  We present a discrete-time model of a spatially structured population and
explore the effects of coupling when the local dynamics contain a strong Allee
effect and overcompensation. While an isolated population can exhibit only
bistability and essential extinction, a spatially structured population can
exhibit numerous coexisting attractors. We identify mechanisms and parameter
ranges that can protect the spatially structured population from essential
extinction, whereas it is inevitable in the local system. In the case of weak
coupling, a state where one subpopulation density lies above and the other one
below the Allee threshold can prevent essential extinction. Strong coupling, on
the other hand, enables both populations to persist above the Allee threshold
when dynamics are (approximately) out-of-phase. In both cases, attractors have
fractal basin boundaries. Outside of these parameter ranges, dispersal was not
found to prevent essential extinction. We also demonstrate how spatial
structure can lead to long transients of persistence before the population goes
extinct.
