  We consider excursions for a class of stochastic processes describing a
population of discrete individuals experiencing density-limited growth, such
that the population has a finite carrying capacity and behaves qualitatively
like the classical logistic model when the carrying capacity is large. Being
discrete and stochastic, however, our population nonetheless goes extinct in
finite time. We present results concerning the maximum of the population prior
to extinction in the large population limit, from which we obtain establishment
probabilities and upper bounds for the process, as well as estimates for the
waiting time to establishment and extinction. As a consequence, we show that
conditional upon establishment, the stochastic logistic process will with high
probability greatly exceed carrying capacity an arbitrary number of times prior
to extinction.
