66 lotka: discontinuous evolution 



INVOLUTION. — Evolution in discontinuous systems. III. 1 

 Alfred J. Lotka. Communicated by J. A. Fleming. 



In our considerations so far we have supposed r i} the fractional 

 rate of increase of any group A i} to be a given function of the 

 general conditions of the system, and in particular of the masses 

 M h and ikf,. 



Let us now look a little more closely at this function r, and 

 examine it in its relation to the physical properties of the living 

 organism. 



In the first place we note that 



r = _L *K = JL (B - Z) (31) 



M dt M K 



If m is the average mass of one individual, we have 



M = Nm (32) 



For our present purpose it will be sufficiently near the truth 

 to assume m to be constant. In that case 



1 dM 1 dN 1 trt Q \ , N /ooX 



r= ■ = = — (G — S) = (q — 8) (33) 



M dt N dt N K y y ' 



where 



G = total number of births per unit of time, and 



S = total number of deaths per unit of time. 



Our problem, then, is to investigate G and S, or g and s as 

 functions of the physical properties of the organism. 



It may appear at first sight the most logical procedure to 

 discuss first of all B or G. On looking into the matter, however, 

 we find that B can always be referred back to Z, and that the 

 latter appears really more directly related to the physical 

 properties of the system, so that its discussion naturally takes 

 first place. 



The statement has just been made, that B can always be 

 referred back to Z. This is evidently true, for whatever material 

 is gained by one group, must be lost by one or more other groups, 

 or to express this in the form of an equation, 



1 See this Journal, 2: pp. 2 and 49, 1912. 



