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.

中文翻译：

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如何避免灭绝时间悖论

理论生态学中的一个基本主题是人口随机种群的灭绝。人口统计随机性的机械模型，例如生死过程，在分析上可能是棘手的，因此经常使用随机微分方程（SDE）进行近似。在这里，我们考虑这种近似方法的两个陷阱。首先，熟悉的确定性模型并不总是适合在SDE中使用。其次，直接从SDE开始而不显式构造机械模型的通用做法使噪声项由建模者自行决定。由于随机模型的稳定性取决于噪声和确定性模型的整体属性，因此，现象学的确定性模型或噪声的启发式选择过多，可能导致模型变得不切实际的稳定。本文的目的是提供一个示例，说明这两种影响如何破坏看似合理的模型。继丹尼斯等。（Theor Ecol 9：323–335 2016）和Levine and Meerson（PRE，87：032127 2013），我们比较了标准逻辑模型和Allee模型的随机扩展的持久性。我们表明，对于噪声的常见选择，随机逻辑模型成指数增长少当引入消光易发生强烈的阿利效应。可以通过认识到Allee效应的常见模型引入过大的补偿来解决这种明显的矛盾，即使补偿了确定性模型以解决过度补偿，过补偿也主导了灭绝的动态。这些问题可以通过对确定性模型和噪声的机械处理来解决。