Other important examples of systems beyond the scope of the physical sciences are ecological systems. A portion of the earth is surface, say a forest, or better still, a body of water such as a lake, can be viewed as a living system in an environment: an ecological system. A state of an ecological system can be taken as the totality of concentrations of organisms of various kinds (including plants) in it, perhaps also their spatial distributions. These quantities change as organisms procreate and die, migrate or emigrate. The rates of change depend on the concentrations because some organisms feed upon others: One can well imagine cyclic fluctuations resulting from predator-prey interactions. As a very simple model, consider a population of foxes and rabbits. The former feed on the latter. As the rabbit population increases, foxes eat better and their population increases also: But as the foxes become more numerous, they catch more rabbits, reducing the rabbit population, which in turn checks the fox population. It is conceivable that, if left alone, an ecological system may attain a steady state -- the concentration of the various organisms being in equilibrium and the equilibrium being preserved by the mutual checks of the populations on each other. (E, p. 20)
Solar System as a dynamic system.
The theory of the solar system is a good example of a dynamic system theory. The state of the solar system at a particular moment is completely described by the relative positions and velocities of the planets relative to the sun. The theory is the system of equations reflecting the rates of change of velocities of these bodies under the influence of their mutual gravitational attractions and all the assertions deduced from these equations. Given the positions and velocities of the planets at one moment of time, they can be deduced for all successive moments. Thus the sequence of states of the system can be deduced from the initial state, that is, from partial information. A dynamic theory permits the determination of the succession of states, once the initial state of the system is known. (E, p. 17)
If two physical systems obey the same mathematical law, they are isomorphic to each other. A famous example of such isomorphism is that between a mechanical harmonic oscillator and an electrical circuit with an inductance, a resistance, and a capacitance. (C, p. 455)
Mathematical formulation for growth in a system.
A system, roughly speaking, is a bundle of relations. For this reason a general system theory, in my opinion, ought to single out purely relational isomorphisms that are abstracted from content.
dm/dt = am1/2 - bm
As an example, consider a mathematical formulation of the growth of some system Specifically, let a physical system be a solid body with a boundary, and let growth be the result of ingestion of substances from the outside through the boundary at a constant rate per unit of surface. Moreover, let the substance which makes up the system break down inside the system as a constant rate per unit mass and be excreted. Then, since the surface is proportional to the two-thirds power of the volume, while mass is proportional to the volume, we have the equation
where m is the mass and a and b are constants (Bertalanffy 1957). Such will be the "law of growth" of all physical systems of this sort, regardless of size or internal organization. On the other hand, if the system is essentially one-dimensional (i.e., grows only at the ends at a constant rate, while breaking down at a constant rate per unit mass), its law of growth will be
dm/dt = a - bm
Evidently not the "level" of the system but, rather, its geometry is likely to be the determinant of its law of growth. If so, then attempts to specify particular laws of growth for "cells," "populations," "corporations," etc., will prove futile. (C, pp. 454-455)
Quantified structure in a matrix.
According to a narrow definition of quantity (a number), the structures as defined by communication channels are not quantities, because they cannot be represented by numbers in the ordinary sense.
A matrix is an array of quantities arranged like a checker board in rows and columns. Such an array can conveniently represent the set of relations between each pair of members in some set. ( 2 - pp. 164-165)
Much of Dr. Rapoport's work was done in the field of game theory, of which the Prisoner's Dilemma is a classic example. He has written one entire book on the problem, with references in many others. The problem as described here, is usually played repeatedly in order to test rational behavior with feedback, since each prisoner knows what the other did last time.
Two prisoners charged with the same crime are held incommunicado. If both confess, both can be convicted. If neither confesses, nither can be convicted. But if one confesses but the other holds out, the first not only goes scot free but gets a reward to boot, while the second gets a more severe punishment than he would have got if both confessed. Should a rational prisoner confess or hold out under these circumstances? (4, p. 290)
Cybeneticians Anatol Rapoport Exemplars
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