Generalized paradox of enrichment: noise-driven rare rarity in degraded ecological systems.

The paradox of enrichment stipulates that increasing the resources available to the prey population can lead to instability and a higher likelihood of population fluctuations. We study the converse situation where the prey's environment is degrading and ask if the dynamical interplay between this degradation and stochasticity can be beneficial to the stabilization of the prey population. The underlying systems are non-autonomous and subject to noise. We uncover a phenomenon pertinent to the paradox of enrichment: rare rarity. In particular, in a slow-fast ecosystem with a sole stable equilibrium, noise can induce dynamical excursions of a trajectory into a region with low species abundance, resulting in rarity. Surprisingly, it is the same noise that can facilitate a rapid recovery of the abundance of the rare species, shortening the duration of the rarity. As the environment continues to degrade, the occurrence of such rarity events can be non-uniform in time and even more rare. The intermittent occurrence of rare rarity is caused by the dynamical interplay between the phase-space distance from the stable equilibrium to the boundary separating two distinct regions of transient dynamics. The rare-rarity phenomenon can also arise in other natural systems such as the climate carbon-cycle system.