The Allee effect is a nonlinear phenomenon exhibited in the population dynamics of sparse populations in which the per capita population growth rate increases with increasing population density. In sufficiently sparse populations, the Allee effect may lead to extinction and is known to generate a threshold in the probability of establishment when presented as a function of introduced population size or density. As introduced populations are generally small, Allee effects are probably common in biological invasions and their consideration is necessary for accurately assessing the risk of invasion by many species, including all sexually reproducing species. Bythotrephes longimanus, an invasive, freshwater, cladoceran zooplankter from Europe, is one such species. Here, I review a previously published model of the Allee effect for continuously sexually reproducing species. Then, I develop a new model for seasonally parthenogenetic species such as Bythotrephes, and thereby demonstrate the potential consequences of Allee effects. This result underscores the importance of considering nonlinear phenomena, including thresholds, when conducting risk analysis for biological introductions.