  Many species are unsustainable at small population densities (Allee Effect),
i.e., below a threshold named Allee threshold, the population decreases instead
of growing. In a closed local population, environmental fluctuations always
lead to extinction. Here, we show how, in spatially extended habitats,
dispersal can lead to a sustainable population in a region, provided the
amplitude of environmental fluctuations is below an extinction threshold. We
have identified two types of sustainable populations: high-density and
low-density populations (through a mean-field approximation, valid in the limit
of large dispersal length). Our results show that patches where population is
high, low or extinct, coexist when the population is close to global extinction
(even for homogeneous habitats). The extinction threshold is maximum for
characteristic dispersal distances much larger than the spatial scale of
synchrony of environmental fluctuations. The extinction threshold increases
proportionally to the square root of the dispersal rate and decreases with the
Allee threshold. The low-density population solution can allow understanding
difficulties in recovery after harvesting. This theoretical framework provides
a novel approach to address other factors, such as habitat fragmentation or
harvesting, impacting population resilience to environmental fluctuations.
