Chance establishment for sexual, semelparous species: overcoming the Allee effect.

We formalize the establishment process for a sexual, semelparous organism through the use of hierarchical probability modeling from parameters of survival, probability of being female, probability of being fertilized, and expected fecundity. We show how to calculate the expected per capita growth rate and probability of extinction. An Allee effect is observed if the expected population growth rate decreases as the initial population size decreases. The model can be further extended as a stochastic process to evaluate the probability of extinction in subsequent generations. One of the novel results is the formulation of an analytical probability distribution for the next generation population size. As case studies, we use the Chinese mitten crab (Eriocheir sinensis) and the apple snail (Pomacea canaliculata), both of which appear on the World Conservation Union's list of 100 worst invaders. We evaluate the strength of the Allee effect and conclude that apple snails experience a weak Allee effect and Chinese mitten crabs experience a strong Allee effect. We emphasize one scenario where the stochastic process reveals that invasion risk can be estimated by the probability of the survival of one fertilized female, because the expected fecundity for one surviving female overwhelms the system such that population persistence is almost certain.