"...The basic parameters of the particular equilibrium curves are
|h1 = (H1 + k1H2)/(1 - r1r2)||h2 = (H2 + k2H1)/(1 - r1r2)|
It is clear that, the greater the initial hostilities and the greater the reaction coefficients, the greater the equilibrium level of hostility. Xenophobia and touchiness both lead to high levels of hostility. As ling as r1r2 < 1, there will be some position of equilibrium, no matter how great the initial hositilities. If the parties are sufficiently touchy, so that r1r2 > 1, there will be no position of equilibrium (except at negative levels of hostility, and this will not be stable). If there are diminishing returns to touchiness (m1 is positive), the equilibrium level of hostility i slikely to be less than th initial touchiness would indicate. In this case, there may be an equilibrium even if levels of touchiness are high. The effect on the equilibrium of a pure change in intitial hostility of one part depends on the sign and slope of the reaction function of the other..." 13
"The processes of social systems which best correspond to determined mechanical dynamic processes are processes which invlove simple growth. Growth at a constant rate can be represented by a simple difference equation of the first degree, such as xt+1 = Kxt, for each term in the sequence is a constant proportion of the term before it. Thus if we leave money in an account growing at compund interest, as long as the rate of interest, that is, its rate of growth, remains constant, the principle sum is highly predictable at any future date into which the system extends.... Similarly, where a country is undergoing steady economic growth at a reasonably constant rate, we can again predict its growth into the future with some confidence. More complex use of systems of this kind might include a model which would relate the rate of growth in some simple way to the existing size of whatever it is that is growing. There is a good deal of evidence, for instance, for the proposition that, for those countries which are having a successgul growth pattern, the rate of growth dimishes as the country get richer. If this is a reasonably stable relationship, however, we can still make predictions with some confidence." 14
"...All real conflict takes place in time and consists of a succesion of states of situation or field. If this succession is governe by fairly simple laws, so that there is a stable relationship between the state of today and the state of tomorrow or, say, betwen yesterday, today, and tomorrow, then there is a dynamic system that can, in general, be solved to predict the state of the system at any future date. One of the great problems in social dynamics is that the dynamic systems are not stable and are frequently subject to unpredictable change. Nevertheless, the succession of states of a social system is not random; some regularites can usually be deteced, and even if the system is not stable enough to permit unconditional prediction of its future states, at least enough can usually be known about it to set limits on the probability of various future positions.
"Where the dynamics of the system results in a succession of identically similar states, the system is said to be in equilibrium. Many dynamic systems move toward an equilibirum; some, however, do not, and move either indefinately onward or else toward system breakdown, some point at which the laws of the system change. An interesting special case is that of cyclical equilibrium, in which the system repeats indefinitely a certain sequence of states. Equilibrium is thus seen to be a special case of a dynamic process; indeed, the stability of an equilibrium connot be known with certainty unless the dynamic process of which it is a special case is also known. In a stable equilibrium, the dynamic paths all coverge on the equilibrium, the dynamic paths all coverge on the equilirium point; in an unstable equilibrium, the dynamic paths diverge away from the equilibrirum point. In a circular equilibrium, the dynamic paths converge on a ring of successive states.
"...The dynamic system that I have described is an example--and only one out of a very large number of possible examples--of a dynamic system in the small. It is described essentially be means of differential or difference equations, which tell us exactly how we get from one point to the next. A difference equation is simply a stable relationship between today and yesterday, and perhaps the day before yesterday or the day before that, depending on the degree of the equation. If we know how to get from yesterday to today and the same law applies, then we know how to get on from today to tomorrow, from tomorrow to the next day, and so on ad infinitum. What might be called mechanical prediction depends on the discovery of stable difference (or differential) equations. This is the secret of the remarkably predictive succes of the astronomers; the movements of the planets are described by very stabe, though complex, differential equations, and so their positions can be predicted indefinitely from the past into the future." 15