lotka: discontinuous evolution 67 



u=co 



B^^\ V Z V (34) 



u=l 



where the coefficients A,,„ have an obvious import. 



But in many cases there is also a simple relation between g 

 and s. Thus for a stationary condition we have 



9 = s (35) 



while, under constant conditions, birth and deathrate tend to 

 approach the relation 



- = fV fg - 8)a p(a)da (36) 2 



g Jo 



where p(a) is the probability, at birth, that an individual picked 

 out at random from among newly born shall reach age a. 



We obtain still another relation between g and s if we suppose 

 that matters have so adjusted themselves, that the birthrate is 

 the one that gives the maximum rate of increase for the popu- 

 lation, a condition which is presumably approached in nature. 

 In such a case we must have 



dr = d(g-sl =0 (37) 



dg dg 



or 



^ = 1 (38) 



dg 



Our immediate task, then, is to discuss the influences which 

 determine the deathrate of a given group. Obviously such 

 deathrate will depend (1) On external conditions; (2) On the 

 the properties of the group. 



As regards the first of these two factors, we note that in general 

 the external influences to which the individual is exposed vary 

 from point to point in space and from instant to instant at a 

 given point. For the purposes of our present discussion we shall 



2 Lotka, Am. Jl. Sci., 24: 201. 1907. Science, 26: 22. 1907. Sharpe and 

 Lotka, Phil. Mag., p. 437, April, 1911. 



