  In this work we construct individual-based models that give rise to the
generalized logistic model at the mean-field deterministic level and that allow
us to interpret the parameters of these models in terms of individual
interactions. We also study the effect of internal fluctuations on the
long-time dynamics for the different models that have been widely used in the
literature, such as the theta-logistic and Savageau models. In particular, we
determine the conditions for population extinction and calculate the mean time
to extinction. If the population does not become extinct, we obtain analytical
expressions for the population abundance distribution. Our theoretical results
are based on WKB theory and the probability generating function formalism and
are verified by numerical simulations.
