68 lotka: discontinuous evolution 



suppose that the mode of variation is given, so that we may 

 restrict our discussion to the second factor, the dependence of 

 the death rate, under given external conditions, on the properties 

 of the group. 



We noted at the outset, that the distinguishing characteristic 

 of the kind of changes with which we are here concerned, is 

 their discontinuity. Let us consider a little more in detail how 

 this discontinuity enters into play in the case of an aggregate 

 of living organisms. If we single out some one particular indi- 

 vidual, and follow up its history, we shall see it exposed, in its 

 travel through time and space, to fluctuating internal and ex- 

 ternal conditions. This will in general lead to changes in the 

 distribution of energy both within the individual and between 

 it and the environment. With regard to such redistribution of 

 energy the following observation is to be made: 



In general an infinitesimal change in the distribution of energy 

 in a material system brings with it an infinitesimal change in the 

 condition of the system. If, however, such change in distri- 

 bution continues in a given direction, sooner or later a point will 

 in general be reached, beyond which any further change, even 

 if infinitesimal, will now cause a finite change in the condition 

 of the system. Thus a discontinuous change takes place. 

 Analytically we might express these facts somewhat as follows: 

 Let P be some one of the parameters which serve to define the 

 state of the system under consideration. Let the distribution 

 of energy in the system suffer a change by the passage, in stated 

 manner, of a quantity dQi of some form of energy E } from one 

 portion of the system into another, or from the surroundings 



dP 

 into the system. Then in general —r- is a finite and continuous 



dQi 



function of Q x . After a certain quantity Q c has been trans- 

 ferred, however, and the parameter P has reached a corresponding- 

 critical value P , any further increment dQ 1} however small, 

 now causes a finite change in P — in other words the function 



— — has a discontinuity at this point. We may speak of P c as 

 the " critical" or "limiting strain" of the system for the par- 



