Chapter 2 introduces discrete-time branching processes. Mathematically, these are much simpler objects than branching processes in continuous time. We have also seen that they occur naturally in many situations, such as generation counting and populations with seasonal regularity in reproduction, and in models for demographic changes recorded annually. Furthermore, it can be argued that data are never recorded continuously, but rather at regular or irregular (albeit sometimes short) intervals. Thus, models in continuous time are not necessarily needed.
The need is more on the conceptual or possibly perceptional side. We certainly conceive of time as a continuous flow, and if mathematical models are to mimic such firsthand conceptions of reality, they should be formulated in continuous time. Similarly, 19th century scientists thought of matter, such as fluids or metals, as self-evidently continuous in the same way as we perceive time. This view has been changed drastically by modern particle-and quantum-based discrete physics.
However, to what extent our perception of time is a cultural, psycho-biological, or physical phenomenon lies outside the scope of this book. We content ourselves with the observation that a continuous-time development of discrete populations is closest to our spontaneous perception of population growth in the flow of time, and that there are good classic mathematical tools for analyzing such situations.
The price to be paid for continuous-time modeling is that the foundation (the rigorous construction of probability spaces and processes) requires more advanced mathematics. We try to conceal this by avoiding explicit construction of the stochastic processes involved. For that, we refer to the mathematical literature.