Institut für Mathematik, Technische Universität Berlin, Berlin, Germany.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico.
J Math Biol. 2021 Apr 28;82(6):53. doi: 10.1007/s00285-021-01596-0.
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.
我们研究了种子库模型的标度极限，其中迁移（进入和离开种子库）与繁殖相比是“缓慢”的。这是受细菌休眠模型的启发，其中休眠期可以比繁殖时间长几个数量级。通过加速时间，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结果，特别是提供了一个原型示例，其中连续扩散的标度极限将是跳跃扩散。对于这种情况，标准的收敛结果通常会失败。虽然这种情况原则上可以通过 Kurtz 的复杂分析方案（J Funct Anal 12:55-67, 1973）来攻击，但这将需要大量的技术努力。相反，在我们的情况下，我们能够通过对偶性以一种令人惊讶的非技术方式识别和显式描述一个明确定义的极限。事实上，我们表明，矩对偶在合适的意义下在极限传递下是稳定的，并允许直接直观地识别极限半群，同时为模型提供概率解释。我们还获得了一种在时间尺度分离情况下连续时间马尔可夫链的一般收敛策略，这具有独立的意义。
