We propose a $\tau$-leaping simulation algorithm for stochastic systems subject to fast environmental changes. Similar to conventional $\tau$-leaping the algorithm proceeds in discrete time steps, but as a principal addition it captures environmental noise beyond the adiabatic limit. The key idea is to treat the input rates for the $\tau$-leaping as (clipped) Gaussian random variables with first and second moments constructed from the environmental process. In this way, each step of the algorithm retains environmental stochasticity to subleading order in the timescale separation between system and environment. We test the algorithm on several toy examples with discrete and continuous environmental states and find good performance in the regime of fast environmental dynamics. At the same time, the algorithm requires significantly less computing time than full simulations of the combined system and environment. In this context we also discuss several methods for the simulation of stochastic population dynamics in time-varying environments with continuous states.

中文翻译：

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在具有快速环境的系统中超越绝热极限：Aτ 跳跃算法

我们提出一个 $\tau$- 受快速环境变化影响的随机系统的跳跃模拟算法。类似于常规$\tau$- 跳跃算法以离散时间步长进行，但作为主要的补充，它捕获了超出绝热极限的环境噪声。关键思想是处理输入速率$\tau$-跳跃为（剪切的）高斯随机变量，具有从环境过程构建的第一和第二时刻。这样，算法的每一步在系统与环境的时间尺度分离上都保持了环境随机性到次导序。我们在几个具有离散和连续环境状态的玩具示例上测试了该算法，并在快速环境动态范围内发现了良好的性能。同时，与组合系统和环境的完全模拟相比，该算法所需的计算时间要少得多。在这种情况下，我们还讨论了在具有连续状态的时变环境中模拟随机种群动态的几种方法。