We present three examples of chemical reaction networks whose ordinary differential equation scaling limit are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences and characterize the invariant measures by Lyapunov function techniques. In particular, we design a natural set of such functions which scale homogeneously to infinity, taking advantage of the same scaling behavior of the reaction rates.

中文翻译：

---

看似稳定的化学动力学可能是稳定的、边缘稳定的或不稳定的

我们展示了三个化学反应网络的例子，它们的常微分方程标度极限几乎相同并且在所有情况下都是稳定的。然而，与这些反应网络相关的马尔可夫跳跃过程显示出全方位的行为：一种是稳定的（正循环），一种是不稳定的（瞬态），一种是边缘稳定的（零循环）。我们研究了这些差异并通过李雅普诺夫函数技术表征了不变性度量。特别是，我们设计了一组自然的此类函数，它们利用反应速率的相同缩放行为均匀地扩展到无穷大。