ABSTRACT
In Chapter 6 we developed a simple Markov process based on multiple immigrations and in Chapter 7 we showed how this model could be exploited to describe the time evolution of a stable variable and generate a stable series of events-characteristics often ascribed to complex systems. The main part of this chapter develops the multiple immigrations model further, based on the considerations of Chapter 6, Section 6.5. There we noted that the multiple immigration coefficients νr could themselves be interpreted as probabilities so that the multiple immigrations term in the rate equation for the generating function Q(s,t) relating to the population of interest could therefore be expressed in terms of the generating function
Q s( ) for the multiple immi-
gration probabilities. One consequence of this was that temporal variations in the immigration coefficients could be included in the model by ascribing a time evolution to
Q through a second process. Thus the number of immi-
grants arriving in the population of interest is effectively governed by the evolution of an ensemble of secondary populations. It is of course possible for the secondary populations to be influenced in the same way by an ensemble of primary ones. Here we shall investigate the consequence of this type of two-way coupling through study of simple death-multiple immigration models. We shall show that models of this kind are exactly solvable under certain conditions and we shall explore their potential for generating stable populations and oscillatory behaviour.
