribution of ages " in the aggregate. 



