Functional-integral based perturbation theory for the Malthus-Verhulst process

TL;DR

This paper develops a functional-integral perturbation approach to analyze the Malthus-Verhulst process, providing asymptotic corrections to mean-field theory and demonstrating good agreement with quasi-stationary states.

Contribution

It introduces a diagrammatic expansion method for the MVP, extending mean-field analysis to include higher-order corrections in the large population limit.

Findings

Good agreement with quasi-stationary moments for large populations

Derived fifth-order correction series for the MVP

Compared results with van Kampen's Omega-expansion

Abstract

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.