  The Verhulst model is probably the best known macroscopic rate equation in
population ecology. It depends on two parameters, the intrinsic growth rate and
the carrying capacity. These parameters can be estimated for different
populations and are related to the reproductive fitness and the competition for
limited resources, respectively. We investigate analytically and numerically
the simplest possible microscopic scenarios that give rise to the logistic
equation in the deterministic mean-field limit. We provide a definition of the
two parameters of the Verhulst equation in terms of microscopic parameters. In
addition, we derive the conditions for extinction or persistence of the
population by employing either the "momentum-space" spectral theory or the
"real-space" Wentzel-Kramers-Brillouin (WKB) approximation to determine the
probability distribution function and the mean time to extinction of the
population. Our analytical results agree well with numerical simulations.
