Bistability and Oscillatory Behaviours of Cyclic Feedback Loops

Abstract

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.