In this paper, we study the stability of an Ordinary Differential Equation (ODE) usually referred to as Cyclic Feedback Loop, which typically models a biological network of \(d\) molecules where each molecule regulates its successor in a cycle (\(A_{1}\rightarrow A_{2}\rightarrow \cdots \rightarrow A_{d-1} \rightarrow A_{d} \rightarrow A_{1}\)). Regulations, which can be either positive or negative, are modelled by increasing or decreasing functions. We make an analysis of this model for a wide range of functions (including affine and Hill functions) by determining the parameters for which bistability and oscillatory behaviours arise. These results encompass previous theoretical studies of gene regulatory networks, which are particular cases of this model.

在本文中，我们研究了常微分方程（ODE）的稳定性，通常被称为循环反馈环，它通常模拟\(d\)个分子的生物网络，其中每个分子在一个循环中调节其后继分子（\ ( A_ {1}\rightarrow A_{2}\rightarrow \cdots \rightarrow A_{d-1} \rightarrow A_{d} \rightarrow A_{1}\) ）。调节可以是积极的，也可以是消极的，通过增加或减少功能来建模。通过确定产生双稳态和振荡行为的参数，我们对该模型的各种函数（包括仿射函数和希尔函数）进行了分析。这些结果涵盖了之前基因调控网络的理论研究，它们是该模型的特例。