# Invasion probabilities, hitting times, and some fluctuation theory for the stochastic logistic process

## TL;DR

This paper analyzes the stochastic logistic process, focusing on invasion probabilities, hitting times, and fluctuation behavior, providing new estimates for extinction, establishment, and population maxima in large populations.

## Contribution

It introduces new theoretical results on the maximum population size, invasion probabilities, and waiting times for a stochastic logistic process with finite carrying capacity.

## Key findings

- High probability of exceeding carrying capacity before extinction
- Establishment probabilities and upper bounds derived
- Estimated waiting times for extinction and establishment

## Abstract

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.