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. In the models we discuss in this chapter and later in Chapter 4 we have scaled the time step to be 1. Models must thus relate the population at time t 1, denoted by Nt+1, in terms of the population Nt at time t. This leads us to study difference equations, or discrete models, of the form

EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa % aaleaacaWG0bGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGobWaaSba % aSqaaiaadshaaeqaaOGaamOraiaacIcacaWGobWaaSbaaSqaaiaads % haaeqaaOGaaiykaiabg2da9iaadAgacaGGOaGaamOtamaaBaaaleaa % caWG0baabeaakiaacMcaaaa!460B! ]></EquationSource><EquationSource Format="TEX"><![CDATA[$${N_{t + 1}} = {N_t}F({N_t}) = f({N_t})$$

((2.1))

where f (Nt) 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 in diverse fields: it is a fascinating area. From a practical point of view if we know the form of f (N _t ) it is a straightforward matter to evaluate Nt+1 and subsequent generations by simply using (2.1) recursively. Of course, whatever the form of f (N), we shall only be interested in non-negative populations.