ABSTRACT
The birth-death-immigration model studied in the last chapter is conventional or classical in the sense that when there are many individuals present, so that the population is ‘dense’, a limit exists in which the population fluctuations are asymptotically characterised by the probability density of a continuous process. In the case of a Poisson distribution, the relative variance of the fluctuations vanishes as the mean becomes large and the asymptotic continuous process is governed by a delta function probability density, that is,
→ δ −P P N N N( ) ~ ( ).N 0 (5.1)
The probability density (5.1) characterises a continuous variable that always takes the value N0, that is, exhibits no fluctuations. Thus the high density limit of the Poisson distribution is the narrowest probability density possible for a continuous variable. However, it is clearly possible to envisage discrete distributions that are narrower than Poisson in the sense that their variance is less than that of a Poisson distribution with the same mean value, that is, with a Fano factor less than unity. These have no continuous counterparts but may be of particular interest in the case of small populations. On the other hand, it should be emphasised that some distributions with a Fano factor larger than unity may also have no continuous counterpart; for instance, distributions that exhibit a difference in character between even and odd numbers of individuals. Many examples of these can be envisaged in biological populations arising from twins and various kinds of pairing processes. Important even-odd effects also occur in quantum systems, for example, when pairs of particles are emitted into a population through nonlinear optics effects, multiple atomic transitions or radioactive decay.
