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 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. We also devise a modified Fokker-Planck approach for this case.