Extinctions in time-delayed population maps, fallings, and extreme forcing

It is known that random population maps with time delay undergo a noise-mediated transition that produces the loss of its structural stability. Such a transition is optimized for a specific value of the noise correlation time of an Ornstein–Uhlenbeck forcing, as a consequence of the coupling of the involved deterministic and stochastic time scales. Here, it is shown that the deterministic time scale is related to the dynamics of the system close to a stability boundary. The escaping process depicts a survival distribution function similar to the one observed in human stick balancing, a task known to involve a truncated Levy forcing. Here, it is shown that such extreme distribution favours the stabilisation of an inverted pendulum, when compared with a normal forcing, and the system parameters are close to a stability boundary. This outcome suggests that an unstable dynamics may temporarily avoid an extinguishing transition if the extreme forcing is able to tune the system parameters at specific time delay values (of physiological significance) close to the stability boundary. These results remark the relevance of feedbacks close to the stability edge on survival and extinction.

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