How to avoid an extinction time paradox

An essential topic in theoretical ecology is the extinction of populations subject to demographic stochasticity. Mechanistic models of demographic stochasticity, such as birth-death processes, can be analytically intractable, so are frequently approximated with stochastic differential equations (SDEs). Here, we consider two pitfalls in this type of approximation. First, familiar deterministic models are not always appropriate for use in an SDE. Second, the common practice of starting directly from an SDE without explicitly constructing a mechanistic model leaves the noise term up to the modeler’s discretion. Since the stability of stochastic models depends on the global properties of both the noise and the deterministic model, overly phenomenological deterministic models, or heuristic choices of noise, can lead to models that are unrealistically stable. The goal of this article is to provide an example of how both of these effects can undermine seemingly reasonable models. Following Dennis et al. (Theor Ecol 9:323–335 2016) and Levine and Meerson (PRE, 87:032127 2013), we compare the persistence of stochastic extensions of standard logistic and Allee models. We show that, for common choices of noise, stochastic logistic models become exponentially less extinction prone when a strong Allee effect is introduced. This apparent paradox can be resolved by recognizing that common models of an Allee effect introduce overcompensation that dominates the extinction dynamics, even when the deterministic model is rescaled to account for overcompensation. These problems can be resolved by mechanistic treatment of the deterministic model and the noise.