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.

中文翻译：

---

具有 Allee 效应的空间结构种群中的多吸引子和长瞬态

我们提出了一个空间结构人口的离散时间模型，并探讨了当局部动态包含强烈的 Allee 效应和过度补偿时耦合的影响。虽然孤立的种群只能表现出双稳态和基本灭绝，但空间结构的种群可以表现出许多共存的吸引子。我们确定了可以保护空间结构种群免于基本灭绝的机制和参数范围，而这在本地系统中是不可避免的。在弱耦合的情况下，一种亚种群密度高于 Allee 阈值而另一种低于 Allee 阈值的状态可以防止基本灭绝。另一方面，当动态（大约）异相时，强耦合使两个种群都能持续高于 Allee 阈值。在这两种情况下，吸引子具有分形盆地边界。在这些参数范围之外，没有发现扩散可以防止基本灭绝。我们还展示了空间结构如何在种群灭绝之前导致持久的长期瞬变。