Understanding how anthropically induced interacting factors may compromise the viability of a particular species or population necessitates expressing them in terms of quantitative effects on population dynamics. The use of mechanistic models to assess these effects is especially helpful to management plans when the causes of species decline are multi-factorial and potentially interacting. Here, we took the opportunity of observed predation by a vagrant falcon on a rare and endemic seabird to develop a population dynamics model encompassing multiple deterministic and stochastic threats. The Macgillivray's prion Pachyptila macgillivrayi, with a world breeding population of c. 540 individuals confined to one rat-free islet off Saint-Paul Island (Southern Indian Ocean), faces high extinction risk from vagrant falcon predation. Extinction is predicted to occur within 200 years if one falcon preys on prions every 5 years, with predation occurring either on breeders only or also on non-breeders. The minimum initial prion population size ensuring a low extinction risk increased linearly with the annual probability of predator occurrence. Therefore, increasing the initial population size is a useful management option to help the prion face stochastic predation. Recent rat eradication on Saint-Paul Island helps the prion to face this threat as it released the carrying capacity of the colony, but the earlier population size might never be recovered if falcons carry on preying on prions frequently. This rare burrowing petrel provides a remarkable case study of an endemic insular species threatened with predation by alien mammals, which reduced historical population size dramatically, and by genuine vagrants as catastrophic events that reduce population growth and increase its temporal variance, and might cause the extinction.