  A dynamical model of an ecological community is analyzed within a "mean-field
approximation" in which one of the species interacts with the combination of
all of the other species in the community. Within this approximation the model
may be formulated as a master equation describing a one-step stochastic
process. The stationary distribution is obtained in closed form and is shown to
reduce to a logseries or lognormal distribution, depending on the values that
the parameters describing the model take on. A hyperbolic relationship between
the connectance of the matrix of interspecies interactions and the average
number of species, exists for a range of parameter values. The time evolution
of the model at short and intermediate times is analyzed using van Kampen's
approximation, which is valid when the number of individuals in the community
is large. Good agreement with numerical simulations is found. The large time
behavior, and the approach to the stationary state, is obtained by solving the
equation for the generating function of the probability distribution. The
analytical results which follow from the analysis are also in good agreement
with direct simulations of the model.
