Discrete Population Models for a Single Species

Abstract

Differential equation models, whether ordinary, delay, partial or stochastic, imply a continuous overlap of generations. Many species have no overlap whatsoever between successive generations and so population growth is in discrete steps. For primitive organisms these can be quite short in which case a continuous (in time) model may be a reasonable approximation. However, depending on the species the step lengths can vary widely. A year is common. With fruit fly emergence from pupae it is a day, for cells it can be a number of hours while for bacteria and viruses it can be considerably less. In the models we discuss in this chapter and later in Chapter 5 we have scaled the time-step to be 1. Models must thus relate the population at time t + 1, denoted by N_t+1, in terms of the population N_t at time t. This leads us to study difference equations, or discrete models, of the form N_t+1 = N_tF(N_t) = f(N_t), (2.1) where f(N_t ) is in general a nonlinear function of N_t. The first form is often used to emphasise the existence of a zero steady state. Such equations are usually impossible to solve analytically but again we can extract a considerable amount of information about the population dynamics without an analytical solution. The mathematics of difference equations is now being studied in depth and applied in diverse fields: it is a fascinating subject having given rise to some totally unexpected phenomena some of which we discuss later. Difference equation models are also proving of use in a surprisingly wide spectrum of biomedical areas such as cancer growth (see, for example, the article by Cross and Cotton 1994), aging (see, for example, the article by Lipsitz and Goldberger 1992), cell proliferation (see, for example, the article by Hall and Levinson 1990) and genetics (see, for example, the chapter on inheritance in the book by Hoppensteadtand Peskin 1992 and the book by Roughgarden 1996.)