We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting using a recent result by Cardin and Teixeira. The existence of invariant manifolds is subject to hyperbolicity conditions, which we test algorithmically using Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

中文翻译：

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多时间尺度生物网络的算法归约

我们提出了一种符号算法方法，该方法允许计算用于微分方程建模的生物网络的不变流形和相应的简化系统，其中包括用于细胞生物化学的化学反应网络和用于药理学、流行病学和生态学的区室模型。基于热带几何，通过缩放获得给定网络的多个时间尺度。使用 Cardin 和 Teixeira 的最新结果，我们的减少在奇异扰动设置中在数学上是合理的。不变流形的存在受双曲线条件的约束，我们使用 Hurwitz 标准进行算法测试。我们最终获得了一系列嵌套不变流形和这些流形上的相应简化系统。我们的理论结果通常伴随着适用于基于现有现成软件系统直接实现的严格算法描述，特别是符号计算库和可满足性模理论求解器。我们展示了使用我们自己的原型实现从著名的 BioModels 数据库中获取的计算示例。