  We apply a functional-integral formalism for Markovian birth and death
processes to determine asymptotic corrections to mean-field theory in the
Malthus-Verhulst process (MVP). Expanding about the stationary mean-field
solution, we identify an expansion parameter that is small in the limit of
large mean population, and derive a diagrammatic expansion in powers of this
parameter. The series is evaluated to fifth order using computational
enumeration of diagrams. Although the MVP has no stationary state, we obtain
good agreement with the associated {\it quasi-stationary} values for the
moments of the population size, provided the mean population size is not small.
We compare our results with those of van Kampen's $\Omega$-expansion, and apply
our method to the MVP with input, for which a stationary state does exist.
