We consider excursions for a class of stochastic processes describing apopulation of discrete individuals experiencing density-limited growth, suchthat the population has a finite carrying capacity and behaves qualitativelylike the classical logistic model when the carrying capacity is large. Beingdiscrete and stochastic, however, our population nonetheless goes extinct infinite time. We present results concerning the maximum of the population priorto extinction in the large population limit, from which we obtain establishmentprobabilities and upper bounds for the process, as well as estimates for thewaiting time to establishment and extinction. As a consequence, we show thatconditional upon establishment, the stochastic logistic process will with highprobability greatly exceed carrying capacity an arbitrary number of times priorto extinction.
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