I have described how selection and transformation can be viewed as the two fundamental processes that can cause changes in the composition of a population in deterministic models. Each of these forces will have its own effect on the population. What I wish to do in this chapter is examine some of the basic properties of the interaction of these two processes.

Without specifying the exact nature of the types in the population or the nature of the transformations or selection acting on them, we can nevertheless examine some general features of their evolution when both selection and transformation are occuring. In this chapter, I will model two basic life cycles, pure branching and pair-mating, where selection and transformation are occurring. By employing the formal representations of selection and transformation developed in the previous chapter, mathematical recursions will be developed giving the frequencies of types in terms of their frequencies in the previous generation, which will allow a rigorous analysis of the interaction of selection and transformation. These recursions will be framed for infinite populations, and selection and transformation will not be fluctuating in time. The results are summarized as follows:

Populations where no transformation is occurring have these properties:

1. The mean fitness of the population increases in time when fitnesses are frequency independent.
2. The frequencies converge to stable equilibrium points, or surfaces in which the population is neutral.
3. At equilibrium, the marginal frequencies of all types present are equal.

In contrast populations also undergoing transformation have the following properties:

1. The mean fitness of the population may decrease in time.
2. The frequencies of types may approach stable limit cycles.
3. The set of types present at stable equilibria or cycles may be different when there is transformation occurring from those present when only selection is acting.
4. Frequency dependence of the marginal fitnesses is not required to produce positive stability of equilibria.
5. The marginal fitnesses of the types present at an equilibrium need not be equal.

It is this last property that is the basic source of selection on transformation types.