THE DIFFERENTIATING SYSTEM 4O9 



rather than open as was the system discussed above, and if the suppHes of 

 raw materials are taken as constant, the equations which result are of the 

 same type as those which arise in the study of the growth of two popula- 

 tions of animals which compete with one another for a limited food 

 supply. Lotka (1934) had discussed the relatively simple situation of two 

 populations (or substances) for which the equations take the form 



dp 



- = m^P- fe^P - k^,PQ 



'p'^qp 



He shows that according as m^kq is greater or less than mjzpq, and nipkg, 

 greater or less than mgkp, so the fmal state of the system is either wholly P 

 or wholly Q, or a certain fixed ratio between them, or fmally the system 

 is one which will finish up either entirely P or entirely Q according to the 

 initial concentrations of these substances, 



Kostitzin (1937) has also discussed shortly the more general case in which 

 there are many competing and interacting substances (or populations), 

 so that we have a large series of simultaneous differential equations, each 

 containing terms of the second order, such as P^ or PQ, etc. He shows that 

 such a system may be expected to exhibit a number of alternative steady 

 states, some at least of which are likely to be stable, and that the particular 

 one which the system actually attains will in many cases depend on the 

 initial conditions. 



Competitive interactions are not only almost necessary consequences of 

 the nature of the situation, but there is definite and direct evidence for their 

 existence. Perhaps the most striking is that produced by Spiegelman (1948, 

 1950) from the study of adaptive enzyme formation in yeasts. If a yeast 

 is grown in the presence of two substrates, for neither of which it 

 originally possesses the appropriate enzyme, it will gradually manufacture 

 the suitable adaptive enzymes, which are of course protein in nature. If 

 the supphes of nitrogen are restricted, these two proteins enter into 

 obvious competition with one another for it, one or both of the enzymes 

 being formed more slowly when the yeast is adapting to two substrates 

 simultaneously than when there is only one new substrate (Fig. 19.2). 

 Spiegelman also shows that the systems producing the adaptive enzymes 

 are self-reinforcing or 'autocatalytic' ; and the way in which a muscle cell, 

 for instance, fills with myosin, suggests that the same is true of the 

 formation of many cytoplasmic proteins. We have thus all the components 



