Both this chapter and the next survey basic branching processes relevant to biological applications. We describe process structure, and give some examples. Underlying assumptions are exhibited and it is indicated when and where a specific model might be applied (which areas, under what conditions). Note that even the simplest models have proved their usefulness to biology in the past, and continue to find intriguing new applications (e.g., in molecular biology, cf. Chapter 7).

The purpose is to give an impression of what is available in the literature, so as to gain an overview of the theory as well as starting points for modeling. Furthermore, we give references in which more details, proofs, and results on these models, and generalizations of them, can be found. When possible we rely on verbal formulations and illustrations, rather than extensive equations, and we do not give the main mathematical results yet. However, we do introduce basic concepts and notation, and also some fundamental facts used in later chapters, in which the results are stated and their relevance to basic biological issues made clear.

This chapter treats discrete-time models. In these, time is represented by integers that indicate reproduction periods. Thus, it is assumed that reproduction can only occur during separate non-overlapping periods. In the simplest forms (described in Sections 2.1 to 2.3) generations cannot overlap either (think of annual plants).

An argument in favor of discrete-time models is that they are mathematically much easier to handle than those in continuous time (see Chapter 3). Furthermore, many species do reproduce only during fixed periods of the year.