2. SYSTEM AND ENVIRONMENT 15 



equations of the epigenetic system, providing the steady state assumption is 

 valid. We will carry out such a reduction in Chapter 4, when the principle 

 will become clearer in terms of particular equations. This procedure is by no 

 means new, and has been used in kinetic studies for a good many years. The 

 only contribution which we make is a formalization of the conditions which 

 must be met in order that the reduction be valid, conditions which we define 

 in terms of relaxation times. 



The possibility that lower-order (shorter relaxation time) variables can be 

 eliminated from the equations of motion of higher-order systems, means that 

 the dynamic description of higher-order systems need not be more complex 

 than that of lower-order systems. Thus there is no necessary relation between 

 the position of a system in a temporal ordering of dynamic activities and its 

 complexity. A biophysical system can and very often does have a much more 

 complicated mathematical description than a population of randomly-mating 

 organisms, regarded as an evolving gene pool. Even more dramatic is the fact 

 that certain epigenetic processes, such as the spiral growth of seeds in the cone 

 of a conifer, can be described in terms of a few initial conditions and a law of 

 growth which follows the Fibonacci number series (Thompson, 1959); whereas 

 the metabolic activities taking place in the same pine cone would require a 

 very complex set of equations to adequately describe their dynamics. Again, 

 the growth of a coral reef could undoubtedly be described in much simpler 

 terms than the metabolic, epigenetic, or genetic processes of the organisms 

 whose skeletons constitute the substance of the coral. Here we have clearly a 

 very great difference in relaxation times, since the reef takes many decades to 

 grow appreciably, while the coral polyps have a generation time of a few 

 months. Therefore the "nesting" properties of systems defined according to 

 relaxation times, whereby one system contains all lower-order systems, carries 

 no implications with regard to the complexity of behaviour which is found in 

 one system compared with another. 



Regarded as in some sense mathematical spaces, the systems form an 

 ordered set defined by the relation of inclusion : 



S, c= . . . c 5*, 



n 



where for example Si = metabolic system, ^2 = epigenetic system, ^3 = genetic 

 system, and so on up through higher-order constructs such as cultural system, 

 geographical system, etc. It is perhaps worth noting that an essential difference 

 between our definition of systems according to relaxation times and the various 

 criteria proposed by Nanney (1958) for identifying cellular systems, lies pre- 

 cisely in the property of inclusion of lower-order systems by higher-order ones. 

 Nanney's criteria were directed towards an exclusive division of the constit- 

 uents and processes making up the two systems, genetic and epigenetic. The 

 difficulties of this approach were discussed by Sand (1961), who suggested 

 that one might more profitably consider a distinction on the basis of process 

 rather than system. The relaxation time criterion defines system in terms of 

 process, and so it perhaps comes closer to achieving a useful distinction for 

 biological phenomena. 



