Stability and Hopf bifurcation analysis in a delayed three-node circuit involving interlinked positive and negative feedback loops

A system involving interlinked positive and negative feedback loops is a flexible motif that can tune itself in gene regulatory networks. It is well known that time delay is inevitable in gene regulatory networks due to transcription and translation not being physically co-located. In this paper, we systematically consider the effect of time delay on the dynamical behavior of the three-node circuit with three time delays. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing positive and negative feedback strengths as well as time delays τ ₁ , τ ₂ , τ ₃ as the bifurcation parameters, respectively. Moreover, stability and direction of the Hopf bifurcation of time delay are studied by using the normal form method and the center manifold theorem. Finally, several examples are performed to illustrate some analytical results we obtained.