We investigate scaling limits of the seed bank model when migration (to and from the seed bank) is ‘slow’ compared to reproduction. This is motivated by models for bacterial dormancy, where periods of dormancy can be orders of magnitude larger than reproductive times. Speeding up time, we encounter a separation of timescales phenomenon which leads to mathematically interesting observations, in particular providing a prototypical example where the scaling limit of a continuous diffusion will be a jump diffusion. For this situation, standard convergence results typically fail. While such a situation could in principle be attacked by the sophisticated analytical scheme of Kurtz (J Funct Anal 12:55–67, 1973), this will require significant technical efforts. Instead, in our situation, we are able to identify and explicitly characterise a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that moment duality is in a suitable sense stable under passage to the limit and allows a direct and intuitive identification of the limiting semi-group while at the same time providing a probabilistic interpretation of the model. We also obtain a general convergence strategy for continuous-time Markov chains in a separation of timescales regime, which is of independent interest.

当迁移（进出种子库）比繁殖“慢”时，我们研究了种子库模型的缩放限制。这是由细菌休眠模型驱动的，其中休眠期可能比繁殖时间长几个数量级。加速时间，我们遇到了时间尺度分离现象，这导致了数学上有趣的观察，特别是提供了一个原型示例，其中连续扩散的缩放限制将是跳跃扩散。对于这种情况，标准收敛结果通常会失败。虽然这种情况原则上可能会受到 Kurtz 复杂的分析方案（J Funct Anal 12:55-67, 1973）的攻击，但这将需要大量的技术努力。相反，在我们的情况下，我们能够以一种令人惊讶的非技术方式通过对偶性来识别和明确描述一个明确定义的限制。事实上，我们证明了矩对偶在达到极限时在合适的意义上是稳定的，并且允许直接和直观地识别极限半群，同时提供模型的概率解释。我们还获得了具有独立兴趣的时间尺度分离机制中连续时间马尔可夫链的一般收敛策略。