The following is written from the perspective of using the Poisson lognormal distribution
to describe community structure (the distribution of species when sampling individuals
from a community of several species).
Under the assumption of random sampling, the number of individuals sampled from a given
species with abundance y, say N, is Poisson distributed with mean \(\code{nu}\,y\)
where the parameter nu expresses the sampling intensity. If ln y is normally distributed
with mean mu and standard deviation sig among species, then the vector of individuals sampled
from all S species then constitutes a sample from the Poisson lognormal distribution
with parameters (mu + ln nu, sig), where mu and sig
are the mean and standard deviation of the log abundances. For nu = 1, this is the Poisson
lognormal distribution with parameters (mu,sig) which may be written in the form
$$P(N=\code{n};\code{mu},\code{sig}) = q(\code{n};\code{mu},\code{sig}) = \int\limits_{-\infty}^{\infty} g_\code{n}(\code{mu},\code{sig},u)\phi(u)\;du,$$
where \(\phi(u)\) is the standard normal distribution and
$$g_\code{n}(\code{mu},\code{sig},u) = \frac{\exp(u\,\code{sig}\,\code{n} + \code{mu}\,\code{n} - \exp(u\,\code{sig} + \code{mu}))}{\code{n}!}$$
Since S is usually unknown, we only consider the observed number of individuals for the observed species.
With a general sampling intensity nu, the distribution of the number of individuals then follows the
zero-truncated Poisson lognormal distribution
$$\frac{q(\code{n};\code{mu},\code{sig})}{1-q(0;\code{mu},\code{sig})}$$